Volume 14 Number 37 EJTP
Electronic Journal of Theoretical Physics
ISSN 1729-5254
This issue is devoted to Stephen Hawking (Oxford, 8 January 1942 -- Cambridge, 14 March 2018)
Editors
Ignazio Licata Ammar Sakaji
http://www.ejtp.com April, 2018 E-mail: [email protected]
Volume 14 Number 37 EJTP
Electronic Journal of Theoretical Physics
ISSN 1729-5254
This issue is devoted to Stephen Hawking (Oxford, 8 January 1942 -- Cambridge, 14 March 2018)
Editors Ignazio Licata Ammar Sakaji
http://www.ejtp.com April, 2018 E-mail: [email protected]
Editor in Chief Co-Editor
Ammar Sakaji Theoretical Condensed Matter, Mathematical Physics Ignazio Licata Center of Theoretical Physics and Astrophysics, Amman- Jordan Foundations of Quantum Mechanics, Tel:+962778195003 Complex System & Computation in Physics and Biology, IxtuCyber for Complex Systems , Email: [email protected] and ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy editor[AT]ejtp.info Email: ignazio.licata[AT]ejtp.info ignazio.licata[AT]ixtucyber.org
Editorial Board
Gerardo F. Torres del Castillo Leonardo Chiatti
Mathematical Physics, Classical Mechanics, Medical Physics Laboratory AUSL VT General Relativity, Via Enrico Fermi 15, 01100 Viterbo (Italy) Universidad Autónoma de Puebla, México, Tel : (0039) 0761 1711055 Email:gtorres[AT]fcfm.buap.mx Fax (0039) 0761 1711055 Torresdelcastillo[AT]gmail.com Email: fisica1.san[AT]asl.vt.it chiatti[AT]ejtp.info
Maurizio Consoli Francisco Javier Chinea Non Perturbative Description of Spontaneous Symmetry Breaking as a Condensation Phenomenon, Emerging Differential Geometry & General Relativity, Gravity and Higgs Mechanism, Facultad de Ciencias Físicas, Dip. Phys., Univ. CT, INFN,Italy Universidad Complutense de Madrid, Spain, Email: Maurizio.Consoli[AT]ct.infn.it E-mail: chinea[AT]fis.ucm.es
Sharmanthie Fernando
Avshalom Elitzur General Theory of Relativity and Black Holes Department of Physics, Geology and Engineering Foundations of Quantum Physics Technology, ISEM, Institute for Scientific Methodology, Palermo, Italy Northern Kentucky University, KY 41099, USA Email: Avshalom.Elitzur[AT]ejtp.info Email: Fernando[AT]nku.edu
Elvira Fortunato Tepper L. Gill Quantum Devices and Nanotechnology: Mathematical Physics, Quantum Field Theory Departamento de Ciência dos Materiais Department of Electrical and Computer Engineering CENIMAT, Centro de Investigação de Materiais Howard University, Washington, DC, USA I3N, Instituto de Nanoestruturas, Nanomodelação e Nanofabricação Email: tgill[AT]Howard.edu FCT-UNL tgill[AT]ejtp.info Campus de Caparica 2829-516 Caparica Portugal Tel: +351 212948562; Directo:+351 212949630 Fax: +351 212948558 Email:emf[AT]fct.unl.pt elvira.fortunato[AT]fct.unl.pt
Alessandro Giuliani Vitiello Giuseppe Mathematical Models for Molecular Biology Senior Scientist at Istituto Superiore di Sanità Quantum Field Theories, Neutrino Oscillations, Biological Roma-Italy Systems Email: alessandro.giuliani[AT]iss.it Dipartimento di Fisica Università di Salerno Baronissi (SA) - 84081 Italy Phone: +39 (0)89 965311 Fax : +39 (0)89 953804
Email: [email protected]
Arbab Ibrahim Richard Hammond Theoretical Astrophysics and Cosmology Department of Physics, Faculty of Science, General Relativity University of Khartoum, High energy laser interactions with charged particles P.O. Box 321, Khartoum 11115, Classical equation of motion with radiation reaction Sudan Electromagnetic radiation reaction forces Email: aiarbab[AT]uofk.edu Department of Physics arbab_ibrahim[AT]ejtp.info University of North Carolina at Chapel Hill, USA Email: rhammond[AT]email.unc.edu
Kirsty Kitto Hagen Kleinert Quantum Theory and Complexity Information Systems | Faculty of Science and Technology Queensland University of Technology Quantum Field Theory Brisbane 4001 Australia Institut für Theoretische Physik, Freie Universit¨at Berlin, Email: kirsty.kitto[AT]qut.edu.au 14195 Berlin, Germany Email: h.k[AT]fu-berlin.de
Beny Neta Wai-ning Mei Applied Mathematics Condensed matter Theory Department of Mathematics Physics Department Naval Postgraduate School University of Nebraska at Omaha, 1141 Cunningham Road Omaha, Nebraska, USA Monterey, CA 93943, USA Email: wmei[AT]mail.unomaha.edu Email: byneta[AT]gmail.com physmei[AT]unomaha.edu
Peter O'Donnell Rajeev Kumar Puri Theoretical Nuclear Physics, General Relativity & Mathematical Physics, Physics Department, Panjab University Homerton College, University of Cambridge, Chandigarh -160014, India Hills Road, Cambridge CB2 8PH, UK E-mail: po242[AT]cam.ac.uk Email: drrkpuri[AT]gmail.com rkpuri[AT]pu.ac.in
Haret C. Rosu Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Zdenek Stuchlik Camino a la Presa San José 2055 Relativistic Astrophysics Col. Lomas 4a. sección, C.P. 78216 Department of Physics, Faculty of Philosophy and Science, San Luis Potosí, San Luis Potosí, México Silesian University, Bezru covo n´am. 13, 746 01 Opava, Email: hcr[AT]titan.ipicyt.edu.mx Czech Republic Email: Zdenek.Stuchlik[AT]fpf.slu.cz
Fabrizio Tamburini S.I. Themelis Atomic, Molecular & Optical Physics Electromagnetic Vorticity, General Relativity, Quantum Foundation for Research and Technology - Hellas mechanics, Astrophysics of compact objects P.O. Box 1527, GR-711 10 Heraklion, Greece Scientist in Residence at Email: stheme[AT]iesl.forth.gr ZKM - Zentrum für Kunst und Medientechnologie Lorenzstraße 19, 76135 Karlsruhe, Germany www.zkm.de/ Email: fabrizio.tamburini[AT]gmail.com
Nicola Yordanov Yurij Yaremko Physical Chemistry Special and General Relativity, Bulgarian Academy of Sciences, Electrodynamics of classical charged particles, BG-1113 Sofia, Bulgaria Mathematical Physics, Telephone: (+359 2) 724917 , (+359 2) 9792546 Institute for Condensed Matter Physics Email: ndyepr[AT]ic.bas.bg of Ukrainian National Academy of Sciences ndyepr[AT]bas.bg 79011 Lviv, Svientsytskii Str. 1 Ukraine Email: yu.yaremko[AT]gmail.com yar[AT]icmp.lviv.ua yar[AT]ph.icmp.lviv.ua
Former Editors:
Ignazio Licata, Editor in Chief (August 2015-)
Ignazio Licata, Editor in Chief (October 2009- August 2012)
Losé Luis López-Bonilla, Co-Editor (2008-2012)
Ammar Sakaji, Founder and Editor in Chief (2003- October 2009) and (August 2012- August2015).
Table of Contents
No Articles Page 1 In memory of Stephen Hawking i
Ignazio Licata
2 Preface iv
Ammar Sakaji
3 Projective Limits of State Spaces: Quantum Field Theory 1 Without a Vacuum
Suzanne Lanery
Centro de Ciencias Matematicas, Universidad Nacional Autonoma de Mexico, Apartado Postal 61-3, 58089 Morelia, Michoacan, Mexico
4 Noncommutative Structure of Massive Bosonic Strings 21
N. Mansour, E.Diaf and M.B. Sedra
SIMO-Lab, faculty of sciences,Ibn Tofail University,Kenitra, Morocco Department of Physics, Pluridisciplinairy Faculty of Nador, OLMAN-RL Pluridisciplinary Faculty of Nador, University Mohammed Premier, Oujda B.P N 300, Selouane, 27000 Nador, Morocco.
5 Minimal Length, Minimal Inverse Temperature, 35 Measurability and Black Hole
Alexander E.Shalyt-Margolin
Institute for Nuclear Problems,Belarusian State University, 11 Bobruiskaya str., Minsk 220040, Belarus
6 Poisson Bracket and Symplectic Structure of Covariant 55 Canonical Formalism of Fields
Yasuhito Kaminaga
School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
7 Neutrino Masses and Effective Majorana Mass from a 73 Cobimaximal Neutrino Mixing Matrix
Asan Damanik
Department of Physics Education, Sanata Dharma University Kampus III USD Paingan, Maguwoharjo, Sleman, Yogyakarta, Indonesia
8 Relativistic Klein-Gordan Equation with Position 79 Dependent Mass for q-deformed Modifed Eckart plus Hylleraas potential
S. Sur and S. Debnath
Department of Mathematics, Jadavpur University, Kolkata - 700 032, India
9 Investigation Fermionic Quantum Walk for Detecting 91 Nonisomorph Cospectral Graphs
M. A. Jafarizadeh, F. Eghbalifamy and S. Namiz
Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran
10 Thermodynamics of Hot Quantum Scalar Field in a (D + 1) 115 Dimensional Curved Spacetime
W. A. Rojas C. and J. R. Arenas S.y
Departamento de Fisica , Universidad Nacional de Colombia Observatorio Astronomico Nacional, Universidad Nacional de Colombia
11 Spin and Zitterbewegung 125 in a Field Theory of the Electron
Erasmo Recami and Giovanni Salesi
Facolta di Ingegneria, Universitia Statale di Bergamo, 24044{Dalmine (BG), Italy; INFN Sezione di Milano, Milan, Italy; and DECOM, Faculty of Electrical Engineering (FEEC), State University at Campinas (UNICAMP), Campinas, Brazil. Facolta di Ingegneria, Universita Statale di Bergamo, 24044{Dalmine (BG), Italy; INFN{Sezione di Milano, Milan, Italy
12 Solutions to the Gravitational Field Equations in Curved 145 Phase-Spaces
Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia. 30314, USA
13 Electromagnetic Media in pp-wave Spacetime 161 Mohsen Fathi
Department of Physics, Payame Noor University (PNU), PO BOX 19395-3697 Tehran, Iran
14 Validation of the Hadron Mass Quantization from 179 Experimental Hadronic Regge Trajectories
Navjot Hothi, Shuchi Bisht
Department of Physics, University of Petroleum and Energy Studies, Dehradun - 248007, India Department of Physics, Kumaun University, Nainital-263002, India
15 Neimark-Sacker and Closed Invariant Curve Bifurcations 195 of A Two Dimensional Map Used For Cryptography
Yaniss Yahiaoui and Nourredine Akrouney
Laboratoire de Mathematiques Appliquees, Faculte des Sciences Exactes, Universite de Bejaia, 06000 Bejaia, Algeria
16 Physics of Currents and Potentials IV. Dirac Space and 213 Dirac Vectors in the Quantum Relativistic Theory
V.A. Temnenko
Tavrian National University, Vernadsky prospect 4, 95022 Simferopol, Crimea
Electronic Journal of Theoretical Physics 14, No. 37 (2018) i
EJTP V14, No 37
He Lived Here In memory of Stephen Hawking Oxford, 8 January 1942 – Cambridge, 14 March 2018 What a strange awakening today. Stephen Hawking – escaped from an infaust diagnosis 21 years ago and from many fatal surgeries- is gone.
My generation has grown up with catchphrases such as “where were you when John Lennon died?” I think that Stephen Hawking death will bring back similar questions in the future. As it has been said for Einstein: “He lived here”. I’m not speaking only of the powerful and empathetic relation he and his work had with the media and audience. All of us admired Eddie Redmayne in The Theory of Everything and, earlier, Benedict Cumberbutch playing the role of Hawking, all of us know something about black holes and their radiation; and A Brief History of Time is surely one of the the most successful book of all time. Actually, there another reason why the Stephen Hawking death will be stuck in our minds. Just for once, the image in the media was really the man behind and beyond the news. You could always perceive he was a man of stature and an intense person, there was something unique between the brightening of his eyes and the lines of his most technical papers. Einstein used to say that a theoretical physicist can appear to be an opportunist with no scruples to epistemologists. The reason lies in the fact that a scientist uses precise tools, mathematics for theoretical physicists, and not the power of interpretations; in no way a scientist worries about giving a frame to a result so mimicking a philosopher. That’s where Hawking was, maybe, the most secular among the scientists. He never married a theory, but he wooed them all, just like he never failed to smile at a pretty woman. He always questioned how far a theory could we lead, and we could actually say about the Universe. Among all the things that gave him everlasting fame there are
Fig. 1 Stephen Hawking with W.J. Kaufmann (1977) ii Electronic Journal of Theoretical Physics 14, No. 37 (2018)
two problems which really stay at the extreme borders of knowledge. One deals with the final stage of massive stars, the famous Black Holes, which, according to Einstein, during the last stage of their lives should enter on infinite collapse, a singularity. Laplace had already saw it, as well as Oppenheimer and Landau later, up to Wheeler and his master Sciama. Nobody had ever investigated it before as Hawking did. Singularity had wandered like a monster in specialized reviews and journals for some years, later – thanks to Hawking and Penrose – it became clear that it was just a structural limitation of Einstein gravitation theory and it was time to give room to a new theory, the quantum gravity, which is still a frontline topic in theoretical physics. Stephen Hawking was the one who reached some milestones in this new field, the black hole radiation and the information analysis of a physical system with Hawking-Bekenstein formula. Black holes were just an exercise, because now Stephen was ready to look at the more mysterious singularity, the Big Bang. It was about in the ’80s, he and Jim Hartle proposed the no-boundary Universe, a charming expression that we can coarsley translate by saying that space and time emerge from a quantum nebulosity; something similar to the Nicola Cusano Universe, where there is no before and no after, where each point is the center. Or, just to be a bit more technical, where time is curved and imaginary before collapsing into what we see and what the Standard Model describes. Similarly to all the other physicists, I happen to quote Stephen thousand times, and every time it was an occasion to read his works again. I admired his ability to build an apparently impenetrable castle of mathematics all around a strong idea. He could have been an excellent chess player. I say “apparently” because Hawking knew very well that mathematics was a sublime form of rethoric which could always be attacked or taken apart. Or started from scratch. What really makes the difference for a physical theory are generalities and the steadiness of its starting points. Sometimes, a weak point could be found in Stephen’s approach (shrewd, very subtle!), but , at the same time, you couldn’t help but notice how the question had been posed with absolute clearness and how it would be really difficult to do it better. Leonardo Chiatti and I started from Hartle-Hawking theory to develop the idea of the Archaic Universe [1,2,3 ]and, recently, Fabrizio Tamburini, Maria Felicia de Laurentis and I have discovered a particular mode of Hawking radiation, the so-called soft hairs. There’s only a case when Hawking admitted to be defeated, in front of a young Don Page, about the end of the Universe, namely about the possibility that the whole wave-function rewinds to go back to origins. Like it happened some years before with Kip Thorne about the possibility to discover a black hole in Cygnus X-1, also in that occasion a stake was paid: a magazine subscription (in the case of Kip, it was a yearly subscription to Playboy). In my opinion, the Hawking idea is well-grounded, so the last word has not been spoken. Maybe, the most don’t know that there is a beautiful theatrical play titled God and Stephen Hawking on Stephen Hawking life and his struggles with his disease and the biggest mysteries of the Universe. The author, Robin Hawdon, was really within “the zone” when wrote it, you can find in the play the same humor which has became the irreducible trait of Stephen. Electronic Journal of Theoretical Physics 14, No. 37 (2018) iii
In the end there are some cues echoing the closing lines of A Brief History of Time, where Stephen reaffirms his faith in a Final Reason, it equates him with giants like Einstein. Let’s listen to it once again: STEPHEN: I do know it is there, inherent to the infinite experiment of the Universe. A solution that – differently from any metaphysical theory and belief – will look to be so clear....so patent...and we will realize it has been with us all the time. Bye Stephen!
Ignazio Licata
[1] Ignazio Licata: Universe Without Singularities. A Group Approach to de Sitter Cosmology, EJTP, vol. 3 nr. 10 (2006), pp. 211-224 [2] Ignazio Licata, Leonardo Chiatti: The Archaic Universe: Big Bang, Cosmological Term and the Quantum Origin of Time in Projective Cosmology, International Journal of Theoretical Physics, vol. 48, nr. 4 (2009), pp. 1003-1018 [3] Ignazio Licata, Leonardo Chiatti: Archaic Universe and Cosmological Model: ”Big- Bang” as Nucleation by Vacuum, International Journal of Theoretical Physics, vol. 49, nr. 10, (2010) pp. 2379-2402 [4] Fabrizio Tamburini, Mariafelicia De Laurentis, Ignazio Licata and Bo Thid´e Twisted Soft Photon Hair Implants on Black Holes, Entropy (2017), 19 (9), 458 Ignazio Licata iv Electronic Journal of Theoretical Physics 14, No. 37 (2018)
Preface
In the first quarter of 2018, we present a collection of fourteen manuscripts covering important topics of theoretical and mathematical physics ranging from quantum walk, gravitational waves, string theory, gauge field theories and canonical formalism, gravi- tational thermodynamics and quantum gravity, neutrino masses and effective Majorana, relativistic Klein-Gordan equation, thermodynamics of hot Quantum scalar field, Spin and Zitterbewegung, solutions to the gravitational field equations in curved phase-spaces, hadron mass quantization, Neimark-Sacker bifurcation and chaotic attractors for discrete dynamical systems, and Dirac space in the Quantum relativistic theory. Lan´ery on his paper presents a self-contained introduction of the projective limits of state spaces: quantum field theory without a vacuum and its relations to other QFT approaches. Mansour et al. addresses in his paper the Faddeev-Jackiw quantization methodology in the noncommutative structure of massive Bosonic strings. Margolin defined the gravitational thermodynamics for minimal length and minimal inverse tem- perature. Kaminaga in paper propose the Poisson bracket for a new canonical theory. Damanik in his work derives a neutrino mass matrix from cobimaximal neutrino mixing matrix in parallel with effective Majorana mass. Debnath address relativistic Klein- Gordan equation for q-deformed modified Eckart plus Hylleraas potential. Jafarizadeh et al. on their work on graph isomorphism problem investigate Fermionic quantum walk for detecting Nonisomorph Cospectral Graphs. Rojas et al. use the brick wall model to calculate of free energy of quantum scalar field in a curved spacetime (D +1) dimensions. Recami et al. in his paper ”Spin and Zitterbewegung” address the classical theory of the electron in parallel with quantum analogue to extend a new non-linear Dirac-like equation. Castro in his paper gives mathematical solutions to the gravitational field equations in curved phase-spaces. Fathi presents dialectic transformation media within gravitational waves. Hothi et al. show the validation of the Hadron mass quantization from experimental Hadronic Regge trajectories. Yahiaoui et al. in their cryptographic work discuss the dynamics and bifurcations of a family of two-dimensional noninvertible maps. Temnenko in his 4th paper of the series of physics of currents and potentials addresses Dirac space and vectors. I want to express my sincere gratitude to the my friend Ignazio Licata for the valu- able discussions, reviewing and excellent editorial work, and thanks to my friend Hanna Sabat from the center of theoretical physics and astrophysics for his help in editing the manuscripts, many thanks to our referees for their valuable comments and notes. We thank all authors who contributed their articles for this issue.
Ammar Sakaji EJTP 14, No. 37 (2018) 1–20 Electronic Journal of Theoretical Physics
Projective Limits of State Spaces: Quantum Field Theory Without a Vacuum
Suzanne Lan´ery∗ Centro de Ciencias Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 61-3, 58089 Morelia, Michoac´an, M´exico
Received 27 December 2017, Accepted 16 March 2018, Published 20 April 2018
Abstract: Instead of formulating the states of a Quantum Field Theory (QFT) as density matrices over a single large Hilbert space, it has been proposed by Kijowski [20] to construct them as consistent families of partial density matrices, the latter being defined over small ’building block’ Hilbert spaces. In this picture, each small Hilbert space can be physically interpreted as extracting from the full theory specific degrees of freedom. This allows to reduce the quantization of a classical field theory to the quantization of finite-dimensional sub-systems, thus sidestepping someofthecommon ambiguities (specifically, the issues revolving around the choice of a ’vacuum state’), while obtaining robust and well-controlled quantum states spaces.The present letter provides a self-contained introduction to this formalism, detailing its motivations as well as its relations to other approaches to QFT (such as conventional Fock-like Hilbert spaces, path-integral quantization, and the algebraic formulation). At the sametime, it can serve as a reading guide to the series of more in-depth articles [27, 28, 29, 30]. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Quantum Field Theory; Vacuum States; Inequivalent Representations; Geometric Quantization; Projective Limits; Algebras of Observables PACS (2010): 02.40.Yy; 03.50.-z; 03.70.+k; 04.62.+v; 04.60.Ds
1. Motivation: Quantization Ambiguities in Quantum Field The- ory
Many choices have to be made in the quantization of a classical theory. Assuming one is following the canonical quantization path (see section 5. for further discussion of the relevance for path-integral approaches of the issues discussed here), the first step is to choose a complete set of basic variables for the theory. Heuristically, these are the vari-
∗ Email: [email protected] 2 1–20
ables for which the semi-classical limit will work best, hence their choice should ideally reflect the observables against which the classical theory of interest has been best tested and confirmed. The next step is to find a representation of these basic variables as operators on a suitable Hilbert space H,namely a mapping f → fˆ such that f,ˆ gˆ = i { f, g} (1)
(where [ · , · ] denotes the commutator of operators, while {·, ·} denotes the Poisson brackets of classical observables). At this point, quantum field theory (in a broad sense, namely quantum theories meant to encompass infinitely many degrees of freedom) differs crucially from quantummechanics (dealing with the quantum counterparts of classical systemsthathavefinitely many degrees of freedom). The tools from geometric quanti- zation [45] (that we will discuss further in subsection 2.2) provide a clear and detailed understanding of the canonical quantization of finite dimensional systems, including a parametrization of available choices (aka. quantization ambiguities). In somecases,it may even turn out that there is no choice at all, because the Poisson-algebra of interest admits only one suitable representation: this is for example the content of the Stone-von- Neumann theorem [41, 42, 39] in the case of linear systems. By contrast, the representation theory for infinite dimensional system tends to be very involved. Even in the simplest case of a free scalar field on Minkowski spacetime, it is known that there exist infinitely many inequivalent representations, and although it has been possible, in this very special case, to fully classify them [15], this classification is so complex that it gives little insight on how to choose one. As a way out, a pragmatic way of selecting a good representation amongthesetoonumerous options is to single out a distinguished quantum state, the vacuum: it is indeed possible, via the so-called GNS construction [11, 38] to ’seed’ a full representation HΩ from a single state Ω (to specify the latter, even before we are equipped with a Hilbert space, we can give the corresponding expectation values of all products of the basic variables, aka. the n-point functions, see [17, part III, def. 2.2.8]). This approach has established itself as the standard way to think about quantum field theory, at least in the context of Minkowski spacetime, where the vacuummay be selected by requiring it to be invariant under all spacetimesymmetries (ie. under the Poincar´e group). However, one should keep in mind that the initial choice of vacuum is deeply imprinted in the thus obtained representation. The only quantum states that can be written as (pure
or statistical) states on HΩ are those that barely differ from the vacuum:atmost discrete quantum excitations on top of the state Ω are allowed. The set of all states living on
the representation HΩ is referred to as the vacuum sector, in acknowledgment of the fact that there are many more quantum states beyond it (falling out of it because they lie too far away from the chosen vacuum), among whose some may actually be interesting for specific purposes [17, part V]. An implication of the relative smallness of the vacuum sector is that the vacuum state need to be closely tailored to the dynamics: otherwise, the
time evolution would immediately kick the states out of HΩ (a precise statement of this 1–20 3
heuristic expectation is given, for Poincar´e-invariant QFTs, by the Haag no-go theorem, [16]). A radical alternative, prompted by the lack of a natural vacuum inthecaseofquantum field theory on curved spacetime, is to use as state space the whole set of possible quantum states over the chosen basic observables (each such state being specified, as explained above, by the expectations values it prescribes for all products of observables). This approach can be followed in the context of Algebraic Quantum Field Theory (AQFT, [17, 18]): by shifting the focus from a particle picture to the local and causal structure of the quantum theory, AQFT provides tools to discuss the properties of quantum fields in the absence of an underlying Hilbert space. The aim of the present letter is to argue that a projective definition of quantum field theory, as was introduced by Jerzy Kijowski [20] and further developed by Andrzej Okol´ow [32, 34, 33], can provide a middle way between the conventional vacuum-based approach and the full algebraic one, retaining a constructive description of the quantum state space (subsections 3.2 and 3.3) while keeping enough flexibility to accommodate a wide class of quantum states (subsection 3.1) and to decouple the subsequent implementation of the dynamics from the initial building of the state space (section 4.). The work summarized in the following sections (and developed in details in [27, 28, 29, 30]) was notably motivated by the specific difficulties encountered when one tries to formulate background independent quantum field theories, rather than theories on a (possibly curved) background spacetime (eg. to quantize general relativity itself in a non- perturbative way [2, 40]). It turns out that for background independent gauge theories (at least those with compact gauge group), there does exist a preferred vacuum state, the Ashtekar-Lewandowski vacuum [3, 4], which is uniquely selected precisely by the re- quirement of background independence [31, 10]. Unfortunately, this vacuum has some unwanted properties. One of them is that it is an eigenstate of the variable conjugate to the gauge field, rather than a coherent state like the usual Fock vacuum. Since states in the vacuum sector cannot differ too much from the vacuum, this makes it difficult to find semi-classical states among them [12, 22]. Another problem is that the GNS representa- tion built on this vacuum lives on a non-separable Hilbert space. This particular issue may or may not go away once we identify quantum states that only differ by a change of coordinates (depending on how precisely this identification is carried out, see [40, 8]) but in any cases it can lead to technical difficulties [5]. Paradoxically, non-separable Hilbert spaces seem too small, because their orthonormal basis need uncountably many basis vectors, making it tempting to consider uncountable linear combinations while only countable ones are allowed: in other words, it is in this case even more likely that phys- ically interesting states will lie out of the vacuum sector. In an effort to overcome these difficulties, the projective quantization techniques that we will review in the next section have been applied to this kind of theories, resulting in a quantum state space that may have applications to the study of the semi-classical and cosmological sectors of quantum gravity [26]. 4 1–20
Mη
Mη πη→η
Mη→η × Mη
Mη πη→η
Mη→η × Mη→η × Mη Mη→η × Mη πη→η
Mη
≈
Fig. 1 Three-spaces consistency for projective systems (left side), reformulated in termsof factorizations (right side)
2. Systematic Quantization of Infinite-dimensional Systems
2.1 Building an Infinite-dimensional Theory from a Collection of Partial Descriptions
The key observation underlying Kijowski’s projective formalism [20, 33] is that a given experiment can only measure finitely many observables. Thus, we never need to consider at once the full, infinite-dimensional phase space M∞ of a field theory: it is sufficient to work in a small, partial phase space Mη that extracts from M∞ just the degrees of freedom (dof.2) relevant for the experiment at hand (throughout the present letter, the symbol η will be used to denote a selection of finitely many dof. out of the full theory, and we will call η a label). M In order to use such a collection of finite-dimensional partial phase spaces η η to completely specify a field theory, we need to ensure that the different partial theories are consistent with each other [27, subsection 2.1]: 1. first, we need a way to express the relations between the dof. in different labels. We will write η η if all dof. contained in η are also contained in η (we will also say that η is coarser as η ,orthatη is finer as η). This means that any observable fη 3 on Mη corresponds to an observable fη on Mη , and, by duality , that there exists a
projection πη →η from Mη to Mη such that
fη = fη ◦ πη →η
2. the predictions for a given experiment, as calculated in a partial theory η, should be independent of the choice of η (provided η is fine enough to hold all relevant dof.).
Thus, in particular, the Poisson brackets between two observables fη and gη on Mη
2 By a dof. we mean a pair of conjugate variables. 3 To see that πη→η is uniquely specified once we know the mapping fη → fη between observables, one can consider a complete set of observables (aka. coordinates) on Mη. 1–20 5
should agree with the Poisson brackets between the corresponding observables on a
finer Mη . Expressed in terms of the just introduced projection πη →η, this reads
{fη ◦ πη →η,gη ◦ πη →η} = {fη,gη}◦πη →η (2)
3. it should be possible to consider composite experiments made of two (or more) sub- experiments, each of which can be described within a different partial theory4.In other words, for any η, η, there should exist η such that η, η η. This property is called directedness of the set of labels.
4. the relation between any two partial theories should be unambiguous. Thanks to the just mentioned directedness property, this can be ensured simply by requiring that the projections defined among three increasingly refined partial theories match as shown on the left part of fig. 1. In mathem atical terms, this list of requirements can be summarized by saying that the collection M forms a projective (aka. inverse) system and it ensures that M∞ can η η M M be reconstructed from η η (more precisely, a space lim can be constructed as the so-called projective limit of this system, that will, in general, be a distributional extension of M∞, see [27, def. 2.6 and prop. 2.7]). A key point of the construction is that each label η corresponds to a selection of con-
jugate position and momentum variables. This ensures that the projection map πη →η between the phase spaces Mη and Mη is unambiguous: as stressed above, it is completely determined by matching the physical interpretation of the observables in η vs. η.By contrast, if the partial theories were labeled by selections of configuration variables only (as is usual eg. when studying coarse graining in the path-integral formalism), we would only be provided with projections between the configuration spaces (since phase spaces can be thought of as cotangent bundles, and forms are naturally pull-backed, rather than push-forwarded, there is no canonical prescription to lift a projection between configu- ration spaces to a projection between the associated phase spaces). Furthermore, our labels do not have to be assembled from pre-assigned pairs of mutually independent, canonically conjugate variables (aka. modes). In particular, different labels can consist of the same selection of configuration variables, but paired with different conjugate impul- sions5: the corresponding partial theories are then interpreted as extracting from the full, continuum field theory information about different (partially overlapping) dof. The di- rectedness requirement (which is much weaker than the requirement of a preferred mode decomposition, and is indeed fulfilled in a range of situations where the latter would not
4 We are discussing the classical theory here. The quantum theory is more subtle, since one could argue that, due to the principle of complementarity, some sub-experiments may be mutually excluding. However, the case for the directedness of the label set can still be made, see [28, section 1]. 5 This is manifest eg. in the treatment of gauge theories in [26]: selections of configuration variables are represented by graphs (generalized lattices) and these need to be decorated with dual surfaces to fully specify the impulsions (aka. electric fluxes) captured by a given partial theory (see the discussion at the beginning of [26, section II]). 6 1–20
Mη
Mη→η t Φη
πη→η t Φη Mη
Fig. 2 Factorization from a Poisson-brackets-preserving projection between phase spaces
be an option; see [26, 25]) guarantees that these partial informations can be assembled into a consistent picture. A useful consequence of eq. (2) is that it ensures that there exists a preferred factoriza- tion of Mη as Mη →η ×Mη (at least locally: there may be some topological obstructions preventing this factorization to hold globally). The way this factorization is obtained is 6 as follow (see [33, section 3.4] and [27, prop. 2.10]). Any observable fη on Mη generates t M t aHamiltonian flow Φη on η,andfη generates a corresponding Hamiltonian flow Φη on Mη .AlltheflowsonMη generated in this manner from all possible observables on
Mη together span joint orbits in Mη (represented as dashed lines on fig. 2), and eq. (2)
ensures that any such orbit can (again, locally) be identified with Mη. All what is left
to do is then to define Mη →η as the corresponding quotient, ie. the set of all such orbits
(see fig. 2). Mη →η can thus be interpreted as holding dof. that are complementary to the
ones retained by Mη (since, by construction the dof. in Mη →η Poisson-commute with
the ones in Mη). If we assume, for simplicity, that, for any η η , the factorization Mη ≈Mη →η ×Mη holds globally, we can translate the consistency condition illustrated on the left part of fig. 1 into a condition written in terms of the factorizations [27, def. 2.11], as shown on the right part of fig. 1. The meaning of this condition is that, when extracting from η the dof. corresponding to a much coarser partial theory η,thecomplementary dof. that we discard should be the same whether we go from η to η inonesteporintwo,hence
there should be a natural identification Mη →η ≈Mη →η ×Mη →η .
6 Again, it is crucial that the η’s are labeling phase spaces: the Poisson-bracket structure is what allows to identify the complementary observables. By contrast, if we were only provided with a projection between configuration spaces, there would be no way to single out a preferred factorization of the larger space. 1–20 7
2.2 Leveraging the Well-developed Procedures for Quantizing Finite-dimensional Systems
Once a collection of classical partial theories has been put in the just mentioned factorized form, constructing a corresponding collection of quantum partial theories is straightfor- ward [28]. All we have to do is to quantize each Mη into a corresponding Hilbert space
Hη, and, since Mη is finite-dimensional, we have at our disposal the full range of tech- niques known for quantizing systems with finitely many dof. From theusualrulesof quantummechanics we expect that, for any η η, the factorization of phase spaces
Mη ≈Mη →η ×Mη will give rise to a tensor-product factorization of Hilbert spaces
Hη ≈Hη →η ⊗Hη. In this way, we can obtain a collection of Hilbert spaces, with relations between them expressed in terms of tensor-products. But where is the promised quantum state space for the full field theory? The crucial realization, that goes back to Kijowski [20], is that we can now construct quantum states as collections ρη η,whereeachρη is a density matrix (aka. mixed or statistical state) on Hη. In order for such a collection to describe aquantum state for the full theory, we need to ensure that the partial states ρη are consistent with each other. Namely, whenever η η , the states ρη and ρη should lead to the same measurement probabilities for all dof. retained by the coarser label η. Using well-known tools from statistical quantummechanics, this means that we need to enforce
ρη =Trη →η ρη (3) where Trη →η denotes the partial trace on the tensor product factor Hη →η. Indeed, the partial trace satisfies the important property ˆ ˆ ˆ ˆ Tr ρη fη =TrTrη →η ρη fη =Trρη 1η →η ⊗ fη =Trρη fη (4) In order to be able to consistently impose eq. (3), the way to reconstruct a coarser partial quantum state from a finer one should be unambiguous, in other words we should have
Trη →η =Trη →η ◦ Trη →η for any three increasingly fine labels η η η. Fortunately, this is automatically guaranteed by the quantum analogue of the classical consistency condition from fig. 1 [28, defs. 2.1 and 2.2]. The quantum state space obtained in this way may be called projective,
since it arises mathematically as a projective limit (like the space Mlim mentioned in subsection 2.1). Note that it is necessary in this construction to use density matrices rather than Hilbert space vectors (aka. pure states). Indeed, the partial trace is the right tool to use when one is interested in the restriction of a quantum state to specific dof. (thanks to the property stressed in eq. (4)) and it cannot be defined as a map between pure states, because the partial trace of a pure state is often a mixed state and, reciprocally, the partial trace of a mixed state can appear pure7. Another way of looking at this is
7 When ρη has the special form ρ˜ ⊗|ψ ψ|, any statistical superpositions inρ ˜ will be traced out. 8 1–20
that the distinction between pure states and mixed ones is always relative to a certain (sub)system, hence it is effectively useless in an infinite-dimensional theory: the only absolute notion of state purity would be the one defined with respect to the full theory, but such an absolute notion cannot have any experimental relevance, because it would require to measure quantum states against a complete set of commuting observables, that is, to perform infinitely many measurements. In [28, section 3] and [25] more specific results have been obtained regarding the implementation of this general strategy. As is apparent from the discussion above, what we need is a consistent quantization scheme, namely a prescription to quantize each
partial theory Mη in such a way that the Cartesian product factorizations, holding on the classical side, are indeed lifted to tensor-product factorizations on the quantum side, ˆ ˆ and that the quantization of observables agree, ie. that we do have fη =1η →η ⊗ fη as was assumed in eq. (4). Such a scheme can be adapted from geometric quantization [45]. The latter formalizes the traditional recipes of canonical quantization into a systematic procedure, which can be understood starting from the following observation: A very simple way of quantizing a 8 finite dimensional phase space Mη would be to set Hη = L2(Mη) and, for any observable
fη on Mη ˆ fηψ = −i {fη,ψ} . Then eq. (1) is simply the Jacobi equation satisfied by the Poisson brackets. This naive ansatz can be improved by incorporating some multiplicative action of fη in the defini- ˆ tion of fη, yielding what is known as pre-quantization [45, section 8.2]. Since the pre- quantization primarily depends on the Poisson bracket structure of the classical theory, and this structure is preserved by the projections πη →η as required in subsection 2.1, it generically provides a consistent quantization scheme in the sense above [28, theorem 3.9 and prop. 3.11]. Pre-quantization is however not fully satisfactory (hence the ’pre’ qualification) as it leads to Hilbert spaces which are simply too big. Physically admissible Hilbert spaces are extracted by restricting ourselves to a subset of the pre-quantized Hilbert space [45, chap. 9]: for example the usual position representation of quantummechanics is re- covered by considering only the pre-quantum states ψ(x, p) which are constant in p, ie. only depend on x, while holomorphic quantization is obtained by keeping only the ψ
which are holomorphic functions (with respect to a suitable complex structure on Mη, eg. z = x + ip). This extra step of the geometric quantization program is called imposing a polarization. For the purpose of constructing projective quantum state spaces, the ques- tion is therefore whether polarizations can be imposed consistently on each pre-quantized
Hη. Building on previous work by Okol´ow [33], this question was examined in two concrete cases (namely, position quantization when the configuration spaces for all partial theories
8 Note that Mη, as a phase space, admits a natural measure: the Liouville measure, which is singled out (up to a multiplicative constant) and has the important property of being preserved under any Hamiltonian flow, which ensures, in particular, that fˆη will be self-adjoint whenever fη real. 1–20 9
are Lie groups and the projective structure from subsection 2.1 is suitably compatible with the group structure, in [28, subsection 3.1], and holomorphic quantization when
all projection maps πη →η are holomorphic, in [28, subsection 3.2]) and the answer was found to be positive. These results suggest that polarizations can be consistently im- posed on the pre-quantized projective system whenever the classical precursors of these
polarizations are compatible with the coarse-graining projections πη →η. Furthermore, thenatureofthecompatibility required turns out to be rather weak, because it only has to be satisfied up to unitary equivalence of the quantizations. For example, if the
small, finite-dimensional phase spaces Mη are linear spaces, the Stone–von Neumann theoremmentioned at the beginning of section 1. ensures that all polarizations which
respect the linear structure lead to unitarily equivalent representations Hη. Leveraging these equivalences, a general quantization scheme for linear (aka. free) field theories was developedin[25],whichisexplicitly independent of any overall choice of polarization on the (infinite-dimensional) classical phase space: specifically, the label set it uses spans all possible choices of polarization at once, so that none has to be singled out, making it suitable, eg. to the study of quantum fields on curved backgrounds (while field theories defined on a flat background spacetimes admits a physically preferred polarization, this is no longer the case on curved backgrounds, especially non-static ones; see [18]). Moreover, it may also be possible to dispense from the requirement that the classical
factorizations Mη ≈Mη →η ×Mη hold globally (see [28, section 4]).
3. Relations to Standard Constructions: Improved Universality and Richer State Space
3.1 Embedding All Vacuum Sectors at Once
Since the program exposed in the previous section is meant to improve on the vacuum- based quantization described in section 1., we need to understand how the quantum state spaces obtained by both methods relate to each other. Suppose that we are given avacuum state Ω, which can be completely specified, as underlined in section 1., by the expectation values it prescribes for any product of observables. Then, we can define
a partial vacuum state Ωη,bysimply forgetting about the expectation value of any product involving an observable that does not live on the partial theory labeled by η. Now, quantum theories describing finitely many dof. do not suffer from the pathologies affecting quantum field theory: the states that can be written as density matrices on the
Hilbert space Hη are likely to span a large part of the space of possible quantum states.
Thus, it is not unreasonable to assumethateachΩη corresponds to a density matrix ρη on
Hη, and, since the Ωη all stem from the same state Ω of the full theory, they are guaranteed to be consistent, ensuring that eq. (3) holds. In other words, the vacuum Ωislikelyto belong to the projective quantum state space constructed in subsection 2.2, and with him all quantum states that only differ marginally from Ω, ie. the whole vacuum sector around Ω [28, prop. 2.8]. This reasoning suggests that the vacuum sectors corresponding 10 1–20
to a large class of vacuums are included for free in the projective state space.
In [28, theorem 2.9], we examined in particular the case where the ρη obtained from Ω are actually pure states for all η. Note that in contrast to the discussion so far, this is a very special situation, in the sense that pure states are fairly ’exceptional’ in the space of all quantum states [17, part III, theorem 2.2.18]; yet, it turns out to apply to the most interesting cases (such as the embedding of the usual Fock space, or of the Ashtekar- Lewandowski state space mentioned in section 1., into their projective counterparts, see [29, prop. 3.17] and [25, subsection 3.3], or [26, theorem 40], respectively). Under this additional assumption, it is possible to prove that the embedding of the corresponding vacuum sector into the projective state space is injective (aka. one-to-one) and its image can be precisely characterized. A different construction that could claim to include at once many vacuum sectors are the so-called infinite tensor products (ITP, [43]). These can be defined as huge direct sums
of GNS Hilbert spaces HΩ (recall section 1.) for Ω running through a suitable class of vac- uums. In other words, the different vacuum sectors are simply put side by side. Besides resulting in a large, non-separable Hilbert space, a drawback of this construction is that it is fairly fragile: it requires a decomposition of the dof. into independent (commuting) subsets, with the slightest modification of this decomposition leading to inequivalent ITP Hilbert spaces. More seriously, it introduces spurious distinctions between physically in- distinguishable states: since the different sectors appear as superselection sectors (ie. the observables are not sensible to quantum correlations between these sectors), quantum superpositions of states in different sectors are indistinguishable from statistical superpo- sitions. By contrast, projective state spaces, while allowing for statistical superpositions of states in different vacuum sectors, do not produce such an unphysical redundancy. This is exhibited by the fact that we can map the states on an ITP to states in a corresponding projective state space, in such a way that ITP states are mapped to the same projective state whenever they are physically indistinguishable [28, theorem 2.11].
3.2 The Case for Countable Collections of Partial Theories
While projective state spaces give access to a large class of states, they also have the advantage of providing a fairly explicit description of these states, in contrast to an approach where the quantum state space is directly defined as the set of all possible quantum states (like in AQFT [17, 18], which we mentioned in section 1.). In the latter case, quantum states have to be specified through the expectations values they prescribe for arbitrary products of observables9. The difficulty is that there are relations that these expectations values together must satisfy (in particular to ensure that all measurement probabilities will come out positive, see [17, part III, def. 2.2.8]) making it non-trivial to actually construct valid states in this form and study their properties. Arguably, quantum states expressed as collections of partial density matrices are also subjected to
9 See however an alternative formulation, advocated in [21], allowing to recast the full state space of AQFT in projective form. 1–20 11
the satisfaction of extra relations (namely, the consistency condition from eq. (3)). Yet, with an additional assumption on the size of the label set, it is possible to obtain an explicit parametrization of all available projective states [30, subsection 2.1]. Suppose first that our partial theories are labeled by an increasing sequence: say, for concreteness, ηn n1, where the partial theory labeled by ηn consists of lattice dof. on a regular lattice with step a/2n. Then, the full quantum state space can be constructed very easily: indeed, each projective state can be constructed via a recursive process, starting H from a density matrix ρη1 on η1 , and choosing, for any n 1, ρηn+1 in the pre-image −1 Trηn+1→ηn ρηn (taking advantage of the fact that the partial trace is always surjective). The class of label sets for which a constructive description of the quantum state space can be obtained in this manner is actually much broader. This follows from the following observation. If we have a projective state space build on a label set L, then restricting it to a subset K does not change the quantum state space, provided K is cofinal in L, ie. that it satisfies the condition ∀η ∈L, ∃κ ∈K η κ.
The projective state spaces are the same because coarser partial states can always be reconstructed from finer ones: if we know ρκ for all κ ∈K, we can reconstruct any ρη as Trκ→η ρκ for some κ η [28, prop. 2.6]. For example, if we had chosen the sequence of lattice steps above to be a/4k,k 1, we would in fact have been describing the same state space, since {2k | k 1} is a cofinal part of {n | n 1}. Now, any countable label set admits a cofinal sequence of increasingly fine labels (this follows easily from the directedness property mentioned in subsection 2.1, see [30, prop. 2.1]). Furthermore, the same observation also implies that it is not really the cardinality of the label set that matters, but rather its cofinality (the smallest possible cardinality of a cofinal part in it). To summarize, constructive descriptions are readily available for all projective quantum state spaces built on label sets having countable cofinality.
3.3 Projective State Spaces Are Robust With Respect to the Selection of Partial Theories
As each label extracts finitely many dof. out of the full theory, expecting the label set to be of countable cofinality effectively requires that the full theory be spanned by countably many basic observables. This is a physically legitimate assumption, since it should be possible to determine and specify which observable has been measured in a given exper- iment using only a finite amount of information. This does not forbid the classical field theory to be formulated in termsofacontinuum of dof. for mathematical convenience, but we should keep in mind that infinitesimally close observables will be indistinguishable in practice, so that a dense and countable subset of basic observables should be sufficient to capture the entire physical content of the theory. 12 1–20
Nevertheless, one might be worried about making the construction of the quantum state space dependent on the selection of a more or less arbitrary subset of observables. In particular, it seems that the universality put forward in section 2. would suffer. Con- tinuing the example started in subsection 3.2 above, we could have chosen the sequence of lattice steps to be, say, a/3p,p 1. In the absence of a background geometry, the choice of a sequence of finer and finer lattices would involve an even greater degree of arbitrariness, since there is no notion of ’regular lattices’ in this case (see the construction in [34, section 3] and [26, section II]). The natural question to ask is therefore whether these various options lead to dramatically different quantum state spaces. Remarkably, there is a simple condition that an increasing sequence of labels should satisfy to ensure that the projective state space built on it will be universal [30, subsec- L(ext) tion 2.2]. Denote by the complete set of all possible labels, among which a sequence L(ext) has to be chosen. As explained above, if a cofinal sequence κk k1 can be found in , the projective state space built on it coincides with the one that could be build on L(ext) L(ext) and is thus universal. But if is simply too big to adm it such a cofinal sequence, we can relax the condition and only require the sequence κk k1 to be quasi-cofinal in the following sense: for any label η in L(ext), it should be possible to slightly deform η so as to make it a sublabel of one of the κk. More precisely, it should be possible [30, def. 2.7]: 1. to let the deformation be arbitrarily small;
2. and, if some of the dof. contained in η are among the dof. captured by the sequence κk k1 (namely, if there exists a κko that happens to be a sublabel of η), to demand that these particular dof. be left untouched by the deformation [30, fig. 2.1]. Inthecaseof1-dimensional lattices (and by extension of square lattices in any dimension), it transpires from the analysis in [30, section 3] that a sequence of finer and finer lattices will be quasi-cofinal if, and only if, the set of lattice nodes becomes dense as k →∞.In thecaseoffreequantum fields on a possibly curved background (using the quantization scheme mentioned at the end of subsection 2.2), quasi-cofinal sequences of labels were found in [25, subsection 3.2] to correspond to dense, countably-generated vector subspaces in the infinite-dimensional classical phase space. Both results are in agreement with the physical discussion at the beginning of the present subsection: they demonstrate how dense subsets of dof. end up being sufficient to capture the physical content of the theory. Let us clarify what we precisely mean by saying that the projective state space on a quasi-cofinal sequence is ’universal’ [30, theorem 2.8]. If we have two different quasi-cofinal sequences, the set of dof. captured by the first one will of course not be exactly the same as the set of dof. captured by the second one. However, an arbitrarily small deformation can map the first set into the second one, and, modulo this approximate identification of the observables, the two quantum state spaces will coincide (ie. any state in the first one will be in one-to-one correspondence with a state in the second one, in such a way that the expectations values of the suitably identified observables match). An important corollary of this property [30, prop. 2.9] is that symmetries of the original field theory can be represented as isomorphisms of the thus constructed quantum state space: indeed, a 1–20 13
symmetry transformation, mapping a sequence κk to a sequence λl ,canbecomposed with a small deformation mapping λl back to κk , to obtain a transformation that stabilizes the state space and observable algebra on κk . Note that this notion of universality should not be confused with the one expressed by
Fell’s theorem [9]. Fell’s theorem is a result stating that the Hilbert space HΩ constructed from avacuum state Ω as discussed in section 1. is approximately universal in the following sense: for any quantum state Ω that can be built on the same algebra of observables
as Ω, any finite set of observables f1,...,fN and any desired precision , there exists a
density matrix ρ on HΩ such that the expectation values of these particular observables, as computed with respect to Ω vs. ρ, will not differ by more than .Themost crucial difference between this result and the one discussed in the present subsection is that,
if one want to fully take advantage of Fell’s theorem to recover from HΩ asufficiently universal quantum state space, one will effectively have to work, not in the space of density matrices on HΩ, but in a suitable completion thereof (namely, the completion with respect to the topology implied by the just spelled out statement of the theorem). By contrast, a projective state space constructed on a quasi-cofinal sequence is already ’complete’ enough: its sole limitation lies in the fact that the quantum states it describes a priori only prescribe the expectations values of the observables in a certain countably- generated subset, safe in the knowledge that all other observables included in the (artificial but technically convenient) continuum classical field theory can be reconstructed from those at an arbitrary level of precision.
4. From Kinematical to Dynamical State Spaces
4.1 The Need for Regularization
We have not, so far, addressed the question of how the equations of motion of the original field theory are to be implemented on a quantum state space constructed along the lines of section 2.. This question raises two distinct kinds of issues: first, we need to understand how the dynamics can be expressed in a classical theory which has been reformulated as a collection of partial theories as described in subsection 2.1; second, we need to suitably quantize this classical dynamics. The second part of the problem will involve the same tools as form usually part of a canonical quantization endeavor [7, 14, 13], however the first part is specific to projective state spaces. The core difficulty is that the dynamics is unlikely to respect the coarse graining introduced in subsection 2.1. In a field theory we cannot, in general, write closed equations involving only finitely many degrees of freedom: the evolution of an infinite-dimensional system, when projected on a finite-dimensional truncation, is likely to be non-deterministic, because it is subject to back-reaction from all the discarded dof. whose actual value is unknown. For the rest of the present section, we will adopt a parametrized description of the dynamics: by adding the time variable explicitly to the configuration space of our theory, solutions of the equations of motion can be represented as orbits or trajectories in the 14 1–20
η Yη
x
η πη →η Yη
η Yη η πη→η η Yη
Yη
Fig. 3 Adapted (left side) versus regularized dynamics (right side)
resulting extended phase space (see [45, section 1.8] as well as the procedure illustrated in [24, 23]). This way of looking at dynamics has the advantage that it generalizes seamlessly to gauge theories (where the solutions correspond to orbits under both time evolution and gauge transformations), and even to theories like general relativity, that do not refer to any external time variable (where time evolution is just another gauge transformation [1]). In this language, the dyna mics of the original field theory are said to be adapted to M a collection of partial theories η η if they induce consistent partial dynamics on each Mη [27, subsection 3.1]. Consistency here means that any orbit of the partial dynamics on a finer label η should project exactly to an orbit on a coarser label η. In this way, we can obtain projective families of orbits, which correspond precisely to the solutions of the full dynamics (left part of fig. 3). As stressed above, we cannot expect this ideal case to apply in any realistic field theory (unless we dispose beforehand of an extensive understanding of the dynamics, that we can take into account when defining the collection of partial theories; this would suffer from the samelimitations as underlined in section 1. in the context of vacuum-based quantization). What we can do instead [27, subsection 3.2], is to define, for each η, approximate partial dynamics, that can be formulated in closed form over Mη at the cost of not exactly matching the desired full dynamics.10 These approximate dynamics should become closer and closer to the exact one as we consider finer and finer labels η. This implies that such a collection of approximate partial dynamics will not be consistent in the sense above, because the partial dynamics on a finer label η are more accurate than the ones on a coarser label η. Still, given some initial data x(t =0)(or,more generally, given a point M on a gauge fixing surface in ∞ that intersects each orbit exactly once), we can consider M the family of approximate orbits Yη η,whereYη denotes the orbit on η originating from (the projection of) x (right part of fig. 3).
10 The approach taken in [27, subsection 3.2] is a slight generalization of this proposal, in which additional parameters can be introduced to control the approximation. This is done by indexing the approximation scheme on a larger label set, namely a certain part of L×E, with the set E holding the extra approximation parameters. 1–20 15
The idea behind this approach is to represent an exact solution of the full dynamics by the family of approximate solutions that converges to it. Of course, not all families of successive approximations will converge to an exact solution: the initial data x will in general have to satisfy suitable regularity conditions for the approximation schemeto converge. But, importantly, the convergence of a given family can be studied completely within the projective setup, without referring to the continuous theory M∞.Letus define, for any η ε ε Yη := πε→η Yε ie. the orbit at a finer level of approximation ε, projected back on Mη. As stressed above the fact that the partial dynamics defined on Mη is only a coarse approximation is ε reflected by the deviation Yη = Yη . Convergence of the approximation is then expressed ε as the requirement that, for each η, the orbit Yη converges, as ε gets finer and finer, to ∞ ∞ (the projection of) an exact orbit Yη .Now,thefamily Yη η is a projective family of orbits by construction, and as in the case of ’adapted dynamics’ discussed above, it can be seen as the projective description of an exact solution of the full dynamics.
4.2 A Framework for Studying Convergence at the Quantum Level
The ability to discuss the convergence of the regularization schemefrom within the pro- jective setup is especially crucial when going over to the quantum theory, where there is no obvious equivalent of M∞. Specifically, we will implement the dynamics on the quantum projective state space by turning the approximate partial dynamics on each
Mη into a corresponding quantum dynamics on Hη, constructing for any given initial data an associated family of approximate solutions, and study its convergence with the same strategy as on the classical side. Note that a difference between classical and quan- tum dynamics is that solutions of the (approximate) quantum dynamics are just special quantum states, not subsets of states as in the classical case: this effect can be observed already at the level of classical statistical mechanics, where ’kinematical’ states would be arbitrary probability distributions on the (extended) phase space, while solutions, or ’dynamical’ states, would be those probability distributions that are constant along the orbits. Accordingly, a converging family of approximated solutions of the quantum dy- ∞ namics will define a projective family of quantum states ρη η; in other words, such a family will provide a special state in the kinematical projective quantum state space. In [29, subsection 3.2], this procedure has been applied to the second quantization of the Schr¨odinger equation (aka. non-relativistic quantum field theory [6]). One-particle quantummechanics, seen as a classical field theory, can be quantized in a traditional way leading to a Fock space which is simply the Fock space describing an arbitrary num- ber of non-interacting particles, each obeying the first-quantized theory. Alternatively, a projective quantum state space can be built (the partial theories being obtained by orthogonally projecting the wavefunction on every finite dimensional subspace of the one-particle Hilbert space) and the dynamics can be regularized on it. Interestingly, a domain of convergence for this regularization of the quantum dynamics can be delimited 16 1–20
which is consistent with the traditional approach, in the sense that the dynamical pro- jective quantum states it describes are precisely the ones in the Fock sector. This sheds light as to why the Fock space is an appropriate arena for describing this non-interacting theory and demonstrates how projective techniques can allow us to get a handle on the quantum dynamics without having to guess beforehand agoodvacuum: instead, a sector on which the dynamics can be well-defined should naturally emerge from the study of the convergence behavior of approximate solutions. This also raises the intriguing open ques- tion of whether suitable domains of convergence will always take the form of a vacuum sector with respect to a certain (dynamics-aware) vacuum.
5. Outlook: Bridging Canonical Quantization and Path-integral Approaches
Looking for a good quantum state space may seem like a preoccupation only relevant if we insist to rely on canonical quantization. The argument here would be that we do not actually need any quantum state space: it is sufficient to know the quantum probabilities for the outcome of any arbitrary experimental protocol, and these prob- ability distributions could themselves be reconstructed from their moments. However, the so-called time-ordered expectation values that can be computed from a path-integral formulation are not the moments corresponding to the successive measurement of the cor- responding observables, notwithstanding the fact that the measurement operations are, of course, causally ordered. For example, to know the probability of measuring successively 11 f1 = a1,...,fn = an, we need to compute
P (f1 = a1,...,fn = an)= Π Ω Π Π Ω Π ...Π Ω = 1 2 1 ... n 1 = Ω Π1Ω Πn−1 ... Π1Ω = Ω | Π1 ...ΠnΠn ...Π1 | Ω (5) ˆ where Πi denotes the spectral projector on the eigenvalue ai of fi. Knowing all time- ordered expectation values is thus a priori not sufficient to recover the physically interest- ing moments (note the mirroring in the formula above: from right to left, the observables appear first in causal-order and are then repeated in reverse causal order). Reciprocally, it turns out that if we had the vacuum-vacuum expectation values for product of oper- ators in arbitrary orders, which would encode all the physically interesting information, we could in fact also reconstruct the quantum state space, via the GNS construction mentioned in section 1. (in this context also known as Wightman reconstruction [44]). Now, on Minkowski spacetime, the expectation values for products of observables in arbitrary order can all be reconstructed from the time-ordered ones (or, equivalently, the quantum expectation values for the different orderings of operators can be all re- constructed via analytic expansion from the Euclidean path-integral: different orderings 11 This is assuming the universe was initially in the vacuum state, but we can easily relax this questionable P (f =a ,...,f =a ) assumption by considering relative probabilities, eg. 1 1 n n /P (f1=a1). 1–20 17
then arise as different prescriptions for going around singularities, as was proved by Osterwalder and Schrader [35, 36]). It is however not clear how the fairly subtle com- plex analysis underlying this result could be generalized to non-static background space- times (although results exist when a timelike Killing vector is available [19]), or even to background-independent theories. This is the situation where disposing of a (kinematical) quantum state space can be valuable even for the path-integral approach. We can then, instead of only computing vacuum-vacuum expectation values, develop the path-integral between arbitrary boundary quantum states(likeisdoneeg.inthespinfoam approach [37] to quantum gravity, leaning against the Ashtekar-Lewandowski Hilbert space men- tioned in section 1.) and this is indeed an alternative way of capturing all the physically interesting information (since we can then insert a decomposition of the identity in the middle of the above formula). Projective quantum state spaces seem particularly well-suited to play the role of supportive state spaces for boundary-aware path-integral formalisms. On one hand, they implement the same kind of coarse-graining/refining which is used in practice to compute, through a limiting process, path-integral expressions. On the other hand, they anticipate the loss of privileged status for the vacuum state that is the natural consequence of allowing non-vacuum boundaries. Reciprocally, techniques that have been developed in the context of path-integral approaches could be applied to the strategy presented in section 4., to help with the study of convergence, and with the inclusion of renormalization (which was not needed for the simple example considered in [29, subsection 3.2] but is likely to come into play when considering interacting theories).
Acknowledgement
I would like to thank Thomas Thiemann for encouraging me to write this overview letter. Financial support through the Programa de Becas Posdoctorales de la UNAM is acknowledged.
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EJTP 14, No. 37 (2018) 21–34 Electronic Journal of Theoretical Physics
Noncommutative Structure of Massive Bosonic Strings
N. Mansour1, E.Diaf 2 andM.B.Sedra1∗
1SIMO-Lab, faculty of sciences,Ibn Tofail University,Kenitra, Morocco 2Department of Physics, Pluridisciplinairy Faculty of Nador, OLMAN-RL Pluridisciplinary Faculty of Nador, University Mohammed Premier, Oujda B.P N 300, Selouane, 27000 Nador,Morocco.
Received 27 December 2017, Accepted 16 March 2018, Published 20 April 2018
Abstract: In this paper we will apply the Faddeev-Jackiw quantization methodology to the massive open strings in the D-brane background with a non-vanishing constant B-field.We shall work in the discrete version, and the reduced phase space is obtained directly by solving the mixed boundary conditions. The non-commutativity is extended to the noncommutativity of pace coordinates and momentum coordinates along the D-brane is reproduced in easy way.We feel that this method of obtaining the noncommutativity in string theory in the massive case is more elegant than previous approaches discussed in the literature. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Noncommutative geometry; string theory; massive bosonic string; Faddeev Jackiw quantization PACS (2010): 11.25.Yb; 11.25.-w; 11.25.Hf; 02.40.Gh; 11.25.Uv
One of the interesting develoment in string theory is the realisation of noncommuta- tive geometry [1] and this appears in the work of Witten on open string field theory [2, 3]. Noncommutative geometry arise in D-branes on constant antisymmetric tensor field [4].The main goal of this paper is giving the essential aspects of the symplectic Faddeev Jackiw Quantization and applying it for quantization of massive bosonic open string in the presence of antisymmetric tensor B-field, we find that phase space coordinates of the massive open string end-points become totally noncommutative. The Faddeev-Jackiw formulation and its equivalence with the Dirac approach is studied in detail by several authors. The fundamental commutators relations of open string coordinates xi on the branes are represented by noncommutativity ,[xi,xj]=iθij where θij is an antisymmetric constant tensor deponding on the background constant fields [7, 8, 9, 10, 11, 12, 13,
∗ Email: [email protected], [email protected], [email protected] 22 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34
14]. Strings attached to branes involve mixed (combination of Dirichlet and Neumann) boundary conditions. This makes the quantization procedure more subtle. The standard canonical commutation relations can not be imposed as quantum commutators as they are not consistent with the boundary conditions. This situation is analogous to that of systems with constraints, where one must build up commutators that are consistent with them. The difference is that the boundary conditions, in the form that comes from the functional variation of the string action, involve velocities. So, they do not correspond to standard Dirac constraints. If we apply directly standard Dirac procedure to the string action we would not find the boundary conditions as constraints. Nevertheless, it has been recently shown in refs.that it is possible to use the Dirac procedure as long as one rewrites the boundary conditions in terms of phase space variables and introduces them as constraints. Then the interesting result of the non commutativity of the phase space coordinates of the string end points emerges. We will see that with the symplectic quantization procedure the boundary conditions arise directly as constraints by means of a discretization of the string worldsheet spatial coordinate.We will show how to find a particular choice of symplectic variables such that the boundary conditions show up naturally as constraints[15].This way we find a straightforward way of building up the phase space coordinates commutators consistent with the mixed boundary conditions and thus reproduce the totaly noncommutativity at the string end points[17,18,19].
1 A brief review of the Faddeev-Jackiw method
In this section we discuss the methodology of FJ approach to quantize the singular sys- tems. In this formalism, we first write the Lagrangian of a singular system into the first-order form as follows:).
0 0 k − L = ak(ξ)∂τ ξ V (ξ)(1) Where ξi is called the symplectic variable, V (ξ) is called the symplectic potential. The
first-order form can be implemented by introducing some auxiliary variables (ak)suchas the canonical momentum. The Euler-Lagrange equations of motion for Lagrangian (15) can be written as ∂V (ξ) f 0 (ξ)ξ˙j = (i =1, 2, 3, ..., n), (2) ij ∂ξi where fij is so-called symplectic matrix with following explicit form: ∂a0 ∂a0 f 0 = j − i (3) ij ∂ξi ∂ξj If it is non singular we define the commutators of the quantum theory (if there is no ordering problem for the corresponding quantum operators) as
∂A 0 −1 ∂B [A(ξ),B(ξ)] = (f )ij (4) ∂ξi ∂ξj Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34 23
0 α If the matrix (17) is singular we find the zero modes that satisfy fijvj =0andthe corresponding constraints:
α α ∂V Ω = vl =0 (5) ∂ξl Now, we modify our original Lagrangian by introducing the constraint term multiplied with some Lagrange multipliers λα as
1 0 ˙ ˙α α − 1 ˜ ˜˙ − L = ak(ξ)ξk + λ Ω V (ξ)=ar(ξ)ξ V (ξ)(6)
where we introduced the new notation for the extended variables: ξ˜r =(ξk,λα)We find 1 now the new matrix fij
∂a1 ∂a1 f 1 = j − i (7) ij ∂ξ˜i ∂ξ˜j
If f 1 is not singular we define the quantum commutators as ∂A ∂B [A(ξ˜),B(ξ˜)] = (f 1)−1 (8) ˜ ij ˜ ∂ξi ∂ξj This process of incorporating the constraints in the Lagrangian is repeated until a non singular matrix is found.
2 Review of bosonic open string in a background B-field
the electromagnetic field couples with charged open string’s endpoints, as the endpoints behave like point particles. this interaction is given by : 1 i dτAi(X)∂τ X (9) 2πα ∂Σ Where ∂Σ is the boundary of the string worldsheet Σ. In this case ∂Σ represents string endpoints at any time. we have put the endpoint charge q =1atσ = π and q = −1at σ =0. The dynamics of strings ending on a p-brane in the background of the antisymmetric
field ,Bμν is [1] : 1 2 ab μ ν ab μ a ν S = dσ [ημνg ∂aX ∂bX + Bμν∂aX ∂ X ] (10) 4πα 1 i + dτAi(X)∂τ X 2πα ∂Σ
α Where Ai,i =0, 1,p is the U(1) gauge field living on the D-brane, σ =(τ,σ) are the 01 01 world-sheet coordinates ,gαβ = diag(−, +), = − = 1 is the anti-symmetric symbol
in two dimensions,Bμν = −Bνμ, ημν = diag(−, +, ..., +), and the length of the string is π. 24 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34
There is an important relation between B-field and electromagnetic potential Ai(X). Let
Fij = ∂iAj(X) − ∂jAi(X) be electromagnetic field strength. Consider
1 2 αβ i j dσ Fij(X)∂αX ∂βX (11) 4πα Σ 1 2 αβ − i j = dσ (∂iAj(X) ∂jAi(X))∂αX ∂βX 4πα Σ 2 2 αβ i j = dσ ∂iAj(X)∂αX ∂βX 4πα Σ 2 ∂A (X) ∂Xα = dσ2αβ j . ∂ Xi∂ Xj α i α β 4πα Σ ∂X ∂X 2 2 αβ j = dσ ∂αAj(X)∂βX 4πα Σ 2 2 j − j = dσ [∂τ Aj(X)∂σX ∂σAj(X)∂τ X ] 4πα Σ 2 2 j − j = dσ [∂τ (Aj(X)∂σX ) ∂σ(Aj(X)∂τ X )] 4πα Σ 2 − j =0 dτAj(X)∂τ X 4πα ∂Σ
Where in the final step we assume that Aj(X) vanishes at initial and final time τ. So the action becomes : 1 − 2 αβ αβ μ ν S = dσ (η ημν + Bμν(X))∂αX ∂βX (12) 4πα Σ 1 − 2 αβ i j dσ Fij(X)∂αX ∂βX 4πα Σ
If we set the component of B-field to be parallel to the D-brane :
αβ μ ν αβ i j Bμν(X)∂αX ∂βX = Bij(X)∂αX ∂βX (13) we call the string coordinates along the brane Xi and the string coordinates normal to the brane Xa. The action becomes : 1 − 2 αβ μ ν αβ i j S = dσ (η ημν∂αX ∂βX + Fij∂αX ∂βX ) (14) 4πα Σ
Where Fij = Fij + Bij.
The resulting field Fij is invariant under gaug transformation of potential Ai :
Ai → Ai + ∂K,
Bij → Bij.
And under gauge transformation of B-field Ai → Ai − Λi,Bij → Bij + ∂iΛj − ∂jΛi Let us Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34 25
now vary the action. The variation of the first part is :
− 2 1 2 αβ μ ν δ( dσ η ημν∂αX ∂βX ) (15) 4πα Σ −1 2 − μ μ = dσ [ ∂τ X ∂τ δXμ + ∂σX ∂σδXμ] 2πα Σ −1 2 − μ | 2 2 − 2 μ μ = [ dσ ( ∂τ X δXμ) τ + dσ (∂τ ∂σ)X δXμ + dτ∂σX δXμ] 2πα Σ Σ δΣ
The first term vanishes due to the requirement that δXμ = 0 at initial and final time τ . Now we vary the B-field part of the action. Inspired by the relation (3) we have :
− 1 αβ i j Fij∂αX ∂αX (16) 4πα Σ 1 i = dτωi(X)∂τ X 4πα Σ 1 i We instead vary : 4πα Σ dτωi(X)∂τ X 1 i δ( dτωi(X)∂τ X ) (17) 2πα ∂Σ 1 i 1 i = dτδωi(X)∂τ X + dτωi(X)∂τ δX 2πα ∂Σ 2πα ∂ Σ 1 ∂ω 1 1 = dτ i δXj∂ Xi + dτ∂ (ω (X)δXi) − dτ∂ ω (X)δXi j τ τ i τ i 2πα ∂Σ ∂X 2πα ∂Σ 2πα ∂Σ 1 1 j i − i = dτ∂jωi(X)δX ∂τ X +0 dτ∂τ ωi(X)δX 2πα ∂Σ 2πα ∂Σ
δXi = 0 at initial and final time 1 1 ∂ω ∂Xj = dτ∂ ω (X)δXj∂ Xi − dτ i δXi j i τ j 2πα ∂Σ 2πα ∂Σ ∂X ∂τ 1 1 j i − j i = dτ∂jωi(X)δX ∂τ X dτ∂jωi(X)∂τ X δX 2πα ∂Σ 2πα ∂Σ 1 1 i j − j i = ∂Σdτ∂iωjδX ∂τ X dτ∂jωi∂τ X δX 2πα 2πα αΣ
We have Fij = ∂iωj − ∂jωi. Then finally we have the reult :
1 i j dτFij(X)δX ∂τ X (18) 2πα ∂Σ 26 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34
Then the variation of the full action is : (7) + (10) −1 1 2 2 − 2 μ − μ δS = dσ (∂τ ∂σ)X δXμ dτ∂σX δXμ (19) 4πα Σ 4πα ∂Σ 1 i j + dτFij(X)δX ∂τ X 2πα ∂σ −1 1 2 2 − 2 μ − i = dσ (∂τ ∂σ)X δXμ dτ∂σX δXi 4πα Σ 4πα ∂Σ 1 1 − a i j dτ∂σX δXa + dτFij(X)δX ∂τ X 4πα ∂Σ 2πα ∂σ 2 − 2 μ The requirement δS = 0 give the equation of motion (∂τ ∂σ)X = 0 and the boundary conditions at σ =0,π for μ = a:
a a a δX =0,X= x0 (20)
− 1 i 1 i j dτ∂σX δXi + Fij(X)δX ∂τ X 4πα ∂σ 4πα ∂σ −1 1 i − i j = dτ∂σX δXi Fji(X)δX ∂τ X 4πα ∂σ 4πα ∂σ −1 1 i j − i = dτFj(X)δXi∂τ X dτ∂τ X δXi 4πα ∂Σ 4πα ∂Σ − 1 i j i = dτ(Fj(X)∂τ X + ∂σX )δXi 4πα ∂Σ So we have the boundary conditions for μ = i
i i j ∂αX + Fj(X)∂τ X = 0 (21)
i i i i Where Fj = Bj + Fj ,we take Fj =0 The boundary conditions become:
j j gij∂σX − Bij∂τ X = 0 (22)
The effect of electromagnetic field is a change in the boundary conditions.
3 Massive Bosonic Open string in the presence of antisymmet- ric tensor filed and Totally Noncommutative Phase Space
Let us now see how the noncommutative structure of massive bosonic string can be calcu- lated using the symplectic Faddeev Jackiw quantization[16,20], and see how this boundary Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34 27
conditions show up as equations of motion by writing down a discretized version of the lagrangian of massive bosonic string in an external B-field associated with the action gived by: 1 − 2 αβ i j 2 i j αβ i j S = dξ (gijη ∂αX ∂βX + gij(∂σm) X X + Bij∂αX ∂βX ) (23) 4πα Σ 1 − 2 ˙ i 2 − i 2 2 i 2 − ˙ i j S = dσ [(X ) (X ) + m (X ) 2BijX X ] 4πα Σ The analysis will show us how to impliment boundary conditions in the symplectic quan- tization dividing the interval 0 <σ<πin N intervals of length and introducing the i i i coordinates of the endpoints of the intervals as X0,X1, ..., XN we find (Xi − Xi )2 m2 (Xj − Xj )2 L()= [(X˙ i )2 − n+1 n + (Xi )2 − 2B X˙ i n+1 n ] (24) 4πα n 2 2 n ij n n Where the overdot means the derivative with respect to the time-like parameter τ. The massless case (m = 0) is studied in different aspects by several authors [7, 8, 9, 10, 11, 12, 13, 14] with the well-known result of non commutativity at the end-points. We develop the expression then we have : (Xi − Xi)2 (Xi − Xi)2 L()= [(X˙ i)2 +(X˙ i)2 + ... +(X˙i )2 − 1 0 − 2 1 (25) 4πα 0 1 N 2 2 n (Xi − Xi − 1)2 m2 m2 m2 − ... − N N + (Xi)2 + (Xi)2 + ... + (Xi )2 2 2 0 2 1 2 N (Xj − Xj) (Xj − Xj) (Xj − Xj ) − 2B X˙ i 1 0 − 2B X˙ i 1 1 − ... − 2B Xi˙ N N−1 ] ij 0 ij 2 ij N−1 And we also discretize the mixed boundary condition as : g ij (Xj − Xj) − B ∂ Xj = 0 (26) 1 0 ij τ 0 Using the Euler Lagrange equations d ( ∂L ) − ∂L =0 dτ ∂X˙ i ∂Xi i i i ≤ ≤ − We get the equation of motion for respectively X0,Xn,XN with 1 n N 1 i Then we have for X0 :
1 (Xi − Xi)2 m2 L()= [(X˙ i)2 − 1 0 + (Xi)2 − 2B ∂ Xi(Xj − Xj)] 4πα 0 0 ij τ 0 1 0 n ∂L 1 i − j − j = ((∂τ X0) Bij(X X0 )) ∂X˙ i 2πα i d ∂L 1 2 i − j j ( )= ((∂τ X0) Bij(∂τ X1 )+Bij∂τ X0 ) dt ∂X˙ i 2πα i − i 2 ∂L 1 (X1 X0) m i i i = ( + X0 + Bij∂τ X0) ∂X0 2πα d ∂L ∂L ( ) − =0 dτ ∂X˙ i ∂Xi ⇔ 28 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34
i Finally we have the equation of motion for X0:
(Xi − Xi) m2 (∂2Xi) − 1 0 − B ∂ Xj +2B ∂ Xj − Xi = 0 (27) τ 0 ij τ 1 ij τ 0 0
i i We do the same computing for Xn and XN then we get :
i i i i 2 2 i Xn+1 − Xn Xn − Xn−1 j j j m i ∂ X − + − Bij∂τ X − Bij∂τX +2Bij∂τ X − X = 0 (28) τ n n−1 n+1 n n i i 2 2 i XN − XN−1 i m i ∂ X + − Bij ∂τ X − X =0. (29) τ N N−1 N → i → i i → i When we take the limit 0 and consider that X1 X0 and XN−1 XN the equations i i for X0 and XN give the open string boundary conditions of eq.(26) i The equations for points Xn give no contribution at order zero in but to order one in (dividing by and then taking the limit → 0) they take the form of the standard equation of motion for the string coordinates :
2 i − 2 i − 2 i ∂τ Xn ∂σXn m Xn = 0 (30)
i It is important to remark that if we consider some coordinate Xn with fixed n and take the limit → 0 this coordinate would tend to the end point σ = 0 Thus when we want to look at points inside the string (with a finite distance to the boundary) we must look at i → some Xn but increase n when we take the limit 0 in order to look at a fixed point. This analysis of the equations of motion shows us that in the discretised version both boundary conditions and string equations of motion show up in the set of generalized equations of motion but at different orders in the discretization parameter .This will help us to find an appropriate definition of the symplectic variables such that the boundary conditions lead to zero modes in the symplectic matrix. The standard way of writing the Lagrangian L of eq. (25) in a first order form would be to introduce the conjugate momenta Π as symplectic variables and eliminate the second order time derivatives. The conjugate momenta associated with the string coordinates are : 1 Πi = (∂ Xi − B (Xj − Xj)), (31) 0 2πα τ 0 ij 1 0 1 Πi = (∂ Xi − B (Xj − Xj )), (32) n 2πα τ n ij n+1 n 1 Πi = (∂ Xi ) (33) N 2πα τ N The symplectic first order Lagrangian is then: for the first term we have
(Xi − Xi)2 (Xj − Xj) m2 L0(first term)= ((∂ Xi)2 − 1 0 − 2B ∂ Xi 1 0 + (Xi)2) 4πα τ 0 2 ij τ 0 0 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34 29
We have the first symplectic variable:
1 Πi = (∂ Xi − B (Xj − Xj)), 0 2πα τ 0 ij 1 0 i i − j − j 2πα Π0 = ∂τ X0 Bij(X1 X0 ) (Xj − Xj) 1 (Xi − Xi)2 L0(first term)= (2παP i + B 1 0 )2 − 1 0 4πα 0 ij 4πα 1 m2(Xi)2 + (P i − ∂ Xi)∂ Xi + 0 0 2πα τ 0 τ 0 4πα (Xj − Xj) 1 (Xi − Xi)2 m2(Xi)2 = η P i∂ Xj − (2παP i + B 1 0 )2 − 1 0 + 0 ij 0 τ 0 4πα 0 ij 4πα 4πα We do the same computing for the other terms , finally we find the symplectic first order lagrangian :
0 i j i j i j i j i j L = ηij(P0∂τ X0 + P1∂τ X1 + ... + Pn∂τ Xn + ... + PN−1∂τ XN−1 + PN ∂τ XN )+V (34)
Where 1 (Xi − Xi)2 (Xi − Xi)2 (Xi − Xi )2 V = − ( 1 0 + 2 1 + ... + N N−1 ) 4πα (Xj − Xj) 2πα (Xj − Xj) − [(2παP i + B 1 0 )2 +( P i + B 2 1 )2... 4πα 0 ij 1 ij 2πα (Xj − Xj ) ( P i + B N N−1 )2 +(2παP i )2 N−1 ij N m2 m2 m2 − (Xi)2 − (Xi)2 − ... − (Xi )2] (35) 0 1 N
We build up the symplectic matrix f0 with our coordinates X, P
j j j j j j j j j j ⎛ X0 P0 X1 P1 ... Xn Pn ... XN−1 PN−1 XN PN ⎞ Xi 0 −δij 00... 00... 0000 0 ⎜ ⎟ P i ⎜ δij 000... 00... 0000⎟ 0 ⎜ ⎟ i ⎜ − ij ⎟ X1 ⎜ 000δ ... 00... 0000⎟ i ⎜ ij ⎟ P1 ⎜ 00δ 0 ... 00... 0000⎟ ⎜ ⎟ ...⎜ ...... ⎟ ⎜ ⎟ Xi ⎜ 0000... 0 −δij ... 0000⎟ f = n ⎜ ⎟ 0 i ⎜ ij ⎟ Pn ⎜ 0000... δ 0 ... 0000⎟ ⎜ ⎟ ...⎜ ...... ⎟ i ⎜ ij ⎟ X − ⎜ 0000... 00... 0 −δ 00⎟ N 1⎜ ⎟ P i ⎜ 0000... 00... δij 000⎟ N−1 ⎜ ⎟ i ⎝ − ij ⎠ XN 0000... 00... 000δ i ij PN 0000... 00... 00δ 0 In the limit → 0 the string theory is recovered the matrix becomes singular with 30 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34
the zero modes. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ i ⎜ u ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ui ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ i ⎜ ⎟ i ⎜ ⎟ i ⎜ ⎟ i ⎜ ⎟ v1 = ⎜ 0 ⎟ v2 = ⎜ 0 ⎟ v3 = ⎜ 0 ⎟ v4 = ⎜ 0 ⎟ (36) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ui ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 ui
i ∂V For v1 the corresponding constraint come from j ∂X0
i − i j − j 2 ∂V 1 (X1 X0) i (X1 X0 ) Bij m i j = δij + (2πα P0 + Bij ) + X0 =0 ∂X0 2πα 2πα 2πα (Xi − Xi) (Xj − Xj) m2 1 0 δ +2παB P i + B 1 0 B + Xiδ =0 ij ij 0 ij ij 0 ij (Xk − Xk) m2 1 0 (δ − B B )+ Xkδ − 2παB P k =0 kj ki ij 0 kj jk 0 Then we find the first constraint: (Xk − Xk) m2 Ωi = 1 0 + XkM −1 − 2παM −1B P k = 0 (37) 1 0 kj kj jk 0 in the → 0 limit this gives the finit result
k − −1 k | (∂σX 2παMkj BjkP0 ) σ=0 = 0 (38)
Where Mkj = δkj − BkiBij. i ∂V For v2 the corresponding constraint come from j ,same computing leads to the second ∂P0 constraint: (Xj − Xj) Ωi = −(2παP i + B 1 0 ) = 0 (39) 2 0 ij So,this actually gives no contraint as → 0 i For v3 the corresponding contraint is :
(Xk − Xk ) m2 Ωi = N N−1 + Xk M −1 − 2παM −1B P k = 0 (40) 3 N kj kj jk N−1 That leads, when → 0 to the finite result :
i − −1 k | ∂σX 2πα Mkj BjkPN σ=0= 0 (41) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34 31
The last constraint will give no contribution when → 0
i − i Ω4 = 2πα PN =0. (42)
i i In order to incorporate the contraints Ω1 and Ω3 into the symplectic formalism we intro- i i duce lagrange multplier variables λ1 and λ3and add a new term to the lagrangian
(Xk − Xk) m2 L = L + λ˙i ( 1 0 + XkM −1 − 2παM −1B P k) 1 0 1 0 kj kj jk 0 (Xk − Xk ) m2 +λ˙i ( N N−1 + Xk M −1 − 2παM −1B P k ) (43) 3 N kj kj jk N−1
Thus, from the symplectic Lagrangian (43) we identify the following set of symplectic i { i i i i i i i i } variables as q = X0,P0, ..., Xn,Pn, ..., XN ,PN ,λ1,λ3 and the components of the sym- plectic 1-forms are : k− k 2 { i i i (X1 X0 ) m k −1 − −1 k ai = P0, 0,Pn, 0, ..., PN , 0, ( + X0 Mkj 2πα Mkj BjkP0 ) (Xk −Xk ) 2 N N−1 m k −1 − −1 k } , ( + XN Mkj 2πα Mkj BjkPN−1)
By performing somme technical commputing we get this result :
j j j j j j j j X0 P0 ... Xn Pn ... X P λ1 λ3 ⎛ N N ⎞ δ 2 −1 Xi 0 − δij ... 00... 00ij + m M 0 0 ⎜ ij ⎟ i ⎜ ij ij ⎟ P0 ⎜ δ 0 ... 00... 00Γ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ...⎜ ...... ⎟ ⎜ ⎟ i ⎜ − ij ⎟ Xn ⎜ 00... 0 δ ... 000 0⎟ ⎜ ⎟ P i ⎜ 00... δij 0 ... 000 0⎟ n ⎜ ⎟ ⎜ ⎟ ...⎜ ...... ⎟ ⎜ 2 ⎟ i ⎜ − ij − δij m −1 ⎟ XN ⎜ 00... 00... 0 δ 0 + Mij ⎟ ⎜ ⎟ i ⎜ ij ij ⎟ PN ⎜ 00... 00... δ 00 Γ⎟ 2 i ⎜ − δij − m −1 ij ⎟ λ1 ⎝ Mij Γ ... 00... 000 0⎠ 2 j δij − m −1 ij λ3 00... 00... Mij Γ 00 ij −1 Where Γ = −2πα (M B)ij
As this symplectic matrix is non singular the invers is
The elements of the inverse will be the corresponding commutators . The relevant ones 32 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34
are : δij [Xi,Pj]= 0 0 2 −Γij [Xi,Xj]= 0 0 ij 2 −1 2(δ + m Mij ) − m2B 1 (Γij)−1 [P i,Pj]=− ij + 0 0 4πα 2 i j [Xn,Xn]=0 i j [Pn,Pn]=0 −δij [P i,Xj]= 0 0 2 − m2B 1 (Γij)−1 [P i ,Pj ]= ij − N N 4πα 2 Γij [Xi ,Xj ]= (44) N N ij 2 −1 2(δ + m Mij )
i Where Xn represents a point that has a finite distance to the end points of the string. We see that on the D-brane the whole phase space becomes noncommutative. We call this kind of noncommutativiy as totally noncommutative phase space.
References
[1] Alain Connes ”Noncommutative Geometry”. [2] E. Witten ”Noncommutative Geometry And String Field Theory” Nucl. Phys. B 268 (1986) 253. [3] Nathan Seiberg and Edward Witten ”String Theory and Noncommutative Geometry” [hep-th/9908142]. [4] A.Connes ,M.R.Douglas, and.Shwarz ”noncommutative geometry matrix theory : compactification on tori” JHEP 02 (1998) 003, [hep-th/9711162]. [5] F.Ardalan, H. Arfaei, and M. M. Sheikh-Jabbari ”Dirac Quantization of Open Strings and Noncommutativity in Branes” [hep-th/9906161]. [6] Won Tae Kim and John J. Oh ”Noncommutative open string from Dirac quantization” [hep-th/9911085]. [7] ] Ralph Blumenhagen ”A Course on Noncommutative Geometry in String ” Theory Max-Planck-Institut fur Physik, Fohringer Ring 6, 80805 Munchen, arXiv:1403.4805v2 [hep-th] 1 Apr 2014. [8] Louise Dolan, Chiara R. Nappi”Strings and Noncommutativity” ,[ hep-th/0302122]. [9] Chong-Sun Chu”noncommutative geometry from strings” , [hep-th/0502167]. [10] C. Hofman and E. Verlinde ”U-Duality of Born-Infeld on the Noncommutative Two- Torus” JHEP 12 (1998) 010, [hep-th/9810116]. [11] C.-S. Chu and P.-M. Ho ”Noncommutative Open String and D-brane” Nucl. Phys. B550 (1999) 151, [hep-th/9812219]. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 21–34 33
[12] Xiao-Jun Wang ”Strings in Noncommutative Spacetime” ,[hep-th/0503111]. [13] Chong-Sun Chu, and Pei-Ming Ho ”Constrained Quantization of Open String in Background B Field and Noncommutative D-brane ”NEIP-99-011[hep-th/9906192]. [14] Farhad Ardalan, Hessamaddin Arfaei and Mohammad M. Sheikh- Jabbari”Noncommutative geometry from strings and branes ” JHEP02(1999)016 , http://iopscience.iop.org/1126-6708/1999/02/016. [15] Nelson R. F. Braga and Cresus F. L. Godinho”Symplectic Quantization of Open Strings and Noncommutativity in Branes” :[hep-th/0110297]. [16] L. Faddeev , R. Jackiw , Phys.Rev.Lett.60 (1988) 1692. [17] Pichet Vanichchapongjaroen ” D-Branes and Noncommutative Geometry in String Theory” [18] A.Shirzad A. Bakhshi and Y. Koohsarian”Symplectic Quantization of Massive Bosonic String in background B-field” arXiv:1112.5781v2 [hep-th] 2 apr 2012. [19] Sunandan Gangopadhyay ”String in pp-wave background and background B-field from membrane and its symplectic quantization” arXiv:0711.0421v2 [hep-th]13 Nov 2007. [20] A. Shirzad , M. Mojiri ”Constraint structure in modified Faddeev-Jackiw method” arXiv:hep-th/0110023v1 2 Oct 2001.
EJTP 14, No.37 (2018) 35–54 Electronic Journal of Theoretical Physics
Minimal Length,Minimal Inverse Temperature,Measurability and Black Holes
Alexander E.Shalyt-Margolin∗ Institute for Nuclear Problems,Belarusian State University, 11 Bobruiskaya str., Minsk 220040, Belarus
Received 27 December 2017, Accepted 16 March 2018, Published 20 April 2018
Abstract: The measurability notion introduced previously in a quantum theory on the basis of a minimal length in this paper is defined in thermodynamics on the basis of a minimal inverse temperature. Based on this notion, some inferences are made for gravitational thermodynamics of horizon spaces and, specifically, for black holes with the Schwarzschild metric. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Minimal Length,Minimal Inverse Temperature, Measurability PACS (2010): 03.65; 05.20
1 Introduction.
This paper is a continuation of the earlier works published by the author [1],[2]. The main idea and target of these works is to construct a correct quantum theory and gravity in terms of the variations (increments) dependent on the existent energies. It is clear that such a theory should not involve infinitesimal space-time variations
dt, dxi,i=1, ..., 3. (1)
Besides, as shown in [2], with the involvement of some universal units of the minimal
length lmin and time tmin, this theory will be discrete and dependent on the explicitly defined discrete parameters but very close to the initial continuous theory at low energies. The notion of measurability introduced by the author in [2] is essential for the above- mentioned discrete theory. This paper demonstrates that an analogous notion may be introduced in thermodynamics on the basis of the minimal inverse temperature. Based on the obtained results, the
∗ Email: [email protected], [email protected] 36 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
inferences for gravitational thermodynamics of horizon spaces and, specifically, for black holes with the Schwarzschild metric are introduced. All aspects of the authors motivation were given in detail in [1] and, in particular, in [2]. In short, the motivation is as follows. According to the present-day views, both quantum theory and gravity in the ultraviolet region is related to some new parameters defined at high (apparently Planck) energies (for example [3]). But at low energies, this relationship is not apparent due to its insignificant effect in this case, on the one hand, and due to the mathematical apparatus of continuous space-time, where the existent theories are considered, on the other hand. By the author’s opinion, the correct definition of a dynamics of the above-mentioned relationship at all the energy scales will enable us to find the key to solutions of all the problems given below: 1.1 ultraviolet and infra-red divergences in a quantum field theory; 1.2 correct transition to the high-energy (quantum) region for gravity; 1.3 and possibility of the existence of ”nonphysical” solutions for the General Relativity (for example, the solutions involving the Closed Time-like Curves (CTC) [6]–[9]). To make this paper maximally self-contained, the author includes all the required earlier obtained results precisely with the corresponding references in Subsection 2.1 and at the beginning of Subsection 2.2. New results are presented in the second half of Subsection 2.2., and also in Subsection 2.3. and in Section 3.
2 Minimal Length, Minimal Inverse Temperature, and Measur- ability
2.1 Generalized Uncertainty Principles in Quantum Theory and Ther- modynamics
In this Subsection the author presents some of the results from Section 2 of the paper [10],because they are important for this work. It is well known that in thermodynamics an inequality for the pair interior energy - in- verse temperature that is completely analogous to the standard uncertainty relation in quantum mechanics [11] can be written [14] – [19]. The only (but essential) difference of this inequality from the quantum mechanical one is that the main quadratic fluctuation is defined by means of the classical partition function rather than by the quantum me- chanical expectation values. In the last years a lot of papers appeared in which the usual momentum-coordinate uncertainty relation has been modified at very high energies on
the order of the Planck energy Ep [20]–[31]. In this note we propose simple reasons for modifying the thermodynamic uncertainty relation at Planck energies. This modification results in existence of the minimal possible main quadratic fluctuation of the inverse tem- perature. Of course we assume that all the thermodynamic quantities used are properly defined so that they have physical sense at such high energies. Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 37
We start with usual Heisenberg Uncertainty Principle (relation) [11] for momentum - coordinate: Δx ≥ . (2) Δp It was shown that at the Planck scale a high-energy term must appear: p Δx ≥ + αl2 (3) Δp p
2 3 −35 where lp is the Planck length lp = G /c 1, 610 m and α is a constant. In [20] this term is derived from the string theory, in [23] it follows from the simple estimates of Newtonian gravity and quantum mechanics, in [27] it comes from the black hole physics, other methods can also be used [26],[28],[29]. Relation (3) is quadratic in Δp
2 2 − 2 ≤ α lp (Δp) ΔxΔp + 0(4)
and therefore leads to the fundamental length
√ Δxmin =2 α lp (5)
Inequality (3) is called the Generalized Uncertainty Principle (GUP) in Quantum Theory. Using relations (3) it is easy to obtain a similar relation for the energy - time pair. Indeed (3) gives Δx Δp ≥ + αl2 , (6) c Δpc p c then l2 Δpc ΔE Δt ≥ + α p = + αt2 , (7) ΔE c2 ΔE p where the smallness of lp is taken into account so that the difference between ΔE and 5 −43 Δ(pc) can be neglected and tp is the Planck time tp = lp/c = G /c 0, 54 10 sec.
Inequality (7) gives analogously to (3) the lower boundary for time Δt ≥ 2tp, determining the fundamental time √ tmin =2 α tp. (8) Thus, the inequalities discussed can be rewritten in a standard form ⎧ ⎪ Δp ⎪ Δx ≥ + α ⎨⎪ Δp Ppl Ppl ⎪ (9) ⎪ ⎪ ⎩ Δt ≥ + α ΔE ΔE Ep Ep 3 where Ppl = Ep/c = c /G. Now we consider the thermodynamic uncertainty relations between the inverse temperature and interior energy of a macroscopic ensemble 1 k Δ ≥ B , (10) T ΔU 38 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
where kB is the Boltzmann constant. N.Bohr [12] and W.Heisenberg [13] first pointed out that such kind of uncertainty principle should take place in thermodynamics. The thermodynamic uncertainty relations (10) were proved by many authors and in various ways [14] – [19]. Therefore their validity does raises no doubts. Nevertheless, relation (10) was proved in view of the standard model of the infinite-capacity heat bath encompassing the ensemble. But it is obvious from the above inequalities that at very high energies the capacity of the heat bath can no longer be assumed infinite at the Planck scale. Indeed, the total energy of the pair heat bath - ensemble may be arbitrary large but finite, merely as the Universe was born at a finite energy. Hence the quantity that can be interpreted as the a temperature of the ensemble must have the upper limit and so does its main quadratic deviation. In other words, the quantity Δ(1/T ) must be bounded from below. But in this case an additional term should be introduced into (10)
1 k Δ ≥ B + η ΔU (11) T ΔU where η is a coefficient. Dimension and symmetry reasons give
k k ∼ B = B (12) η 2 or η α 2 Ep Ep
As in the previous cases, inequality (11) leads to the fundamental (inverse) temperature √ √Ep √Tp Tmax = = = = , 2 α tpkB 2 α kB 2 α tminkB 1 tmin βmin = = . (13) kBTmax
In the work [32] the black hole horizon temperature has been measured with the use of the Gedanken experiment. In the process the Generalized Uncertainty Relations in Thermodynamics (11) have been derived also. Expression (11) has been considered in the monograph [33] within the scope of the mathematical physics methods. Thus,we obtain a system of the generalized uncertainty relations in the symmetric form ⎧ ⎪ Δp ⎪ Δx ≥ + α + ... ⎪ Δp Ppl Ppl ⎪ ⎪ ⎪ ⎨⎪ Δt ≥ + α ΔE + ... (14) ⎪ ΔE Ep Ep ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ Δ 1 ≥ kB + α ΔU kB + ... T ΔU Ep Ep
or in the equivalent form Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 39
⎧ ⎪ Δp ⎪ Δx ≥ + α l2 + ... ⎪ Δp p ⎪ ⎪ ⎪ ⎨⎪ Δt ≥ + αt2 ΔE + ... ⎪ ΔE p (15) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Δ 1 ≥ kB + α 1 ΔU + ..., T ΔU 2 Tp kB
where the dots mean the existence of higher order corrections as in [34]. Here Tp is
the Planck temperature: Tp = Ep/kB. In literature the relation (10) is referred to as the Uncertainty Principle in Thermody- namics (UPT). Let us call the relation (11) the Generalized Uncertainty Principle in Thermodynamics (GUPT). In this case, without the loss of generality and for symmetry, it is assumed that a dimen- sionless constant in the right-hand side of GUP (formula (3)) and in the right-hand side of GUPT (formula (11)) is the same – α.
2.2 Minimal Length and Measurability Notion in Quantum Theory
First, we consider in this Subsection the principal definitions from [1],[2]which are re- quired to derive the key formulae in the second part of the Subsection and to obtain further results.
Definition I. Let us call as primarily measurable variation any small variation (increment) Δxμ of any spatial coordinate xμ of the arbitrary point xμ,μ =1, ..., 3in some space-time system R if it may be realized in the form of the uncertainty (stan-
dard deviation) Δxμ when this coordinate is measured within the scope of Heisenberg’s Uncertainty Principle (HUP) [11] (formula (2) in the general case):
Δxμ =Δxμ, Δxμ ,μ=1, 2, 3 (16) Δpμ for some Δpμ =0. Similarly, at μ = 0 for pair “time-energy” (t, E), let us call any small variation (increment) the primarily measurable variation in the value of time Δx0 = Δt0 if it may be realized in the form of the uncertainty (standard deviation) Δx0 =Δt and then Δ t =Δt, Δt (17) ΔE for some ΔE = 0. Formula (17) is nothing else but as formula (7) for ΔE Ep. Here HUP is given for the nonrelativistic case. In the relativistic case HUP has the distinctive features [35] which, however, are of no significance for the general formulation of Definition I.,being associated only with particular alterations in the right-hand side 40 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
of the second relation Equation (17).
It is clear that at low energies E EP (momenta P Ppl) Definition I. sets a lower bound for the primarily measurable variation Δxμ of any space-time coordinate xμ. At high energies E (momenta P ) this is not the case if E (P ) has no upper limit. But, according to the modern knowledge, E (P ) is bounded by some maximal quantities Emax,
(Pmax)
E ≤ Emax,P ≤ Pmax, (18) where,in general, Emax,Pmax may be on the order of the Planck quantities Emax ∝
EP ,Pmax ∝ Ppl and also may be the trans-Planck’s quantities.
In any case the quantities Pmax and Emax lead to the introduction of the minimal
length lmin and of the minimal time tmin.
Supposition II. There is the minimal length lmin as a minimal measurement unit for all primarily measurable variations having the dimension of length, whereas the minimal time tmin = lmin/c as a minimal measurement unit for all quantities or primarily measurable variations (increments) having the dimension of time, where c is the speed of light.
lmin and tmin are naturally introduced as Δxμ,μ =1, 2, 3andΔt in Equations (16) and (17) for Δpμ = Pmax and ΔE = Emax.
For definiteness, we consider that Emax and Pmax are the quantities on the order of the Planck quantities, then lmin and tmin are also on the order of the Planck quantities
lmin ∝ lP , tmin ∝ tP . Definition I. and Supposition II. are quite natural in the sense that there are no physical principles with which they are inconsistent. The combination of Definition I. and Supposition II. will be called the Principle of Bounded Primarily Measurable Space-Time Variations (Increments) or for short Principle of Bounded Space-Time Variations (Increments) with abbrevia- tion (PBSTV).
As the minimal unit of measurement lmin is available for all the primarily measurable variations ΔL having the dimensions of length, the “Integrality Condition” (IC) is the case
ΔL = NΔLlmin, (19)
where NΔL > 0 is an integer number. In a like manner, the same “Integrality Condition” (IC) is the case for all the primarily measurable variations Δt having the dimensions of time. And similar to Equation (19), we get the following expression for any time Δt:
Δt ≡ Δt(Nt)=NΔttmin, (20) where similarly NΔt > 0 is an integer number too. Definition 1 (Primary or Elementary Measurability.) (1) In accordance with the PBSTV, let us define the quantity having the dimensions of length or time as primarily (or elementarily) measurable, when it satisfies the Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 41
relation Equation (19) (and respectively Equation (20)). (2)Let us define any physical quantity primarily (or elementarily) measurable, when its value is consistent with points (1) of this Definition.
It is convenient to use the deformation parameter αa. This parameter has been introduced earlier in the papers [36],[10],[37]–[40] as a deformation parameter (in terms of paper [41]) on going from the canonical quantum mechanics to the quantum mechanics at Planck’s scales (Early Universe) that is considered to be the quantum mechanics with the minimal length (QMML): 2 2 αa = lmin/a , (21)
where a is the measuring scale. It is easily seen that the parameter αa from Equation (21) is discrete as it is nothing else but
l2 1 α = l2 /a2 = min = . (22) a min 2 2 2 Na lmin Na
At the same time, from Equation (22) it is evident that αa is irregularly discrete. It should be noted that physical quantities complying with Definition 1 are inadequate for the research of physical systems. Indeed, such a variable as
l2 α (N l )=p(N ) min = l /N , (23) Nalmin a min a min a
where αN l = αa is taken from formula (22 at a = Nalmin,andp(Na)= is a min Nalmin the corresponding primarily measurable momentum), is fully expressed in terms of only Primarily Measurable Quantities of Definition 1 and that’s why it hence may appear at any stage of calculations, but apparently does not complying with Definition 1. Because of this it is necessary to introduce the following definition generalizing Defi- nition 1:
Definition 2. Generalized Measurability We shall call any physical quantity as generalized-measurable or for simplicity mea- surable if any of its values may be obtained in terms of Primarily Measurable Quan- tities of Definition 1.
In what follows, for simplicity, we will use the term Measurability instead of Gen- eralized Measurability. It is evident that any primarily measurable quantity (PMQ) is measurable.Gen- erally speaking, the contrary is not correct as indicated by formula (23). The generalized-measurable quantities follow from the Generalized Uncertainty
Principle (GUP) (formula (3)) that naturally leads to the minimal length lmin [20]– [31]: √ . Δxmin =2 α lp = lmin, (24) 42 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
For convenience, we denote the minimal length lmin =0by and tmin =0by τ = /c. Solving inequality (3), in the case of equality we obtain the apparent formula ± 2 − 2 (Δx (Δx) 4α lp) Δp± = . (25) 2 2α lp
Next, into this formula we substitute the right-hand part of formula (19) for L = x. Considering (24), we can derive the following: 2 (NΔx ± (NΔx) − 1) Δp± = = 1 2 2 2(N ± (N )2 − 1) = Δx Δx . (26)
But it is evident that at low energies E Ep; NΔx 1 the plus sign in the nominator (26) leads to the contradiction as it results in very high (much greater than the Planck) valuesofΔp. Because of this, it is necessary to select the minus sign in the numerator (26). 2 − Then, multiplying the left and right sides of (26) by the same number NΔx + NΔx 1, we get 2 Δp = . (27) 2 − (NΔx + NΔx 1) Δp from formula (27) is the generalized-measurable quantity in the sense of Defini- tion 2. However, it is clear that at low energies E Ep, i.e. for NΔx 1, we have 2 − ≈ NΔx 1 NΔx.Moreover,wehave 2 − lim NΔx 1=NΔx. (28) NΔx→∞
Therefore, in this case (27) may be written as follows:
. Δp =Δp(N ,HUP)= ≈ = ; N 1, (29) Δx 2 − N Δx Δx 1/2(NΔx + NΔx 1) Δx . in complete conformity with HUP. Besides, Δp =Δp(NΔx,HUP), to a high accuracy, is a primarily measurable quantity in the sense of Definition 1.
And vice versa it is obvious that at high energies E ≈ Ep, i.e. for NΔx ≈ 1, there is no way to transform formula (27) and we can write
. Δp =Δp(N ,GUP)= ; N ≈ 1. (30) Δx 2 − Δx 1/2(NΔx + NΔx 1) . At the same time, Δp =Δp(NΔx,GUP)isaGeneralized Measurable quantity in the sense of Definition 2. Thus, we have GUP → HUP (31) Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 43
for
(NΔx ≈ 1) → (NΔx 1). (32) Also, we have
Δp(NΔx,GUP) → Δp(NΔx,HUP), (33)
where Δp(NΔx,GUP) is taken from formula (30), whereas Δp(NΔx,HUP) – from for- mula (29).
Comment 2*. From the above formulae it follows that, within GUP, the primarily measurable vari- ations (quantities) are derived to a high accuracy from the generalized-measurable
variations (quantities) only in the low-energy limit E EP
Next, within the scope of GUP, we can correct a value of the parameter αa from for- 2 − mula (22) substituting a for Δx in the expression 1/2(NΔx + NΔx 1). Then at low energies E Ep we have the primarily measurable quantity αa(HUP) . 1 1 αa = αa(HUP)= ≈ ; Na 1, (34) 2 − 2 2 [1/2(Na + Na 1)] Na that corresponds, to a high accuracy, to the value from formula (22).
Accordingly, at high energies we have E ≈ Ep . 1 αa = αa(GUP )= ; Na ≈ 1. (35) 2 − 2 [1/2(Na + Na 1)]
When going from high energies E ≈ Ep to low energies E Ep,wecanwrite
(Na≈1)→(Na1) αa(GUP ) −→ αa(HUP) (36) in complete conformity to Comment 2*.
2.3 Minimal Inverse Temperature and Measurability
Now, let us return to the thermodynamic relation (11) in the case of equality: 1 k Δ = B + η ΔU, (37) T ΔU that is equivalent to the quadratic equation 1 η (ΔU)2 − Δ ΔU + k =0. (38) T B The discriminant of this equation, with due regard for formula (12), is equal to
1 1 k2 D =(Δ )2 − 4ηk =(Δ )2 − 4α B ≥ 0, (39) B 2 T T Ep 44 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
1 leading directly to (Δ T )min 1 √ kB (Δ )min =2 α (40) T Ep or, due to the fact that kB is constant, we have √ 1 2 α (Δ )min = . (41) kBT Ep
1 It is clear that (Δ T )min corresponds to Tmax from formula (13)
Tmax ≈ Tp 0. (42)
1 ≈ 1 In this case Δ T T and, of course, we can assume that
1 . 1 ( )min = τ = . (43) T Tmax
Trying to find from formula (43) a minimal unit of measurability for the inverse tem- perature and introducing the “Integrality Condition” (IC) in line with the conditions (19),(20) 1 = N τ, (44) T 1/T where N1/T > 0 is an integer number, we can introduce an analog of the primary mea- surability notion into thermodynamics.
Definition 3 (Primary Thermodynamic Measurability) (1) Let us define a quantity having the dimensions of inverse temperature as primarily measurable when it satisfies the relation (44). (2)Let us define any physical quantity in thermodynamics as primarily measurable when its value is consistent with point (1) of this Definition.
Definition 3 in thermodynamics is analogous to the Primary Measurability in a quantum theory (Definition 1). Now we consider the quadratic equation (38) in terms of measurable quantities in the sense of Definition 3. In accordance with this definition and with formula (44) Δ(1/T ), we can write 1 Δ = N τ, (45) T Δ(1/T ) where NΔ(1/T ) > 0 is an integer number. The quadratic equation (38) takes the following form:
2 η (ΔU) − NΔ(1/T )τ ΔU + kB =0. (46) Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 45
Then, due to formula (41), we can find the ”measurable” roots of equation (46) for ΔU as follows: ± 2 − [NΔ(1/T ) NΔ(1/T ) 1]τ (ΔU)meas,± = = 2η ± 2 − 2kB[NΔ(1/T ) NΔ(1/T ) 1]τ = = τ 2 ± 2 − 2kB[NΔ(1/T ) NΔ(1/T ) 1] . (47) τ
2 The last line in (47) is associated with the obvious relation 2η = τ . 2kB In this way we derive a complete analog of the corresponding relation (26) from a quantum theory by replacement
Δp± ⇒ ΔUmeas,±; NΔx ⇒ NΔ(1/T ); ⇒ kB. (48)
As, for low temperatures and energies, T Tmax ∝ Tp,wehave1/T 1/Tp and
hence Δ(1/T ) 1/Tp and NΔ(1/T ) 1. Next, in analogy with Subsection 2.2, in formula (47) we can have only the minus-sign root, otherwise, at sufficiently high NΔ(1/T ) 1for(ΔU)meas,+ we can get (ΔU)meas,+
Ep . But this is impossible for low temperatures (energies). On the contrary, the minus sign in (47) is consistent with high and low energies. So, taking the root value in (47) corresponding to this sign and multiplying the nominator 2 − and denominator in (47) by NΔ(1/T ) + NΔ(1/T ) 1, we obtain
2kB (ΔU)meas = (49) 2 − (NΔ(1/T ) + NΔ(1/T ) 1)τ to have a complete analog of the corresponding relation from (27) in a quantum theory by substitution according to formula (48). Then it is clear that, in analogy with Subsection 2.2, for low energies and temperatures
NΔ(1/T ) 1 (49) may be rewritten as
. 2kB (ΔU)meas =(ΔU)meas(T Tmax)= ≈ 2 − (NΔ(1/T ) + NΔ(1/T ) 1)τ
kB ≈ ,NΔ(1/T ) 1, (50) NΔ(1/T )τ i.e. the Uncertainty Principle in Thermodynamics (UPT, formula (10)) is involved. In this case, due to the last formula, ΔUmeas represents a primarily measurable thermo- dynamic quantity in the sense of Definition 3 to a high accuracy. 46 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
Of course, at high energies the last term in the formula (50) is lacking and, for T ≈
Tmax; NΔ(1/T ) ≈ 1, we have:
. kB (ΔU)meas =(ΔU)meas(T ≈ Tmax)= , 2 − 1/2(NΔ(1/T ) + NΔ(1/T ) 1)τ
NΔ(1/T ) ≈ 1. (51)
From (51) it follows that at high temperatures (energies) (ΔU)meas could hardly be a primarily measurable thermodynamic quantity. Because of this, it is expedient to use a counterpart of Definition 2.
Definition 4. Generalized Measurability in Thermodynamics Any physical quantity in thermodynamics may be referred to as generalized-measurable or, for simplicity, measurable if any of its values may be obtained in terms of the Pri- mary Thermodynamic Measurability of Definition 3.
In this way (ΔU)meas from the formula (51) is a measurable quantity. Based on the preceding formulae, it is clear that we have the limiting transition
(NΔ(1/T )≈1)→(NΔ(1/T )1) (ΔU)meas(T ≈ Tmax) −→ (ΔU)meas(T Tmax ∝ Tp), (52) that is analogous to the corresponding formula (36) in a quantum theory. Therefore, in this case the analog of Comment 2*. in Subsection 2.2 is valid. Comment 2* Thermodynamics From the above formulae it follows that, within GUPT (11), the primarily measur- able variations (quantities) are derived, to a high accuracy, from the generalized- measurable variations (quantities) only in the low-temperature limit T Tmax ∝ Tp. To conclude this Section, it seems logical to make several important remarks.
R2.1 It is obvious that all the calculations associated with measurability of inverse 1 1 . temperature are valid for β = as well. Specifically, introducing βmin = β = τ/k B, T kB T we can rewrite all the corresponding formulae in the ”measurable” variant with appro- priate replacement.
R2.2. Naturally, the problem of compatibility between the measurability definitions in quantum theory and in thermodynamics arises: is there any contradiction between Definition 1 from Subsection 2.2 and Definitions 3 from Subsection 2.3 ? On the basis of the formulae (13) from Subsection 2.1 and (43) from Subsection 2.3 we can state: measurability in quantum theory and thermodynamic measurability are completely compatible and consistent as the minimal unit of inverse temperature τ is nothing else but
the minimal time tmin = τ up to a constant factor. And hence N1/T , (NΔ(1/T )) is nothing else but Nt, (NΔt) in (20). Then it is clear that Nt = Na=tc. Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 47
R2.3 Finally, from the above formulae (50), (51) it follows that the measurable tem- perature T is varying as follows:
Tmax T = ,T Tmax ∝ Tp,N1/T 1; N1/T Tmax T = ,T ≈ Tmax ∝ Tp,N1/T ≈ 1. (53) 2 − 1/2(N1/T + N1/T 1)
In such a way measurable temperature is a discrete quantity but at low energies it is almost constantly varying,so the theoretical calculations are very similar to those of the well-known continuous theory. In the reality, discreteness manifests itself in the case of high energies only.
3 Black Holes and Measurability
Now let us show the applicability of the results from Section 2 to a quantum theory of black holes. Consider the case of Schwarzschild black hole. It seems logical to support the idea suggested in the Introduction to the recent overview presented by seven authors [42]: ”Since for (asymptotically flat Schwarzschild) black holes the temperatures increase as their masses decrease, soon after Hawkings discovery, it became clear that a complete description of the evaporation process would ultimately require a consistent quantum theory of gravity. This is necessary as the semiclassical formulation of the emission process breaks down during the final stages of the evaporation as characterized by Planckian values of the temperature and spacetime curvature”. Naturally, it is important to study the transition from low to high energies in the indicated case.
In this Section consideration is given to gravitational dynamics at low E Ep and at
high E ≈ Ep energies in the case of the Schwarzschild black hole and in a more general case of the space with static spherically-symmetric horizon in space-time in terms of measurable quantities from the previous Section. It should be noted that such spaces and even considerably more general cases have been thoroughly studied from the viewpoint of gravitational thermodynamics in remarkable works of professor T.Padmanbhan [43]–[54] (the list of references may be much longer). First, the author has studied the above-mentioned case in [55] and from the suggested viewpoint in [1]. But, proceeding from Section 2 of the present paper, it is possible to extend the results from [1]. In what follows we use the symbols from [54] which have been also used in [1]. The case of a static spherically-symmetric horizon in space-time is considered, the horizon being described by the metric
ds2 = −f(r)c2dt2 + f −1(r)dr2 + r2dΩ2. (54)
The horizon location will be given by a simple zero of the function f(r), at the radius r = a. 48 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
Then at the horizon r = a Einstein’s field equations ([54], eq.(117)) take the form c4 1 1 f (a)a − =4πPa2 (55) G 2 2
r where P = Tr is the trace of the momentum-energy tensor and radial pressure. There- with, the condition f(a)=0andf (a) = 0 must be fulfilled. On the other hand, it is known that for horizon spaces one can introduce the temperature that can be identified with an analytic continuation to imaginary time. In the case under consideration ([54], eq.(116)) cf (a) k T = . (56) B 4π In [54] it is shown that in the initial (continuous) theory the Einstein Equation for horizon spaces in the differential form may be written as a thermodynamic identity (the first principle of thermodynamics) ([54], formula (119)): cf (a) c3 1 1 c4da 4π d 4πa2 − = Pd a3 , (57) 4 π G 4 2 G 3 − kBT dS dE PdV where, as noted above, T – temperature of the horizon surface, S –corresponding entropy, E– internal energy, V – space volume. It is impossible to use (57) in the formalism under consideration because, as follows from the results given in the previous section and in [1], da, dS, dE, dV are not measurable quantities. First, we assume that a value of the radius r at the point a is a primarily measurable quantity in the sense of Definition 1 from Subsection 2.2., i.e. a = ameas = Na,
where Na > 0 - integer, and the temperature T from the left-hand side of (56) is the measurable temperature T = Tmeas in the sense of Definition 3 from Subsection 2.2.3. Then, in terms of measurable quantities, first we can rewrite (55) as c4 2πk T 1 B a − =4πPa2 . (58) G c meas 2 meas
We express a = ameas in terms of the deformation parameter αa (formula (21)) as
−1/2 a = αa ; (59) the temperature T is expressed in terms of Tmax ∝ Tp from (53).
Then, considering that Tp = Ep/kB, equation (58) may be given as c4 πE 1 √ p 1/2 − 2 [ αa αa]=4πP . (60) G α N1/T c 2 √ Because =2 αl and l = c ,wehave p p Ep c4 2πE 1 c4 2π 1 p 1/2 − 1/2 − 2 [ lpαa αa]= [ αa αa]=4πP . (61) G N1/T c 2 G N1/T 2 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 49
Note that in its initial form [54] the equation (55) has been considered in a continuous
theory, i.e. at low energies E Ep. Consequently, in the present formalism it is im- plicitly meant that the ”measurable counterpart” of equation (55) – (58) (or the same
(60),(61)) is also initially considered at low energies, in particular, Na 1,N1/T 1. Let us consider the possibility of generalizing (60),(61) to high energies taking two dif- ferent cases.
3.1. Measurable case for low energies: E Ep. Due to formula (29), a = ameas = Na,
where the integer number is Na 1 or similarly N1/T 1. In this case GUP, to a high accuracy, is extended to HUP (formula (31),(32)).
As this takes place, αa = αa(HUP)isaprimarily measurable quantity (Definition ≈ −2 1), αa Na , though taking a discrete series of values but varying smoothly, in fact 1/2 ≈ −1 continuously. (60) is a quadratic equation with respect to αa Na having the two −1 parameters N1/T and P . In this terms, the equation (61) may be rewritten as
c4 2π 1 1/2 − 2 [ αa (HUP) αa(HUP)] = 4πP . (62) G N1/T 2
So, at low energies the equation (61) (or (62)) written for the discretely-varying αa may be considered in a continuous theory. As a result, in the case under study we can use the basic formulae from a continuous theory considering them valid to a high accuracy. In particular, in the notation used for Schwarzschild’s black hole [56], we have
2GM N c2 r = N = ; M = a . (63) s a c2 2G As its temperature is given by the formula
c3 TH = , (64) 8πGMkB at once we get 1/2 c cαa TH = = . (65) 4πkBNa 4πkB
Comparing this expression to the expression with high N1/T (N1/T 1) for temperature
from the equation (53) that is involved in (58), we can find that at low energies E Ep,
due to comment R2.2. from Subsection 2.3, the number N1/T is actually coincident with the number Na: −1/2 N1/T = Na = αa (HUP). (66) The substitution of the last expression from formula (65) into the quadratic equation (60) 1/2 for αa makes it a linear equation for αa with a single parameter P .
3.2.Measurable case for high energies:: E ≈ Ep. Then, due to (30), a is the generalized 50 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
2 − ≈ measurable quantity a = ameas =1/2(Na + Na 1), with the integer Na 1. The quantity 2 − − 2 − − Δameas(q)=1/2(Na + Na 1) Na =1/2( Na 1 Na) (67) may be considered as a quantum correction for the measurable radius r = ameas,that is infinitesimal at low energies E Ep and not infinitesimal for high energies E ≈ Ep.
In this case there is no possibility to replace GUP by HUP. In equation (60) αa =
αa(GUP )isageneralized measurable quantity (Definition 2). As noted in formula (53) of Comment R2.3, in this case the number N1/T in equation 2 − (61) is replaced by 1/2(N1/T + N1/T 1), i.e. the equation is of the form
c4 2π 1 1/2 − 2 [ αa (GUP ) αa(GUP )] = 4πP . (68) G 2 − 2 1/2(N1/T + N1/T 1)
In so doing the theory becomes really discrete, and the solutions of (68) take a discrete series of values for every Na or (αa(GUP )) sufficiently close to 1.
In this formalism for a ”quantum” Schwarzschild black hole (i.e. at high energies E ≈ Ep) formula (65) is replaced by
1/2 c cαa (GUP ) TH (Q)= = . (69) 2 − 4πkB1/2(Na + Na 1) 4πkB
We should make several remarks which are important.
Remark 3.3.
As noted in [1], the parameter αa = αa(HUP), within constant factors, is coincident with the Gaussian curvature Ka [57] corresponding to primary measurable a = Na:
2 α = = 2K . (70) a a2 a
Because of this, the transition from αa(HUP)toαa(GUP ) may be considered as a
basis for ”quantum corrections” to the Gaussian curvature Ka in the high-energy region
E ≈ Ep:
2 1 1 αa(GUP ) − αa(HUP)= [ − ]= 2 − 2 2 2 2 1/4(Na + Na 1) Na 2 Q − = (Ka Ka), (71)
Q where the ”measurable quantum Gaussian curvature ” Ka is defined as . 1 KQ = . (72) a 2 − 2 2 1/4(Na + Na 1) Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54 51
In a similar way, with the use of formulae (65) and (64), we can derive a ”measurable quantum correction ” for the mass M of a Schwarzschild black hole at high energies. Remark 3.4.
It is readily seen that a minimal value of Na =1isunattainable because in formula (30) we can obtain a value of the length l that is below the minimum l<for the momenta and energies above the maximal ones, and that is impossible. Thus, we always have
Na ≥ 2. This fact was indicated in [36],[10], however, based on the other approach.
Remark 3.5. It is clear that we have the following transition:
(Na≈1)→(Na1) Eq.(68)(E ≈ Ep) −→ Eq.(62)(E Ep).
Remark 3.6. So, all the members of the gravitational equation (61) (and (68), respec- tively), apart from P , are expressed in terms of the measurable parameter αa. From this
it follows that P should be also expressed in terms of the measurable parameter αa, i.e.
P = P (αa): E Ep, P = P [αa(HUP)] at low energies and E ≈ Ep,P = P [αa(GUP )] at high energies. Then, due to the above formulae, we can have for a ”quantum” Schwarzschild black hole the horizon gravitational equation in terms of measurable quantities c4 (4π − 1) α (GUP )=8πP[α (GUP )]2, (73) G a a 2 − −2 where αa(GUP ) takes a discrete series of the values αa(GUP )=(1/2(Na + Na 1)) ; Na ≥ 2 is a small integer.
4 Conclusion
Taking a simple case as an example, in this paper the author has successfully expressed almost all of the members in the gravitational equation (excepting P )intermsofmea- surable quantities. In the general case the problem at hand is as follows: the formulation of Gravity in terms of measurable quantities and also the derivation of a solution in terms of measurable quantities. Proceeding from the results obtained in [1], [2], such a ”measurable” Gravity – dis-
crete theory that is practically continuous at low energies E Ep and very close to the Einstein theory, though with some principal differences. By authors opinion, in the low- energy ”measurable” variant of Gravity we should have no solutions without physical meaning, specifically Godel’s solution [6].
At high energies E ≈ Ep this ”measurable” Gravity should be really a discrete theory enabling the transition to the low-energy ”measurable” variant of Gravity. Still it is obvious that, to construct a measurable variant of Gravity at all the en- ergy scales, in the general case we need both the primarily measurable variations
Δp(NΔx,HUP) (formula (29)) and the generalized-measurable variations Δp(NΔx,GUP) 52 Electronic Journal of Theoretical Physics 14, No.37 (2018) 35–54
from formula (30). The author believes that such construction should be realized jointly with a construction of a measurable variant for Quantum Theory (QT). Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
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Poisson Bracket and Symplectic Structure of Covariant Canonical Formalism of Fields
Yasuhito Kaminaga∗ School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Received 20 March 2017, Accepted 20 December 2017, Published 20 April 2018
Abstract: The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity are not only manifestly Lorentz covariant but also gauge covariant or diffeomorphism covariant. A mathematical peculiarity of the covariant canonical formalism is that its canonical coordinates are differential forms on a manifold. In the present paper, we find a natural Poisson bracket of this new canonical theory, and study symplectic structure behind it. The phase space of the theory is identified with a ringed space with the structure sheaf of the graded algebra of “differentiable” differential forms on the manifold. The Poisson and the symplectic structure we found can be even or odd, depending on the dimension of the manifold. Our Poisson structure is an example of physical application of Poisson structure defined on the graded algebra of differential forms. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Differential Form; Analytical Mechanics; Gauge Theory; Gravity; Graded Lie Algebra; Graded Manifold; Poisson Bracket; Symplectic Structure PACS (2010): 02.40.-k, 03.50.-z, 04.20.-q, 04.20.Fy, 11.10.Ef, 11.15.-q
∗ Permanent Address: Department of Mathematics, National Institute of Technology, Gunma College, Toriba, Maebashi, Gunma 371–8530, Japan E-mail: [email protected] 56 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72
1 Introduction
In the previous paper [1], the present author developed a new analytical mechanics of fields which treats space and time on an equal footing [2–5]. A similar theory has been studied by Th´eophile De Donder [6], Hermann Weyl [7] and others [8–13], but there is a crucial difference between theirs and ours. That is, they adopt components of tensors (not tensors themselves) as canonical variables while we adopt differential forms themselves (not their components) as canonical variables. This difference causes drastic changes to the resultant theory at least formally. The De Donder-Weyl theory including multisymplectic formalism [6–13] is more similar to the traditional analytical mechanics of fields than ours. For example, in their theory one needs gauge fixing or Dirac bracket or something like that to obtain their canonical equations of gauge theories or those of gravity, as it is the case with the traditional canonical formalism. In our theory, however, one can obtain canonical equations of gauge theories or those of gravity without fixing a gauge nor introducing Dirac bracket nor any other artificial tricks. Our canonical equations are not only manifestly Lorentz covariant but also keep any kind of gauge freedom including diffeomorphism. Our new analytical mechanics of fields is impressive with its simplicity, straightforward nature and mathematical beauty. The purpose of the present paper is to study Poisson structure of our canonical formalism, and to clarify symplectic structure behind it. This paper is organised as follows. In Section 2 we give a brief review of our covariant canonical formalism [1]. In Section 3, we find a natural definition of our Poisson bracket on the basis of physical motivations and heuristic speculations. In Section 4, we collect important formulae of our Poisson bracket. The heuristic part ends here and purely mathematical part begins at Section 5. In Sections 5 and 6, we develop mathematical tools needed for the study of our symplectic structure. In Section 7, we study symplectic structure of our covariant canonical formalism using tools prepared in Sections 5 and 6, and show that it leads to our Poisson bracket found in the heuristic part.
2 Covariant Canonical Formalism of Fields ∞ n r Let M be a C -manifold of dimension n, and let Ω(M)= r=0 Ω (M)be the Z2-graded commutative algebra of differential forms on M. For a map φ : Ωp1 (M) × ··· × Ωpk (M) → Ωr(M), we put β = φ(α1, ··· ,αk) ∈ Ωr(M)with i pi r−pi α ∈ Ω (M). If there exists a differential form ωi ∈ Ω (M) such that β behaves Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 57
1 ··· k i ∧ under variations δα , ,δα as δβ = i δα ωi, then we say the differential form β is differentiable with respect to another differential form αi, and define the derivative of β by αi to be ∂β/∂αi = ωi. In what follows, we assume sum over repeated indices unless otherwise stated. Suppose physical fields are represented with even-forms ϕa ∈ Ω(M) and/or odd-forms ψα ∈ Ω(M). We define a Lagrangian n-form L ∈ Ω(M)sothatthe covariant Lagrange equations ∂L ∂L ∂L ∂L − d =0, + d =0 ∂ϕa ∂dϕa ∂ψα ∂dψα coincide with field equations. Here, we have implicitly assumed, of course, that L is differentiable with respect to ϕa, dϕa, ψα and dψα. Then, we define the conjugate momentum forms pa,πα ∈ Ω(M)as ∂L ∂L p = ,π= , (1) a ∂dϕa α ∂dψα and the Hamiltonian n-form H ∈ Ω(M)as
a α H = dϕ ∧ pa + dψ ∧ πα − L. Suppose that the system is non-singular, that is, (1) can be solved conversely and a α a α dϕ and dψ are represented uniquely with canonical variables ϕ , pa, ψ and
πα. Then, H is differentiable with respect to canonical variables, and we get our covariant canonical equations ∂H ∂H ∂H ∂H dϕa = −(−1)n ,dp= − ,dψα = ,dπ= . (2) a a α α ∂pa ∂ϕ ∂πα ∂ψ Surprisingly, the non-singular assumption is satisfied not only by gauge theories but also by gravity [1]. Indeed, our canonical equations (2) are manifestly Lorentz covariant as well as gauge/diffeomorphism covariant. In the following sections, we will denote by Λr(M) the set of r-forms, on M, which are sufficiently many times differentiable with respect to canonical variables a α n r Z ϕ , pa, ψ and πα.Λ(M)= r=0 Λ (M)formsa 2-graded commutative algebra. Needless to say, it is a subalgebra of Ω(M).
3 Finding Poisson Bracket
Let us find a natural definition of the Poisson bracket of the covariant canonical formalism. To avoid complexity, we first restrict ourselves to n = even case only, 58 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 where n is the dimension of the manifold M.Whenn is even, it follows that ϕa is α even, pa is odd, ψ is odd, πα is even and H is even. Our canonical equations (2) reduce to ∂H ∂H ∂H ∂H dϕa = − ,dp= − ,dψα = ,dπ= . (3) a a α α ∂pa ∂ϕ ∂πα ∂ψ Let F ∈ Λ(M) be an arbitrary, even or odd, differentiable form on the “phase space”. At the moment, we naively consider our phase space to be something a α like a space of which “coordinates” are differential forms ϕ , pa, ψ and πα.Its sophisticated definition will be given in Section 5. We then obtain ∂F ∂F ∂F ∂F dF = dϕa ∧ + dp ∧ + dψα ∧ + dπ ∧ a a α α ∂ϕ ∂pa ∂ψ ∂πα ∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F = − ∧ − ∧ + ∧ + ∧ , a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα where we have used (3). Now, we postulate dF = −{H, F} (4) to define our Poisson bracket as ∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F {H, F} = ∧ + ∧ − ∧ + ∧ . a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα Remembering that H is an even-form, we generalise the definition as ∂E ∂F ∂E ∂F ∂E ∂F ∂E ∂F {E ,F} = 1 ∧ + 1 ∧ − 1 ∧ + 1 ∧ . (5) 1 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα
Here, E1 ∈ Λ(M) denotes an arbitrary differentiable even-form. In what follows, unless otherwise mentioned, E, E1, E2, E3 ∈ Λ(M) are all even-forms and O, O1,
O2, O3 ∈ Λ(M) are all odd-forms. Putting F = E2, O2 in (5), we obtain ∂E ∂E ∂E ∂E ∂E ∂E ∂E ∂E {E ,E } = 1 ∧ 2 + 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 , (6) 1 2 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα ∂E ∂O ∂E ∂O ∂E ∂O ∂E ∂O {E ,O } = 1 ∧ 2 + 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 . (7) 1 2 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα
Next, let us put E1 = O1 ∧ o in (6) and (7), in which o is a constant odd-form, an invariant odd-form under variations of canonical variables. Then, we obtain ∂O ∂E ∂O ∂E ∂O ∂E ∂O ∂E {O ∧ o, E } = 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 − 1 ∧ 2 ∧ o, 1 2 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂π α ∂ψ ∂ψ ∂πα ∂O ∂O ∂O ∂O ∂O ∂O ∂O ∂O {O ∧ o, O } = − 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 − 1 ∧ 2 ∧ o. 1 2 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 59
These results suggest us for putting additional definitions as ∂O ∂E ∂O ∂E ∂O ∂E ∂O ∂E {O ,E } = t 1 ∧ 2 − 1 ∧ 2 + t 1 ∧ 2 − 1 ∧ 2 , 1 2 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα (8) ∂O ∂O ∂O ∂O ∂O ∂O ∂O ∂O {O ,O } = t 1 ∧ 2 − 1 ∧ 2 + t 1 ∧ 2 − 1 ∧ 2 . 1 2 a a α α ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα (9)
Here, t and t are yet unknown real constants, which we assume to be 1 or −1. Now notice that the Poisson bracket we have just defined in (6)(7)(8)(9) is an odd-bracket;thatis,{E1,E2} is odd, {E1,O2} is even, {O1,E2} is even, and {O1,O2} is odd. We can fix t and t postulating “Leibniz rule” in the form of {A, {B,C}} = {{A, B},C}±{B,{A, C}}. Our bracket, for example, satisfies
{E,{E1,O2}} = −t {{E,E1},O2}−{E1, {E,O2}}, {E,{O1,E2}} = −t{{E,O1},E2} + tt {O1, {E,E2}}.
Hence, we get t = t = −1. Then, our bracket has been totally fixed. Now, let us turn our attention to n = odd case. We here outline points only since we can consider it along the same lines in the above. A difference occurs after equation (4) because H is an odd-form when n is odd. That is, when n is odd, the bracket we obtain from (4) is {O1,F}, instead of {E1,F} in (5). Other consideration goes well. After lengthy calculation, we finally get a unique Poisson bracket for n = odd case, too. Again, notice that when n is odd, our bracket becomes an even-bracket;thatis,{E1,E2} is even, {E1,O2} is odd, {O1,E2} is odd, and {O1,O2} is even. We will summarise important formulae of our Poisson bracket in the next section. Before closing this section, note that our Poisson bracket is defined on the space of differential forms. Research on the generalisation of the traditional Poisson bracket to a graded Lie bracket on the space of differential forms has been a long- standing topic of mathematics [14–20]. Our bracket provides a concrete example of such a generalised bracket with physical motivation.
4 List of Poisson Bracket Formulae
In this section, we list, without verbose description, important formulae of our Poisson bracket heuristically found in the previous section. One can prove all 60 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 the formulae listed here by straightforward calculation using definitions, though it is tedious. We will give an alternative elegant proof in Section 7 in this paper. Although we have omitted from the following list, the formula (4) holds, of course, by construction; it is the field equation of a differentiable form F ∈ Λ(M)with our Poisson bracket and the Hamiltonian form H.
4.1 Poisson Bracket in Even Dimensions
Definition ∂E ∂E ∂E ∂E ∂E ∂E ∂E ∂E {E ,E } = 1 ∧ 2 + 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 1 2 a a α α odd ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα ∂E ∂O ∂E ∂O ∂E ∂O ∂E ∂O {E ,O } = 1 ∧ 2 + 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 1 2 a a α α even ∂p a ∂ϕ ∂ϕ ∂pa ∂π α ∂ψ ∂ψ ∂πα ∂O ∂E ∂O ∂E ∂O ∂E ∂O ∂E {O ,E } = − 1 ∧ 2 − 1 ∧ 2 − 1 ∧ 2 − 1 ∧ 2 1 2 a a α α even ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα ∂O ∂O ∂O ∂O ∂O ∂O ∂O ∂O {O ,O } = − 1 ∧ 2 − 1 ∧ 2 − 1 ∧ 2 − 1 ∧ 2 1 2 a a α α odd ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα (10)
Fundamental Brackets
{ a } a { a} − a { α } α { α} − α ϕ ,pb = δb , pb,ϕ = δb , ψ ,πβ = δβ , πβ,ψ = δβ , others = 0 (11)
Symmetry
{E2,E1} = {E1,E2}, {O, E} = −{E,O}, {O2,O1} = −{O1,O2} (12)
Leibniz Rule I
{E,E1 ∧ E2} = {E,E1}∧E2 + E1 ∧{E,E2}
{E,E1 ∧ O2} = {E,E1}∧O2 + E1 ∧{E,O2}
{E,O1 ∧ E2} = {E,O1}∧E2 − O1 ∧{E,E2} {E,O ∧ O } = {E,O }∧O − O ∧{E,O } 1 2 1 2 1 2 (13) {O, E1 ∧ E2} = {O, E1}∧E2 + E1 ∧{O, E2}
{O, E1 ∧ O2} = {O, E1}∧O2 + E1 ∧{O, O2}
{O, O1 ∧ E2} = {O, O1}∧E2 + O1 ∧{O, E2}
{O, O1 ∧ O2} = {O, O1}∧O2 + O1 ∧{O, O2} Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 61
Jacobi identity
{E1, {E2,E3}} + {E2, {E3,E1}} + {E3, {E1,E2}} =0 {E , {E ,O}} − {E , {O, E }} + {O, {E ,E }} =0 1 2 2 1 1 2 (14) {E,{O1,O2}} + {O1, {O2,E}} + {O2, {E,O1}} =0
{O1, {O2,O3}} + {O2, {O3,O1}} + {O3, {O1,O2}} =0 Leibniz Rule II (Another Form of Jacobi Identity)
{E,{E1,E2}} = {{E,E1},E2}−{E1, {E,E2}}
{E,{E1,O2}} = {{E,E1},O2}−{E1, {E,O2}}
{E,{O1,E2}} = {{E,O1},E2} + {O1, {E,E2}} {E,{O ,O }} = {{E,O },O } + {O , {E,O }} 1 2 1 2 1 2 (15) {O, {E1,E2}} = {{O, E1},E2} + {E1, {O, E2}}
{O, {E1,O2}} = {{O, E1},O2} + {E1, {O, O2}}
{O, {O1,E2}} = {{O, O1},E2} + {O1, {O, E2}}
{O, {O1,O2}} = {{O, O1},O2} + {O1, {O, O2}}
4.2 Poisson Bracket in Odd Dimensions
Definition ∂E ∂E ∂E ∂E ∂E ∂E ∂E ∂E {E ,E } = − 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 + 1 ∧ 2 1 2 a a α α even ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα ∂E ∂O ∂E ∂O ∂E ∂O ∂E ∂O {E ,O } = − 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 + 1 ∧ 2 1 2 a a α α odd ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα ∂O ∂E ∂O ∂E ∂O ∂E ∂O ∂E {O ,E } = − 1 ∧ 2 − 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 1 2 a a α α odd ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα ∂O ∂O ∂O ∂O ∂O ∂O ∂O ∂O {O ,O } = − 1 ∧ 2 − 1 ∧ 2 − 1 ∧ 2 + 1 ∧ 2 1 2 a a α α even ∂pa ∂ϕ ∂ϕ ∂pa ∂πα ∂ψ ∂ψ ∂πα (16) Fundamental Brackets { a } a { a} − a { α } − α { α} − α ϕ ,pb = δb , pb,ϕ = δb , ψ ,πβ = δβ , πβ,ψ = δβ , others = 0 (17) Symmetry
{E2,E1} = −{E1,E2}, {O, E} = −{E,O}, {O2,O1} = {O1,O2} (18) 62 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72
Leibniz Rule I
{E,E1 ∧ E2} = {E,E1}∧E2 + E1 ∧{E,E2}
{E,E1 ∧ O2} = {E,E1}∧O2 + E1 ∧{E,O2}
{E,O1 ∧ E2} = {E,O1}∧E2 + O1 ∧{E,E2} {E,O ∧ O } = {E,O }∧O + O ∧{E,O } 1 2 1 2 1 2 (19) {O, E1 ∧ E2} = {O, E1}∧E2 + E1 ∧{O, E2}
{O, E1 ∧ O2} = {O, E1}∧O2 + E1 ∧{O, O2}
{O, O1 ∧ E2} = {O, O1}∧E2 − O1 ∧{O, E2}
{O, O1 ∧ O2} = {O, O1}∧O2 − O1 ∧{O, O2} Jacobi identity
{E1, {E2,E3}} + {E2, {E3,E1}} + {E3, {E1,E2}} =0 {E , {E ,O}} + {E , {O, E }} + {O, {E ,E }} =0 1 2 2 1 1 2 (20) {E,{O1,O2}} + {O1, {O2,E}} − {O2, {E,O1}} =0
{O1, {O2,O3}} + {O2, {O3,O1}} + {O3, {O1,O2}} =0 Leibniz Rule II (Another Form of Jacobi Identity)
{E,{E1,E2}} = {{E,E1},E2} + {E1, {E,E2}}
{E,{E1,O2}} = {{E,E1},O2} + {E1, {E,O2}}
{E,{O1,E2}} = {{E,O1},E2} + {O1, {E,E2}} {E,{O ,O }} = {{E,O },O } + {O , {E,O }} 1 2 1 2 1 2 (21) {O, {E1,E2}} = {{O, E1},E2} + {E1, {O, E2}}
{O, {E1,O2}} = {{O, E1},O2} + {E1, {O, O2}}
{O, {O1,E2}} = {{O, O1},E2}−{O1, {O, E2}}
{O, {O1,O2}} = {{O, O1},O2}−{O1, {O, O2}}
5 Phase Space
Let us change our subject slightly. In the preceding sections, we have heuristically introduced the Poisson bracket. In that context, our bracket seems merely an assembly of ad hoc definitions, but nevertheless it has a profound geometrical meaning as is shown below. In the remaining part of the paper, we turn our attention to mathematical structure behind the Poisson bracket. After preparing necessary tools in the present and the next sections, we will get back to the issue of the Poisson bracket in Section 7. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 63
Let us start our consideration from the definition of phase space. The phase space of our canonical theory, intuitively, is a space of which “coordinates” are forms on an n-dimensional C∞-manifold M. Obviously, we need a sophisticated definition to it to go further. Remember that for any open subset U ⊂ M,there exists a short exact sequence
0 −→ Λ(U) −→ Λ(U) −→ C∞(U) −→ 0,
where Λ (U) is the ideal of nilpotent elements in Λ(U), a Z2-graded commutative algebra of differentiable differential forms on U. Considering this in mind, we identify our phase space with a ringed space (M,OM ) with the structure sheaf
OM : U → Λ(U). Notice that (M,OM ) is a kind of what is called a supermanifold [21–25]. In the following, we think of xi ∈ Λ(U) as any of canonical variables a α i i i r ϕ ,pa,ψ ,πα. We define Z2-graded degree of x as |x | = r (mod2)forx ∈ Λ (U).
In our construction, the set of “functions” on (M,OM ) is identified with Λ(M). A vector field X on (M,OM )isanR-linear derivation of Λ(U)
∂ X = Xi ∧ :Λ(U) → Λ(U) ∂xi with Xi ∈ Λ(U) such that Xf = Xi ∧ ∂f/∂xi for f ∈ Λ(U). The set of all vector fields on (M,OM ) is denoted DerΛ(M), which is a Z2-graded left Λ(M)- i module. |X| implies Z2-graded degree of X. We define the dual basis dx ∈ i HomΛ(U)(DerΛ(U), Λ(U)) of ∂/∂x ∈ DerΛ(U)as ∂ ∂xi ; dxi = = δi . ∂xj ∂xj j
Differential forms on (M,OM ) can be defined in a straightforward manner following generalised techniques of the usual manifold theory so as to accommodate Z2- gradings of the structure sheaf OM . The exterior product ∧ on (M,OM ), which we write in baldface to distinguish from the exterior product ∧ on M, is defined so as to satisfy on (U, OU )
dxi ∧ dxj = dxi ⊗ dxj − (−1)xixj dxj ⊗ dxi, dxi ∧ dxj ∧ dxk = dxi ⊗ dxj ⊗ dxk +(−1)xi(xj +xk)dxj ⊗ dxk ⊗ dxi +(−1)(xi+xj )xk dxk ⊗ dxi ⊗ dxj − (−1)xixj dxj ⊗ dxi ⊗ dxk − (−1)xj xk dxi ⊗ dxk ⊗ dxj − (−1)xj xk+xi(xj +xk)dxk ⊗ dxj ⊗ dxi, 64 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 and so on. Throughout the present paper, we abbreviate the degree symbol ||in the exponent of (−1) for simplicity. As the readers have anticipated, we will use boldfaced d for the exterior derivative on (M,OM ) distinguishing from d on M. O It is important to notice that differential forms on (M, M ), which we denote by O n r O Z ×Z Ω(M, M )= r=0 Ω (M, M ), form a ( 2 2)-bigraded right Λ(M)-module.⎛ We⎞ ⎜r⎟ will say that a differential form λ on (M,OM )hasa(Z2 ×Z2)-bidegree |λ| = ⎝ ⎠ s r (mod 2) if it is an r-form λ ∈ Ω (M,OM ) which satisfies r X1,...,Xr; λ = |Xi| + s (22) k=1 for all vector fields X1⎛, ···⎞, Xr ∈ DerΛ(⎛M).⎞ Arbitrary forms λ1, λ2 ∈ Ω(M,OM )
⎜r1⎟ ⎜r2⎟ with bidegrees |λ1| = ⎝ ⎠ and |λ2| = ⎝ ⎠ satisfy s1 s2
r1r2 s1s2 λ1 ∧ λ2 =(−1) (−1) λ2 ∧ λ1.
r For a differential r-form λ ∈ Ω (M,OM )(r ≥ 1), and a vector field X ∈ DerΛ(M), r−1 we define the interior product of λ and X, iX λ ∈ Ω (M,OM ), by
X1, ··· , Xr−1; iX λ = X1, ··· , Xr−1, X; λ ⎛ ⎞ ⎜r⎟ with X1, ··· , Xr−1 ∈ DerΛ(M). If λ has a bidegree ⎝ ⎠,theniX λ has a bidegree s ⎛ ⎞ ⎜ r − 1 ⎟ ⎝ ⎠ (mod 2). s + |X|
6 Lie Derivatives
The Lie bracket of two vector fields X, Y ∈ DerΛ(M) is the unique vector field defined as
[X, Y ]=XY − (−1)XY YX ∈ DerΛ(M). Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 65
The real vector space structure of DerΛ(M) endowed with the Lie bracket [ , ]on
DerΛ(M)formaZ2-graded Lie algebra. That is, [ , ]isR-bilinear and, for any X, Y , Z ∈ DerΛ(M), there exist symmetry
[Y , X]=−(−1)XY [X, Y ] and Jacobi identity
(−1)XZ[X, [Y , Z]] + (−1)YX[Y , [Z, X]] + (−1)ZY [Z, [X, Y ]] = 0.
For a given vector field X ∈ DerΛ(M), we define a map LX :DerΛ(M) → DerΛ(M), the Lie derivative on DerΛ(M), as
LX Y =[X, Y ](Y ∈ DerΛ(M)). (23)
The following are trivial restatements of the properties just mentioned above. R- bilinearlity
LX (c1Y1 + c2Y2)=c1LX Y1 + c2LX Y2,
Lc1X+c2X2 Y = c1LX1 Y + c2LX2 Y , where c1,c2 ∈ R, symmetry
XY LY X = −(−1) LX Y , and Leibniz rule (Jacobi identity)
XY LX [Y , Z]=[LX Y , Z]+(−1) [Y ,LX Z].
We can naturally generalise the above mentioned Lie derivative LX on DerΛ(M) to obtain another kind of Lie derivative LX :Λ(M) → Λ(M), the Lie derivative on Λ(M), so that
Xf LX (f ∧ Y )=LX f ∧ Y +(−1) f ∧ LX Y (24) for any f ∈ Λ(M)andX, Y ∈ DerΛ(M). The resultant definition of LX : Λ(M) → Λ(M)is
LX f = Xf (f ∈ Λ(M)). (25)
The Lie derivative on Λ(M), just defined, is a derivation on the Z2-graded com- mutative algebra Λ(M). That is, it is R-linear and satisfies Leibniz rule
Xf LX (f ∧ g)=(LX f) ∧ g +(−1) f ∧ (LX g) (26) 66 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 for f,g ∈ Λ(M). The following formula is easily verified.
L[X,Y ]f =[LX ,LY ]f (27)
Similarly, there is a natural extension of the Lie derivatives LX on DerΛ(M) and on Λ(M)toLX :Ω(M,OM ) → Ω(M,OM ), the Lie derivative on Ω(M,OM ). r r That is, for λ ∈ Ω (M,OM ), we define LX λ ∈ Ω (M,OM )sothat
XY1 LX Y1, Y2, ··· , Yr; λ = LX Y1, Y2, ··· , Yr; λ +(−1) Y1,LX Y2, ··· , Yr; λ
X(Y1+Y2+···+Yr−1) + ···+(−1) Y1, Y2, ··· ,LX Yr; λ X(Y1+Y2+···+Yr) +(−1) Y1, Y1, ··· , Yr; LX λ for any Y1, ··· , Yr ∈ DerΛ(M). The resultant definition of LX :Ω(M,OM ) →
Ω(M,OM ) turns out to be
r r LX λ =(−1) (iX d − diX )λ (λ ∈ Ω (M,OM )) (28)
⎛with⎞ r ≥ 1. Note for the unusual sign in⎛ this formula.⎞ If λ has a bidegree ⎜r⎟ ⎜ r ⎟ ⎝ ⎠, its Lie derivative LX λ has a bidegree ⎝ ⎠ (mod 2). Lie derivative s s + |X| on Ω(M,OM ) satisfies XY i (LX Y ) λ = LX iY − (−1) iY LX λ (29) where i(Z)λ implies iZ λ.
7 Symplectic Structure
2 We now define our symplectic form ω ∈ Ω (M,OM )sothat
a α ω = −dϕ ∧ dpa − dψ ∧ dπα (30)
⎛on (U, O⎞U ). ω is a closed, non-degenerate 2-form with a (Z2 × Z2)-bidegree |ω| = ⎜ 2 ⎟ ⎝ ⎠ (mod 2). Note that when n is even, ω becomes what is called an odd n +1 symplectic form, and when n is odd, an even symplectic form [24]. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 67
The symplectic structure ω leads to natural identification between a vector field on (M,OM )anda1-formon(M,OM ). Indeed, if we define a map :DerΛ(M) → 1 Ω (M,OM ), X → iX ω,namely
(X)=iX ω, (31)
1 then the map gives a linear isomorphism between DerΛ(M)andΩ(M,OM ). We denote by the inverse map of .
Definition 1 The Hamiltonian vector field Xf ∈ DerΛ(M) generated by an arbitrary differentiable differential form f ∈ Λ(M) is the unique vector field on
(M,OM ) determined by Xf = df.
Note that the Hamiltonian vector field Xf has a Z2-graded degree |Xf | = |f|+n+1 (mod 2). By definition, it satisfies
df = iXf ω. (32)
a α With canonical coordinates ϕ , pa, ψ , πα ∈ Λ(U), Xf can be written as ∂f ∂ ∂f ∂ X =(−1)(n+1)(f+1) ∧ − ∧ f a a ∂pa ∂ϕ ∂ϕ ∂pa ∂f ∂ ∂f ∂ +(−1)n(f+1) ∧ +(−1)f+n ∧ . (33) α α ∂πα ∂ψ ∂ψ ∂πα
Definition 2 The Poisson bracket of two differentiable differential forms f,g ∈ Λ(M) is the differentiable differential form {f,g} ∈ Λ(M) defined as
{f,g} := − X , X ; ω = − X ; iX ω = −iX iX ω f g f g f g (34) − − − − = iXf dg = Xf ; dg = Xf g = LXf g.
The Poisson bracket {f,g} has a Z2-graded degree |{f,g}| = |f| + |g| + n + 1 (mod 2). The real vector space structure of Λ(M) endowed with the Poisson bracket { , } form a Z2-graded Lie algebra, as it will be obvious from formulae in what follows. The definition {f,g} = −Xf g of the Poisson bracket together with (33) directly yields ∂f ∂g ∂f ∂g {f,g} = −(−1)(n+1)(f+1) ∧ + ∧ ∂p ∂ϕa ∂ϕa ∂p a a (35) ∂f ∂g ∂f ∂g − (−1)n(f+1) ∧ − (−1)f+n ∧ . α α ∂πα ∂ψ ∂ψ ∂πα 68 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72
This result reproduces the heuristic definitions of our Poisson bracket, all of (10) and (16), completely.
Proposition 3 The assignment f → Xf yields a Z2-graded Lie algebra homo- morphism from Λ(M) to DerΛ(M), that is,
X{f,g} = − [Xf , Xg] . (36)
Proof. Noticing − − − LXf ω = iXf d diXf ω = diXf ω = d(df)=0, (37) LX dg = − (iX d − diX ) dg = diX dg = d X; dg = dXg = dLX g, (38) we obtain i (−[X , X ]) ω = −i ([X , X ]) ω = −i LX X ω f g f g f g X X = − LX iX − (−1) f g iX LX ω = −LX iX ω f g g f f g − − { } = LXf dg = dLXf g = d f,g = i X{f,g} ω.
This completes the proof.
Now, it is an easy task to give concise proofs to many formulae listed in Section 4. Symmetry (12)(18):
{g, f} = − Xg, Xf ; ω (f+n+1)(g+n+1) = −(−1) Xf , Xg; ω = −(−1)(f+n+1)(g+n+1){f,g}. (39)
Leibniz rule I (13)(19):
{f,g ∧ h} = −LX (g ∧ h) f − ∧ − (f+n+1)g ∧ − = LXf g h +( 1) g LXf g = {f,g}∧h +(−1)(f+n+1)gg ∧{f,h}. (40)
Leibniz rule II (15)(21):
Xf Xg {{f,g},h} = −X{f,g}h =[Xf , Xg]h = Xf Xgh − (−1) XgXf h
Xf Xg Xf Xg = −Xf {g, h} +(−1) Xg{f,h} = {f,{g, h}} − (−1) {g, {f,h}}, (41) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 69 from which we obtain Leibniz rule II, that is,
{f,{g, h}} = {{f,g},h} +(−1)(f+n+1)(g+n+1){g, {f,h}}. (42)
Putting this formula into a symmetric form, we get
(−1)(f+n+1)(h+n+1){f,{g, h}} +(−1)(g+n+1)(f+n+1){g, {h, f}} +(−1)(h+n+1)(g+n+1){h, {f,g}} =0, (43) which, of course, is Jacobi identity (14)(20).
Acknowledgements
The author would like to thank Prof Jos´e Miguel Figueroa-O’Farrill for useful comments as well as for his warm hospitality at the University of Edinburgh. He is also grateful to the National Institute of Technology for the financial support.
References
[1] Kaminaga Y 2012 Covariant Analytic Mechanics with Differential Forms and Its Application to Gravity Electronic Journal of Theoretical Physics 9 199–216 [2] Nakamura T 2002 Electromagnetism From a Viewpoint of Differential Forms (Original Titile in Japanese) Bussei Kenkyu 79 2–42 [3] Chen C-M, Nester J M and Tung R-S 2015 Gravitational Energy for GR and Poincar´e Gauge Theories: A Covariant Hamiltonian Approach International Journal of Modern Physics D 24 1530026 (73 pages) [4] Nakajima S 2016 Application of Covariant Analytic Mechanics with Differential Forms to Gravity with Dirac Field Electronic Journal of Theoretical Physics 13 95–114 [5] Nakajima S 2016 Reconsideration of De Donder-Weyl Theory by Covariant Analytic Mechanics arXiv:1602.04849 [gr-qc] [6] De Donder Th 1935 Th´eorie Invariantive du Calcul des Variations Nouvelle Edition´ (Paris: Gauthier-Villars) [7] Weyl H 1935 Geodesic Fields in the Calculus of Variations for Multiple Integrals Annals of Mathematics 36 607–629 [8] Kastrup H A 1983 Canonical Theories of Lagrangian Dynamical Systems in Physics Physics Reports 101 1–167 70 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72
[9] Kanatchikov I V 1998 Canonical Structure of Classical Field Theory in the Polymomentum Phase Space Reports on Mathematical Physics 41 49–90 [10] Crnkovi´c C and Witten E 1987 Covariant Descripton of Canonical Formalism in Geometrical Theories, in Hawking S W and Israel W (eds) Three Hundred Years of Gravitation (Cambridge: Cambridge University Press) [11] Forger M and Romero S V 2005 Covariant Poisson Brackets in Geometric Field Theory Communications in Mathematical Physics 256 375–410 [12] Forger M and Salles M O 2015 On Covariant Poisson Brackets in Classical Field Theory Journal of Mathematical Physics 56 102901 [13] Sharapov A A 2014 On Covariant Poisson Brackets in Field Theory arXiv:1412.2902 [hep-th] [14] Abraham R and Marsden J E 1978 Foundations of Mathematics Second Edition (Reading MA: Benjamin/Cummings Publishing) [15] Koszul J-L 1985 Crochet de Schouten-Nijenhuis et cohomologie, in Elie´ Cartan et les math´ematiques d’aujourd’hui - the mathematical heritage of Elie´ Cartan, Proceedings of the seminar held at Lyon, 25–29 juin 1984, Soci´et´e Math´ematique de France, Ast´erisque, num´ero hors s´erie 257–271 [16] Michor P W 1985 A Generalization of Hamiltonian Mechanics Journal of Geometry and Physics 2 67–82 [17] Cantrijin F and Ibort L A 1991 Introduction to Poisson supermanifolds Differential Geometry and its Applications 1 133–152 [18] Beltr´an J V and Monterde J 1995 Graded Poisson Structures on the Algebra of Differential Forms Commentarii Mathematici Helvetici 70 383–402 [19] Grabowski J 1997 Z-graded Extensios of Poisson Brackets Reviews in Mathematical Physics 9 1–27 [20] Monterde J M and Vallejo J A 2002 Poisson Brackets of Even Symplectic Forms on the Algebra of Differential Forms Annals of Global Analysis and Geometry 22 267–289 [21] Kostant B 1977 Graded Manifolds, Graded Lie Theory and Prequantization, in Bleuler K and Reetz A (eds) Differential Geometical Methods in Mathematical Physics Proceedings of the symposium held at the University of Bonn, 1–4 July 1975, Lecture Notes in Mathematics Vol. 570 (Berlin: Springer-Verlag), pp. 177–306 [22] Bartocci C, Bruzzo U and Hern´andez-Ruip´erez D 1991 The Geometry of Supermanifolds, Mathematics and Its Applications Vol. 71 (Dordrecht: Kluwer Academic Publishers) [23] Tuynman G M 2004 Supermanifolds and Supergroups – Basic Theory, Mathematics and Its Applications Vol. 570 (Dordrecht: Kluwer Academic Publishers) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 55–72 71
[24] Rogers A 2007 Supermanifolds: Theory and Applications (Singapore: World Scientific Publishing) [25] Carmeli C, Caston L and Fioresi R 2011 Mathematical Foundations of Supersymmetry, EMS Series of Lectures in Mathematics (Z¨urich: European Mathematical Society)
EJTP 14, No. 36 (2018) 73–78 Electronic Journal of Theoretical Physics
Neutrino Masses and Effective Majorana Mass from a Cobimaximal Neutrino Mixing Matrix
Asan Damanik∗ Department of Physics Education, Sanata Dharma University Kampus III USD Paingan, Maguwoharjo, Sleman, Yogyakarta, Indonesia
Received 14 August 2017, Accepted 25 December 2017, Published 20 April 2018
Abstract: Now, we have confidence that neutrino has a tiny mass and mixing does exist among the neutrino flavors as one can see from the experimental data that have already been reported by many collaborations. Based on the experimental facts, we derive a neutrino mass matrix from cobimaximal neutrino mixing matrix. By constraining the obtained neutrino mass matrix with texture zero, we evaluate its predictions on the neutrino mass relation, the neutrino masses, and the effective Majorana mass. By using the advantages of the experimental data of neutrino oscillations, then we obtain neutrino masses in normal hierarchy: m1 =0.02403 eV, m2 =0.02554 eV, and m3 =0.04957 eV, and the effective Majorana mass: mββ =0.02271 eV that can be tested in the future neutrinoless double beta decay experiments. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Neutrino Masses; Cobimaximal Mixing Matrix; Effective Majorana Mass PACS (2010): 14.60.Lm; 14.60.Pq
1 Introduction
One of the unsolved long standing problems in neutrino physics till today is the explicit form of the neutrino mixing matrix and the absolute values of neutrino masses that can be used to explain the recent experimental data of neutrino oscillations. We have already known three types of neutrino mixing matrix i.e. bimaximal mixing, tribimaximal mixing, and democratic mixing, but now all of them become incompatible anymore when confronted to the recent experimental data, especially about the fact that mixing angle
θ13 = 0 as one can read from the results of the experimental data reported by T2K [1] and Daya Bay [2] collaborations. In order to obtain a mixing matrix which can proceed a consistent predictions with the experimental data, Ma [3] proposed a new
∗ Email: [email protected] 74 Electronic Journal of Theoretical Physics 14, No. 36 (2018) 73–78
mixing matrix which is known as the cobimaximal mixing (CBM) by assuming the mixing
angle θ13 =0, θ23 = π/4, and the Dirac phase δ = ±π/2. In Ref. [3] also claimed that the cobimaximal neutrino mixing matrix is achieve rigorously in a renormalizable model of radiative charged-lepton and neutrino masses.
To explain the evidence of nonzero and relatively large mixing angle θ13,somemodels and methods have already been proposed by several authors. The simple way to accom-
modate nonzero mixing angle θ13 is to modify the neutrino mixing matrix by introducing a perturbation matrix into known mixing matrix such that it can proceeds the nonzero
mixing angle θ13 [4, 5, 6, 7, 8] and the other is to build the model by using some discrete
symmetries [9, 10]. The nonzero mixing angle θ13 is also known relate to the Dirac phase δ as one can see in the standard parameterization of the neutrino mixing matrix. Thus, nonzero mixing angle θ13 gives a clue to the possible determination of CP violation in neutrino sector. Perturbation of neutrino mixing matrix in order to accommodate both nonzero mixing angle θ13 and CP violation have also been reported [11, 12, 13, 14, 15, 16]. In this paper, we evaluate the neutrino mass matrix which is obtained from a co- bimaximal neutrino mixing matrix. The obtained neutrino mass matrix to be used to predict the neutrino masses and an the effective Majorana mass that can be tested in the future neutrinoless double beta decaya experiments. According to the aim of this paper as stated previously, then in section 2 we derive the neutrino mass matrix from a cobimaximal neutrino mixing matrix. In section 3, we evaluate the power prediction of the resulted neutrino mass matrix, which is constrained by texture zero, on the neutrino masses and the effective Majorana mass of neutrinoless double beta decay. Finally, the section 4 is devoted for conclusions.
2 Neutrino mass matrix from a CBM
Theoretically, the neutrino flavor eigenstates (νe,νμ,ντ ) relate to the neutrino mass eigen-
states (ν1,ν2,ν3) via a neutrino mixing matrix V as follow
να = Vαβνβ, (1) where the indexes α = e, μ, τ, β =1, 2, 3,andVαβ are the elements of the neutrino mixing V . The standard parameterization of the mixing matrix (V ) read [17]:
⎛ ⎞ −iδ c12c13 s12c13 s13e ⎜ iδ iδ ⎟ V = ⎝ −s12c23 − c12s23s13e c12c23 − s12s23s13e s23c13 ⎠, (2) iδ iδ s12s23 − c12c23s13e −c12s23 − s12c23s13e c23c13 where cij is the cos θij, sij is the sin θij, θij are the mixing angles, and δ is the Dirac CP-violating phase.
If we put the values for the mixing angle θ23 = π/4 and the Dirac phase δ = π/2, Electronic Journal of Theoretical Physics 14, No. 36 (2018) 73–78 75
then the neutrino mixing matrix of Eq. (2) reads
⎛ ⎞ c c s c is ⎜ √ 12 13 √ 12 13 √ 13 ⎟ ⎝ − 2 − 2 2 ⎠ V = √2 (s12 ic12s13) 2√ (c12 + is12s13) √2 c13 , (3) 2 − 2 − 2 2 (s12 + ic12s13) 2 (c12 is12s13) 2 c13 which is known as a cobimaximal neutrino mixing matrix. In the basis where the charged lepton mass matrix is already diagonalized, the neutrino mass matrix defined by the mass term in the Lagrangian is given by
T Mν = VMV , (4)
where M is neutrino mass matrix in mass basis
⎛ ⎞ m 00 ⎜ 1 ⎟ M = ⎝ 0 m2 0 ⎠. (5)
00m3
By inserting Eqs. (3) and (5) into Eq. (4) we have the following neutrino mass matrix [18]
⎛ ⎞ ab+ iβ −(b − iβ) ⎜ ⎟ Mν = ⎝ b + iβ c − iγ d ⎠, (6) −(b − iβ) dc+ iγ where
2 2 2 2 − 2 a = c12c13m1 + s12c13m2 s13m3, 1 2 2 2 b = −√ s c c (m1 − m2) , 2 12 12 13 1 c = (s2 − c2 s2 )m +(c2 − s2 s2 )m + c2 m , 2 12 12 13 1 12 12 13 2 13 3 1 d = − (s2 + c2 s2 )m +(c2 + s2 s2 )m − c2 m , (7) 2 12 12 13 1 12 12 13 2 13 3 1 2 2 2 2 β = √ s c c m1 + s m2 − m3 , 2 13 13 12 12 γ = −s12c12s13 (m1 − m2) .
It is clear from Eq. (7) if we put b =0orγ = 0, then we have the neutrino mass
matrix with μ − τ symmetry. We cannot put b =0orγ = 0 because it implies m1 = m2 2 which is contrary to the experimental fact that Δm21 > 0. 76 Electronic Journal of Theoretical Physics 14, No. 36 (2018) 73–78
3 Neutrino masses and effective Majorana mass
By knowing the explicit form of the obtained neutrino mas matrix from a cobimaximal mixing matrix as shown in Eq. (6), now we are in position to evaluate the neutrino mass matrix predictions on neutrino masses and the effective Majorana mass. To reduce the number of parameters in neutrino mass matrix, we impose the texture zero into neutrino mass matrix of Eq. (6). By inspecting the neutrino mass matrix in Eq. (6) one can see that the realistic neutrino mass matrix with texture zero is by putting the elements of neutrino mass matrix as follow
Mν(1, 1) = a =0, (8) and
Mν(2, 3) = Mν(3, 2) = d =0, (9)
By imposing the texture zero into neutrino mass matrix of Eq. (6), where the elements of neutrino mass matrix are put to be zero i.e. the elements of neutrino mass matrix as chosen in Eqs. (8) and (9), then the neutrino mass matrix reads
⎛ ⎞ 0 b + iβ −(b − iβ) ⎜ ⎟ Mν = ⎝ b + iβ c − iγ 0 ⎠. (10) −(b − iβ)0 c + iγ
From the neutrino mass matrix of Eq. (10) we can obtain the relations of neutrino masses as function of the mixing angles and it can be used to determine the neutrino masses and its hierarchy. By using the experimental data of neutrino oscillation especially the experimental value of squared mass differences as input. From Eqs. (7), (8), and (9) we can obtain the following relations
2 2 2 2 2 s13m3 = c12c13m1 + s12c13m2, (11) and 2 2 2 2 2 2 2 c13m3 = s12 + c12s13 m1 + c12 + s12s13 m2, (12) which proceed ”neutrino mass sum rule”
m3 = m1 + m2. (13)
It is apparent from Eq. (13) that the neutrino mass hierarchy must be in normal hierarchy. The neutrino mass relation in Eq. (13) is a new result for neutrino mass relation when the obtained neutrino mass matrix from a cobimaximal mixing is constrained by two texture zero as dictated in Eq. (10). Electronic Journal of Theoretical Physics 14, No. 36 (2018) 73–78 77
From Eq. (13) we can have the following relation 2 − 2 2 m3 m2 = m1 +2m2m1, (14) or 2 − 2 m1 +2m2m1 Δm32 =0, (15) 2 2 − 2 where Δm32 = m3 m2 is the squared mass difference of atmospheric neutrino. The global analysis of squared mass difference for atmospheric neutrino and solar neutrino read [19] 2 × −3 2 Δm32 =2.457 10 eV , (16) and 2 × −5 2 Δm21 =7.50 10 eV , (17) respectively. If we insert the value of squared mass difference of Eq. (16) into Eq. (15), then we have − 2 m1 =( m2 +0.001 m2 + 2457) eV. (18) From Eqs. (15), (16), (17), and (18) we can have the neutrino masses as follow
m1 =0.02403 eV,
m2 =0.02554 eV, (19)
m3 =0.04957 eV. By knowing the absolute values of neutrino masses and its hierarchy, we can evaluate the prediction of cobimaximal neutrino mixing matrix on the effective Majorana mass. The effective Majorana mass is a parameter of interest in neutrinoless double beta decay
experiment because the effective Majorana mass mββ is the combination of neutrino mass in eigenstates basis and the neutrino mixing matrix terms as follow [20] 2 mββ = ΣVeimi , (20)
where Vei is the i-th element of the first row of neutrino mixing matrix, and mi is the i-th of the neutrino mass eigenstate. From Eqs. (3) and (20) we obtain the effective Majorana mass as follow 2 2 2 2 − 2 mββ = c12c13m1 + s12c13m2 s13m3 . (21) 0 0 For example, if we take the central values of mixing angle θ13 =9 [2] and θ12 =35 [19] and the neutrino masses as shown in Eq. (19) into Eq. (21), then we have the effective Majorana mass as follow
mββ =0.02271 eV, (22) which can be tested in the future neutrinoless double beta decay experiments. The value of the effective Majorana mass in this paper is still far below the upper bound of the experimental results from isotope Xenon and Germanium experiments [21]
mββ ≤0.3eV, (23) 78 Electronic Journal of Theoretical Physics 14, No. 36 (2018) 73–78
4 Conclusions
We have used a cobimaximal neutrino mixing matrix to obtain a neutrino mass matrix. If
the obtained neutrino mass matrix to be constrained by two texture zero i.e. Mν(1, 1) = 0
and Mν(2, 3) = Mν(3, 2) = 0, then we can obtain the neutrino mass sum-rule: m3 = m1 + m2 which imply that the hierarchy of neutrino mass is in normal nierarchy. By using the advantages of the experimental data of squared mass difference as input, then we obtain
the neutrino masses: m1 =0.02403 eV, m2 =0.02554 eV, and m3 =0.04957 eV. By using the central values of the experimental data for mixing angle θ13 and θ23 and the obtained
neutrino masses, then we obtain the effective Majorana mass: mββ =0.02271 eV which can be tested in the future neutrinoless double beta decay experiments.
References
[1] K. Abe et al. (T2K Collab.), Phys. Rev. Lett. 107 (2011) 041801. [2] F.P.Anet al. (Daya Bay Collab.), Phys. Rev. Lett. 108 (2012) 171803. [3] E. Ma, Phys. Lett. B755 (2016) 348. [4] S. Boudjemaa and S. F. King, Phys. Rev. D79 (2009) 033001. [5] X.-G. He and A. Zee, Phys. Rev. D84 (2011) 053004. [6] A.Damanik,Int.J.Mod.Phys. A27 (2012) 1250091. [7] A. Damanik, arXiv:1206.0987v1 [hep-ph]. [8] W. Rodejohann and H. Zhang, arXiv: 1207.1225v1 [hep-ph]. [9] Q.-H. Cao, A. Damanik, E. Ma, and D. Wegman, Phys. Rev. D83 (2011) 093012. [10] F. Bazzocchi and L. Merlo, arXiv: 1205.5135 [hep-ph]. [11] P. F. Harrison and W. G. Scott, Phys. Lett. B547 (2002) 219. [12] P. F. Harrison and W. G. Scott, Phys. Lett. B594 (2004) 324. [13] W. Grimus and L. Lavoura, Phys. Lett. B579 (2004) 113. [14] R. Friedberg and T. D. Lee, arXiv: hep-ph/0606071. [15] Z. Z. Xing, H. Zhang, and S. Zhou, Phys. Lett. B641 (2006) 189. [16] S. Zhou, arXiv: 1205.0761v1 [hep-ph]. [17] P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett. B530 (2002) 167. [18] X.-G. He, Chinese J. Phys. 53 (2015) 1002021. [19] M. Gonzales-Garcia, M. Maltoni, and T. Schwetz, JHEP 11 (2014) 052. [20] G. Benato, Eur. Phys. J. C75 (2015) 563. [21] H. Pas and W. Rodejohann, New J. Phys. 17 (2015) 115010. EJTP 14, No. 37 (2018) 79–90 Electronic Journal of Theoretical Physics
Relativistic Klein-Gordan Equation with Position Dependent Mass for q-deformed modified Eckart plus Hylleraas potential
S. Sur∗ andS.Debnath Department of Mathematics, Jadavpur University, Kolkata - 700 032, India
Received 10 May 2017, Accepted 25 December 2017, Published 20 April 2018
Abstract: Relativistic Klein-Gordan equation with Position Dependent Mass has been solved analytically for the q-deformed modified Eckart plus Hylleraas potential. A generalised series is used to obtain the bound state solutions of the K-G equation using the Frobenious Method . The one dimensional K-G equation for the mass dependent modified Eckart plus Hylleraas potential in absence of scalar potential are studied in this paper. The exactly normalized bound state wave function and energy expressions are obtained by using N-U method. Also, the bound state solutions are found for the Hulth´en and Rosen-Morse potential. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Klein-Gordan equation; q-deformed modified Eckart plus Hylleraas potential; position dependent mass; N-U method; Frobenious Method PACS (2010): 03.65.Ge; 03.65.Fd
1 Introduction
Quantum Mechanical phenomena are described by Schr¨odinger equation which dictates the dynamics of quantum systems represented by Hamiltonian Operator. Solutions of Klein-Gordan Equation for some physical potential have important applications in Molec- ular Physics, Quantum Chemistry, Nuclear physics, condensed matter Physics, high en- ergy physics. The study of potentials such as Hulth´en [1], Morse [2], Rosen-Morse [3], Pseudo-harmonic [4], Poschl-Teller [5, 6], Kratzer-Fuez [7], generalized Wood Saxon [8], ring-shaped Hartmann [9] and the corresponding wave functions has been performed using various methods. Recently, there has been renewed interest in solving Quantum Mechanical systems within the frame work of Nikiforov-Uvarov method[10-14].This technique is successfully
∗ Email: [email protected] 80 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90 used to solve Schr¨odinger, Klein-Gordan, Dirac and Duffin-Kemmer-Petieu Equqtions. In nuclear physics, the shape form of the potential also plays an important role partic- ularly when studying the structure of deformed nuclei or the interaction between them. Therefore , our aim , in the present work is to investigate analytical bound state so- lutions of the Klein-Gordon equation with q-deformed modified Eckart plus Hylleraas potential[15-19] in the Frobenius method [20] as well as in N-U method. Also, we will show that, when the deformation parameter q takes a particular value (q = 1), the ob- tained results lead to the solutions of the same problem for modified Eckart plus Hylleraas potential. In recent years , the solutions of the non-relativistic wave equation with position- dependent mass have been a topic of great interest[21-25] , but there are only few papers that give the solution of the relativistic wave equation with position-dependent mass in quantum mechanics. Exact solution of the Dirac equation with position-dependent mass in the Coulomb field [26], Kepler problem in Dirac theory for a particle whose potential and mass are inversely proportional to the distance from the force center [27], the approximate solution of the one-dimensional Dirac equations with spatially dependent mass for the generalized Hulthen potential [28], the exact solution of the one-dimensional K-G equation with spatially dependent mass for the inversely linear potential [29] are some papers on relativistic wave equations with position dependent mass. Our focus is to study the quantum systems with Position Dependent Effective Mass (PDEM). PDEM Klein-Gordan Equation plays an important role in the study of elec- tronic properties of semi-conductors in homogeneous crystals,quantum dots,He clusters, quantum liquids etc. Exact solutions of effective mass Klein-Gordan Equations are dif- ficult to obtain, as such, approximate numerical techniques are often used. Our work is generalised as follows:- In section 2 we have discussed the NU method and Frobenious method . We give a brief discussion of Klein-Gordan Equation with position-dependent mass in section 3. In section 4 we discuss the solutions of Klein-Gordan Equation by using both the methods and section 5 is left for conclusion. 2. Overview of Nikiforov-Uvarov and Frobenius Method Method A. Overview of Nikiforov-Uvarov Method The N-U method is based on solving a second order linear differential equation by reducing it to a generalized hypergeometric type. In both relativistic and non-relativistic quantum mechanics, the wave equation with a given potential can be solved by this method by reducing the one dimensional K-G equation to an equation of the form :
τ˜(x) σ˜(x) Ψ(x)+ Ψ(x)+ Ψ(x)=0 (1) σ(x) σ2(x)
Where σ(x)and˜σ(x) are polynomials of degree atmost 2 andτ ˜(x) is a polynomial of degree atmost 1 . In order to find a particular solution to equation(1) , we set the following wave function as a multiple of two independent parts
Ψ(x)=Φ(x)y(x)(2) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90 81
Thus equation (1) reduces to a hyper-geometric type equation of the form :
σ(x)y(x)+τ(x)y(x)+λy(x)=0
Where τ(x)=˜τ(x)+2π(x) satisfies the condition τ (x) < 0andπ(x) is defined as σ(x) − τ˜(x) σ(x) − τ˜(x) π(x)= ± ( )2 − σ˜(x)+Kσ(x)(3) 2 2 in which K is a parameter . Determining K is the essential point in calculation of π(x). Since π(x) has to be a polynomial of degree at most one, the expression under the square root sign in Eq. (3) can be put into order to be the square of a polynomial of first degree [10], which is possible only if its discriminant is zero. So, we obtain K by setting the discriminant of the square root equal to zero . Therefore, one gets a general quadratic equation for K . By using n(n − 1) λ = K + π(x)=−nτ (x) − σ(x)(4) 2 The values of K can be used for the calculation of energy eigenvalues . Polynomial solutions yn(x) are given by the Rodrigues relation B d y (x)= n ( )n[σn(x)ρ(x)] (5) n ρ(x) dx
in which Bn is a normalization constant and ρ(x) is the weight function satisfying 1 τ(x) ρ(x)= exp dx (6) σ(x) σ(x) on the other hand , second part of the wave function φ(x) in relation (2) is given by π(x) φ(x)=exp dx (7) σ(x)
B.Overview of Frobenius Method This method finds the solutions of a differential equation in the form of series,either a whole series,a Laurent series, or even a series involving contribute exhibitors. The differ- ence between these situations is the properties of regularity of the equation coefficients. To do this you must put the equation in the form:
y(x)+P (x)y(x)+Q(x)y(x)=0 (8)
Suppose a regular singular point x0,singular functions P(x) and Q(x) and using the Fuck’s theorem, we can write the solutions of the differential equation in the form: ∞ k+r y(x)= ak(x − x0) (9) 0 82 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90
The indicial equation is obtained for
r(r − 1) + P (0)r + Q(0) = 0 (10)
For each found values r, we determine the values ak and then the solutions of the differ- ential equation.
3. Brief discussion of Klein Gordan Equation with position de- pendent mass
The one dimensional K-G equation for a spinless particle of mass m in the natural units = c = 1 can be expressed
Ψ(x)+[(E − V (x))2 − (m + S(x))2]Ψ(x) = 0 (11)
where E , V(x) and S(x) are the relativistic energy of the particle , vector and scalar potentials respectively. Now considering the q-deformed modified Eckart plus Hylleraas Potential of the form: V a − e−2αx e−2αx e−2αx V (x)= 0 − V + V (12) b 1 − qe−2αx 1 1 − qe−2αx 2 (1 − qe−2αx)2
Where q is the shape parameter.
4.0
3.5
3.0
2.5
2.0
1.5
1.0 4 2 0 2 4 Fig.1.The modified Eckart plus Hylleraas Potential with unit value of α, a, b, q. We prefer to use the mass function equals to the rest mass along with the vector part of the potential as V a − e−2αx e−2αx e−2αx m(x)=m + 0 − V + V (13) 0 b 1 − qe−2αx 1 1 − qe−2αx 2 (1 − qe−2αx)2 to obtain an exactly solvable Schr¨odinger-like equation in absence of scalar potential . The mass function should also be a physical distribution , so we restrict ourself in the Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90 83
range 0 ≤ x<∞ , which gives the finite mass values as follows : ⎧ ⎪ ⎨ V0 − − → → m0 + b (a 1) V1 + V2 (forq 0) ,x 0 m(x)=⎪ ⎩ V0a → ∞ m0 + b ,x
Actually, this distribution corresponds to shifted scalar potential function in the problem. Substituting equation (13) in equation (11) we have V a − e−2αx Ψ(x)+ (E2 − m2) − 2(E + m ) 0 − V 0 0 b 1 − qe−2αx 1 e−2αx e−2αx + V Ψ(x) = 0 (14) 1 − qe−2αx 2 (1 − qe−2αx)2
4A.Application of Nikiforov-Uvarov Method
Introducing a new variable s = e−2αx it is straight forward to show that (14) takes the form: 1 − qs 1 Ψ(s)+ Ψ(s)+ s2q2(2 − γ2 − ζ2) s(1 − qs) s2(1 − qs)2 +2qs(γ2 − 2)+(2 − ω2) Ψ(s) = 0 (15)
2− 2 E m0 2 2 E+m0 { V0 − } 2 Whereweusethenotations 4α2 = ,γ = 4α2q 2V1 +2b (aq +1) 2V2 , ζ = E+m0 { − V0 − } V0 E+m0 2 4α2q (V1 +V2) b (aq 1) and 2 b 4α2 = ω comparing equation (15) with equation (1) we have
τ˜(s)=1− qs; σ(s)=s(1 − qs); σ˜(s)=s2q2(2 − γ2 − ζ2)+2qs(γ2 − 2)+(2 − ω2); (16)
Substituting equation (16) the relation (3) we get qs 1 π(s)=− ± q2s2( + γ2 + ζ2 − 2 − k )+qs(k − 2γ2 +22)+(ω2 − 2) (17) 2 4 1 1
where k1 satisfies the relation k = k1q Further the discriminant of the upper expression under the square root has to be set equal to zero. Therefore, we obtain 1 Δ=q2(k +22 − 2γ2)2 − 4q2( + γ2 + ζ2 − 2 − k )(ω2 − 2) (18) 1 4 1
2 Solving equation (18) for constant k1 , we obtain the double roots as , k1 ,k1 =2(γ − 2 ± 2 2 − 2 2 1 2 2 − 2 ω ) 2ξη ,whereξ = ω and η =(4 + ζ + ω γ ). 84 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90
Thus substituting these values for each k1 into equation (17) , we obtain ⎧ ⎪ ⎨ 2 2 qs (ξ − η)qs − ξ; for k1 =2(γ − ω )+2ξη π(s)=− ± (19) 2 ⎩⎪ 2 2 (ξ + η)qs − ξ; for k1 =2(γ − ω ) − 2ξη
By choosing an appropriate value for k in π(s) which satisfies the condition τ (s) < 0, − 1 2 − 2 − one gets π(s)= qs(ξ + η + 2 )+ξ for k =2(γ ω ) 2ξη ; giving the function:
τ(s)=1− 2qs[1+(ξ + η)] + 2ξ (20)
If we consider λ = k +Π defined in (4) we obtain 1 λ = q[2(γ2 − ω2) − 2ξη − − (ξ + η)] (21) 2 Again using equation (4) , we have:
2 λn = q[n + n +2n(ξ + η)] (22)
Using the condition λ = λn one obtains the eigen values of from the following equation: 8(γ2 − ω2) − (2n +1)2 − 1 − 2η(2n +1) 2 ω2 − 2 = (23) 4(2n +1)+2η
From (6) it can be shown that the weight function ρ(s)isρ(s)=s2ξ(1 − qs)2η and by substituting ρ(s) into the Rodrigues relation (5) one gets
B d B y (s)= n ( )n[sn(1 − qs)ns2ξ(1 − s)2η]= n P (2ξ,2η)(s) (24) n s2ξ(1 − qs)2η ds s2ξ(1 − qs)2η n
(2ξ,2η) where Pn (s) stands for Jacobi polynomial [30] and Bn is the normalizing constant. The other part of the wave function is simply found from (7) as ,
1 φ(s)=sξ(1 − qs)( 2 +η) (25)
Finally , the wave function is obtained as follows
− − 1 ξ − ( η+ 2 ) (2ξ,2η) ψ(s)=Bns (1 qs) Pn (s) (26)
4B.Application of Frobenius Method consider the same Klein-Gordan equation and the same Eckart plus modified Hylleraas Potential given in section 3. After development , we get the following equation:
1 1 V a − s s s ψ(s)+ ψ(s)+ 2 − 2β2 0 − V + V Ψ(s) = 0 (27) s s2 b 1 − qs 1 1 − qs 2 (1 − qs)2 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90 85
2 2 E −m0 2 E+m0 2 where we use the notations 4α2 = and 4α2 = β Comparing (27) with the equation (8) we have, P (s)=1and
2 − 2 − s V0 a−s s Q(s)= 2β V1 1−qs + b 1−qs + V2 (1−qs)2 Putting these values the equation (27) becomes ,
P (s) Q(s) ψ(s)+ ψ(s)+ ψ(s) = 0 (28) s s2 By using Fuck’s theorem , we can write : ∞ k+r ψ(s)= aks ,witha0 = 0 (29) k=0
Differentiation gives us: ∞ ∞ k+r−2 k+r−1 ψ (s)= (k + r − 1)(k + r)aks and ψ (s)= (k + r)aks (30) k=0 k=0
Putting equation (30) in equation (28) one obtains:
∞ V V a sk{[(k + r)2 + 2 − 2 0 aβ2]+s2[q2{(k + r)2 + 2}−2qV β2 − 2q 0 β2] k b 1 b k=0 V +s[−2q(k + r)2 − 2q2 +2V β2 +2 0 (qa +1)β2 − 2V β2]} = 0 (31) 1 b 2 By effecting a change of variable we obtain:
∞ V a [(q2 +1)(r2 + 2) − 2qV β2 − 2 0 β2(a + q)] + sn[a {(q2 +1){(n + r)2 + 2} 0 1 b n n=1 2 V0 2 2 V0 2 2 −2V β q − 2 β (a + q)} + a − {2V β +2 (aq +1)β − 2V β 1 b n 1 1 b 2 −2q{(n + r − 1)2 + 2}}] = 0 (32)
2 2 2 − 2 − V0 2 By solving the indicial equation I = a0[(q +1)(r + ) 2qV1β 2 b β (a + q)] ,we obtain
2 2 2 2 V0 2 (q +1)(r + ) − 2qV1β − 2 β (a + q)=0 b −2(q2 +1)+2qV β2 +2V0 β2(a + q) i.e. r = ± 1 b = ±ν (33) q2 +1
For r = ν we have:
n { − 2 2}− 2 − V0 2 2 2q (i + ν 1) + 2V1β 2 b (aq +1)β +2V2β an = a0 ,n=1, 2, ....(34) (q2 +1){(i + ν)2 + 2}−2V qβ2 − 2 V0 (a + q)β2 i=1 1 b 86 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90
So it gets a representation of the solution
k { − 2 2}− 2 − V0 2 2 2q (i + ν 1) + 2V1β 2 b (aq +1)β +2V2β ak = a0 ,k=1, 2, ...... (35) (q2 +1){(i + ν)2 + 2}−2V qβ2 − 2 V0 (a + q)β2 i=1 1 b
Using the relations (33) and (23) , we obtain the energy eigenvalue associated with the wave function. We can express the solutions obtained based on the Jacobi poly- nomial [31]:this result is more accurate.The coefficients of the solution being assessed explicitly,we seek the bounded solutions.We will only retain the negative value .
5. Discussion In this subsection we consider some special cases of the potential in consideration: (I) Hulthen Potential:
If we set V0 = V2 =0anda =0andb = 1 ,the potential in (12) reduces to
e−2αx V (x)=−V (36) 1 1 − qe−2αx which is the Hulthen potential.
2
1
0
1
4 2 0 2 4 Fig.2.The Hulth´en Potential with unit value of α, q. Furthermore we get the eigen values from the equation 8γ2 − (2n +1)2 − 1 − 2η(2n +1) 2 2 = − (37) 4(2n +1)+2η and the eigen function is
− − 1 ξ − ( η+ 2 ) (2ξ,2η) ψ(s)=Bns (1 qs) Pn (s) (38)
2 2 2 2 1 − 2 2 − 2 where γ =2ζ ,ω =0,η =(4 ζ ),ξ = . Again, applying Frobenius method we obtain
k 2q{(i + ν − 1)2 + 2}−2V β2 a = 1 a ,k=1, 2, .... (39) k (q2 +1){(i + ν)2 + 2}−2V qβ2 0 i=1 1
(II) Rosen-Morse Potential: Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90 87
If we set V1 = V2 =0anda = −1andb = 1 ,the potential in (12) reduces to
1+e−2αx V (x)=−V (40) 0 1 − qe−2αx which is the Rosen-Morse potential.
4
2
0
2
4
4 2 0 2 4 Fig.3.The Rosen-Morse Potential with unit value of α, q. Furthermore we get the eigen values from the equation 8(γ2 − ω2) − (2n +1)2 − 1 − 2η(2n +1) 2 2 = ω2 − (41) 4(2n +1)+2η and the eigen function is
− − 1 ξ − ( η+ 2 ) (2ξ,2η) ψ(s)=Bns (1 qs) Pn (s) (42)
2 V0(aq+1) E+m0 2 −V0(aq−1) E+m0 2 V0 E+m0 2 1 2 2 − 2 where γ =2 b 4α2q ,ζ = b 4α2q , ω =2b 4α2q , η =(4 + ω + ζ γ ) , ξ2 = ω2 − 2 . Again, applying Frobenius method we obtain
k { − 2 2}− V0 2 2q (i + ν 1) + 2 b (aq +1)β ak = a0 ,k=1, 2, .... (43) (q2 +1){(i + ν)2 + 2}−2 V0 (a + q)β2 i=1 b
(III) shape parameter q =1: For N-U method we have the wave function as
− − 1 ξ − ( η+ 2 ) (2ξ,2η) ψ(s)=Bns (1 s) Pn (s) (44) one obtains the eigen values of from the following equation: 8(γ2 − ω2) − (2n +1)2 − 1 − 2η(2n +1) 2 ω2 − 2 = (45) 4(2n +1)+2η
For Frobenius method, we have
k { − 2 2}− 2 − V0 2 2 (i + ν 1) + V1β b (a +1)β + V2β ak = a0 ,k=1, 2, .... (46) {(i + ν)2 + 2}−V β2 − V0 (a +1)β2 i=1 1 b 88 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 79–90
5. Conclusion In this article , the exact solution of the effective mass K-G equation for the modified Eckart plus Hylleraas potential in absence of Lorentz scalar potential. The eigen values and eigen functions are obtained using the Frobenius method as well as Nikiforov-Uvarov method. We gave a schematic graphical representation of the modified Eckart plus Hyller- aas potential with a shape parameter ‘q’ and also the graphical representation of Hulth´en and Rosen-Morse Potential. The eigen values of the potential reduces to that of well known potentials viz., Hulth´en Potential in equation (36) and Rosen-Morse Potential in equatioon (40), when we make appropriate choices of parameter a, b, V0,V1,V2 . Finally we also obtain the wave function which is expressed in terms of the Jacobi Polynomials.
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EJTP 14, No. 37 (2018) 91–114 Electronic Journal of Theoretical Physics
Investigation Fermionic Quantum Walk for Detecting Nonisomorph Cospectral Graphs
M. A. Jafarizadeh∗, F. Eghbalifam† and S. Nami‡ Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran.
Received 4 June 2017, Accepted 20 August 2017, Published 20 April 2018
Abstract: The graph isomorphism (GI) is investigated in some cospectral networks. Two graphs are isomorphic when they are related to each other by a relabeling of the graph vertices. The GI in two scalable (n+2)-regular graphs G4(n; n+2) and G5(n; n+2), is studied analytically by using the multiparticle quantum walk. These two graphs are a pair of non-isomorphic connected cospectral regular graphs for any positive integer n. In order to investigation GI in these two graphs, the adjacency matrices of graphs have been rewritten in the antisymmetric fermionic basis. These fermionic basis are in a form that the adjacency matrices in these basis will be 8×8 for all amounts of n. Then it is shown that the multiparticle quantum walk is able to distinguish pairs of non-isomorph graphs. Rewriting the adjacency matrices of graphs in these basis reduces the complexity of calculations. Also we construct two new graphs T4(n; n +2) and T5(n; n + 2) and repeat the same process of G4 and G5 to study the GI problem by using multiparticle quantum walk. Finally the GI has been discussed in some examples of cospectral graphs. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Fermionic Quantum Walk; Graph Isomorphism Problem; Cospectral Graphs PACS (2010): 98.80.Cq; 98.80. Hw; 04.20.Jb; 04.50+h
1 Introduction
One of the important problems about networks is the graph isomorphism (GI) problem [1]. Two graphs are isomorphic, if one can be transformed into the other by a relabeling of vertices (i.e. , if two graphs with the same number of vertices and edges, can not be transformed into each other by relabeling of vertices, then they will be non-isomorph).
∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] 92 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
Classical algorithm which runs in a time polynomial in the number of vertices of the graphs, is able to distinguish many graph pairs, but some pairs are distinguished com-
putationally√ difficult. Currently, the best general classical algorithm has a run time O(c N log N ), where c is a constant and N is the number of vertices in the two graphs. Typical instances of graph isomorphism (GI) can be solved in polynomial time because two randomly chosen graphs with identical numbers of vertices and edges typically have different degree and eigenvalue distributions. Moreover, GI is solved efficiently for a few classes of graphs, such as trees[2], planar graphs[3], graphs with bounded degree[4], bounded eigenvalue multiplicity[5], and bounded average genus[6]. Researchers have also recently solved GI using various methods by using physical systems. Rudolph mapped the GI problem onto a system of hard-core atoms [7]. Gudkov and Nussinov proposed a physically motivated classical algorithm to distinguish non-isomorphic graphs [8]. One of the useful tools in detecting non-isomorphic graphs is quantum walk (QW). Many researchers are interested in the field of quantum walks (QWs) [9-12] which it can be implemented experimentally on a circles with single photon [13]. This interesting field has many applications in other quantum processes such as quantum search [14], quantum algorithm [15] and measuring network vertex centrality [16]. Quantum random walks (QRW)s [17-19] is the Markov process in which, at every time step, a particle moves to one of the neighboring sites as a result of the random outcome of a coin toss. Two references [20,21] are about open quantum random walks and the reference [22] is the continuous time two particle quantum walk on a one-dimensional noisy lattice. Some researchers used quantum random walks to investigate the capability of quantum walks to distinguish nonisomorphic graphs. Shiau et al. proved that the simplest classical algorithm fails to distinguish some pairs of nonisomorphic graphs and also proved that continuous-time one-particle QRWs cannot distinguish some non-isomorphic graphs [23]. Douglas and Wang modified a single-particle QRW by adding phase inhomogeneities, altering the evolution as the particle walked through the graph [24]. Emms et al. used discrete-time QRWs to build potential graph invariants [25,26]. Berry et al. studied discrete-time quantum walks on the line and on general undirected graphs with two interacting or noninteracting particles [27]. For strongly regular graphs, they showed that noninteracting discrete-time quantum walks can distinguish some but not all non- isomorph graphs with the same family parameters. Gamble et al. extended these results, proving that QRWs of two noninteracting particles will always fail to distinguish pairs of nonisomorphic SRGs with the same family parameters [28]. Then Rudinger et al. numerically demonstrated that three-particle noninteracting walks have distinguishing power on pairs of SRGs [29,30]. In [31] the authors proposed a new algorithm based on a quantum walk search model to distinguish strongly similar graphs. In our previous paper [32] we investigated GI problem in strongly regular (SRG) graphs by using the entanglement entropy. We obtained the adjacency matrix of SRG in the stratification basis, then we calculated the entanglement entropy in non-isomorph SRGs and showed that the entanglement entropy can distinguish the non-isomorph pairs of SRGs. In this paper we use quantum walk to distinguish non-isomorph cospectral graphs. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 93
Cospectral graphs are graphs that share the same graph spectrum. The non-isomorph
cospectral scalable pairs G4(n, n +2)andG5(n, n + 2) are introduced in [33]. We use n-particle quantum walk for GI problem in these graphs. to this aim we rewrite the adjacency matrices of these two graphs in the new basis. The new basis are obtained by fermionization of n-particle standard basis. The adjacency matrices of these graphs in thenewbasisis8× 8. So the dimension of adjacency matrices reduces to 8 × 8from (8n + 12) × (8n + 12). Therefore the complexity of calculations reduces significantly by using this n-particle quantum walk. We give the amplitudes of n-particle quantum walk on these graphs and show that it has the ability to distinguish these non-isomorph pairs.
Also we use the adjacency matrices of G4 and G5 to construct two new graphs which we
call T4(n, n +2)and T5(n, n + 2). These two graphs are Cospectral and non-isomorph for any positive integer n. We use the antisymmetric fermionic basis again and rewrite the adjacency matrices of two new graphs in these basis. By repeating the previous method
for T4 and T5, from the difference between the amplitudes of n-particle qauntum walk, one can conclude that they are non-isomorph . The paper is structured as follows. In Section 2 we give some preliminaries in four subsections. First we explain some interpretation about the graph and the stratification techniques in 2.1. Then in 2.2 we briefly clarify quantum walk. In section 3, first we introduce two non-isomorph graphs G4(n, n +2)andG5(n, n + 2) and prove that they are cospectral. Then in 3.1 we investigate GI in these two graphs by using quantum walk in fermionic basis. The results show that the n-particle quantum walk has power of
distinguishing pairs of non-isomorph graphs G4(n, n +2)and G5(n, n + 2). In 3.1.1 we do the same process for two new non-isomorph graphs T4(n, n+2) and T5(n, n+2). In section 4 we give some examples of non-isomorph cospectral graphs which are distinguished by using single particle quantum walk. We discuss our conclusions in Section 5.
2 Preliminaries
2.1 Graphs and their Stratification techniques
A graph is a pair G =(V,E), where V is a non-empty set and E is a subset of {(i, j); i, j ∈ V,i = j}. Elements of V and of E are called vertices and edges, respectively. Two vertices i, j ∈ V are called adjacent if (i, j) ∈ E, and in that case we write i ∼ j. A finite sequence i0; i1; ...; in ∈ V is called a walk of length n (or of n steps) if ik−1 ∼ ik for all k =1, 2, ..., n. A graph is called connected if any pair of distinct vertices is connected by a walk. The degree or valency of a vertex x ∈ V is defined by κ(x)=|y ∈ V : y ∼ x|. The graph structure is fully represented by the adjacency matrix A defined by ⎧ ⎪ ⎨ 1 i ∼ j (A)ij = ⎪ (1) ⎩ 0 otherwise 94 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
Obviously, (i) A is symmetric; (ii) an element of A takes a value in 0, 1; (iii) a diagonal
element of A vanishes. Let l2(V ) denote the Hilbert space of square-summable functions on V ,and|i ; i ∈ V becomes a complete orthonormal basis of l2(V ). The adjacency
matrix is considered as an operator acting in l2(V )insuchawaythat A|i = |j i ∈ V. (2) j∼i
For i = j let ∂(i, j) be the length of the shortest walk connecting i and j. By definition ∂(i, i) = 0 for all i ∈ V . The graph becomes a metric space with the distance function ∂. Note that ∂(i, j) = 1 if and only if i ∼ j.Wefixapointo ∈ V as an origin of the graph. Then, a natural stratification for the graph is introduced as: ∞ V = Vi(o) Vi(o):={j ∈ V : ∂(o, j)=i} (3) i=0
If Vk(o)=∅ happens for some k ≥ 1, then Vl(o)=∅ for all l ≥ k. With each stratum Vi,
we associate a unit vector in l2(V ) defined by 1 |φi = √ |k (4) κi k∈Vi(o) where, κi := |Vi(o)| and |k denotes the eigenket of k-th vertex at the stratum i.The
closed subspace of l2(V ) spanned by |φi is denoted by Γ(G). Since |φi becomes a complete orthonormal basis of Γ(G), we often write Γ(G)= ⊕C|φk (5) k
In this stratification for any connected graph G,wehave V1(β) ⊆ Vi−1(α) Vi(α) Vi+1(α)(6)
for each β ∈ Vi(α). Now, recall that the i-th adjacency matrix of a graph G =(V,E)is defined as ⎧ ⎪ ⎨ 1 if ∂(α, β)=i (Ai)α,β = ⎪ (7) ⎩ 0 otherwise
Then, for reference state |φ0 (|φ0 = |o ), with o ∈ V as reference vertex), we have Ai|φ0 = |β . (8)
β∈Vi(o)
Then by using (4) and (10), we have √ Ai|φ0 = κi|φi . (9) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 95
Then, for reference state |φ0 (|φ0 = |o ), with o ∈ V as reference vertex), we have Ai|φ0 = |β . (10)
β∈Vi(o)
Then by using (4) and (10), we have √ Ai|φ0 = κi|φi . (11)
For more details you can see [34-36].
2.2 Continuous time quantum walk
The continuous-time quantum walk is defined by replacing Kolmogorovs equation with
Schrodingers equation. Let |φi(t) be a time-dependent amplitude of the quantum process on graph Γ. The wave evolution of the quantum walk is
d i |φ(t) = H|φ(t) (12) dt
where we assume =1and|φ0 is the initial amplitude wave function of the particle. The solution is given by −iHt |φ0(t) = e |φ0 (13) Where elements of amplitudes between strata are calculated
−iHt φi(t)|φ0(t) = φi(t)|e |φ0 (14)
Obviously the above result indicates that the amplitudes of observing walk on vertices belonging to a given stratum are the same. Actually one can straightforwardly the transition probabilities between the vertices depend only on the distance between the vertices irrespective of which site the walk has started. So, if stratification of two non- isomorphism graph is different, the quantum walk on these graphs are different.
3 Investigation of graph isomorphism (GI) problem in G4(n, n + 2) and G5(n, n +2)
In this section, the graphs G4(n, n+2)and G5(n, n+2)with8n+12 vertices are defined.
The (n+2)-regular graphs G4(n, n+2) and G5(n, n+2) are a pair of connected cospectral integral regular graphs for any positive integer n. We prove that these two graphs are
non isomorphic by using the fermionic quantum walk. The adjacency of G4(n, n +2)are defined as ⎛ ⎞
⎜ A0 A1 ⎟ A(G4(n, n +2))=⎝ ⎠ (15) A1 A0 96 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
where ⎛ ⎞ ⎜ 0 Jn×(n+2) 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J(n+2)×n 0 I(n+2) 0 ⎟ ⎜ ⎟ A0(G4)=⎜ ⎟ (16) ⎜ 0 I 0 B ⎟ ⎝ (n+2) (n+2) ⎠
00B(n+2) 0 and ⎛ ⎞ ⎜ 0000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 I(n+2) 00⎟ ⎜ ⎟ A1(G4)=⎜ ⎟ (17) ⎜ 0000⎟ ⎝ ⎠
000I(n+2)
and ⎛ ⎞
⎜ 1 J1,n 0 ⎟ ⎜ ⎟ ⎜ ⎟ B = ⎜ J J − I J ⎟ (18) ⎝ n,1 n n n,1 ⎠
0 J1,n 1
After some relabeling, the total adjacency matrix for G4(n, n +2)is ⎛ ⎞
⎜ 00Jn×(n+2) 00000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00I B 0000⎟ ⎜ (n+2) (n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J(n+2)×n I(n+2) 00I(n+2) 000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 B(n+2) 000I(n+2) 00⎟ ⎜ ⎟ A(G4(a, b)) = ⎜ ⎟ ⎜ 00I 000J × I ⎟ ⎜ (n+2) (n+2) n (n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000I(n+2) 000B(n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000J × 000⎟ ⎝ n (n+2) ⎠
0000I(n+2) B(n+2) 00 (19)
The adjacency matrix for G5(n, n +2)is ⎛ ⎞
⎜ A0 A1 ⎟ A(G5(a, b)) = ⎝ ⎠ (20) A1 A0 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 97
where A0 and A1 for G5(n, n +2)are ⎛ ⎞ ⎜ 0 Jn×(n+2) 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J(n+2)×n 0 I(n+2) I(n+2) ⎟ ⎜ ⎟ A0(G5)=⎜ ⎟ (21) ⎜ 0 I 00⎟ ⎝ (n+2) ⎠
0 I(n+2) 00 and ⎛ ⎞ ⎜ 00 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00 0 0 ⎟ ⎜ ⎟ A1(G5)=⎜ ⎟ (22) ⎜ 00B 0 ⎟ ⎝ (n+2) ⎠
00 0 B(n+2)
and the matrix B isthesameastheG4(n, n + 2). After some relabeling, the adjacency matrix of G5(n, n +2)is ⎛ ⎞
⎜ 0 Jn×(n+2) 0000 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J × 0 I I 00 0 0⎟ ⎜ (n+2) n (n+2) (n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 I(n+2) 00B(n+2) 00 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 I(n+2) 000B(n+2) 00⎟ ⎜ ⎟ A(G5(a, b)) = ⎜ ⎟ ⎜ 00B 000I 0 ⎟ ⎜ (n+2) (n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000B(n+2) 00I(n+2) 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000I I 0 J × ⎟ ⎝ (n+2) (n+2) (n+2) n ⎠
0 0 0000Jn×(n+2) 0 (23)
Now we want to show that two graphs G4(n, n +2)andG5(n, n + 2) are Cospectral. The adjacency matrices of these graphs can be written as
A = I2 ⊗ A0 + σx ⊗ A1 So the eigenvalues of adjacency matrices of these two graphs will be the eigenvalues of
two matrices A0 ± A1. ⎛ ⎞ ⎜ 0 Jn×(n+2) 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J(n+2)×n ±I(n+2) I(n+2) 0 ⎟ ± ⎜ ⎟ (A0 A1)(G4)=⎜ ⎟ (24) ⎜ 0 I 0 B ⎟ ⎝ (n+2) (n+2) ⎠
00B(n+2) ±I(n+2) 98 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
'$ '$ '$ '$
' s1 ab20 s $ ' s1 20 s $ J &% &%J J &% &%J # I # # # §s2 11 s¤ '§s2 11 s¤$ s3 12 s ' s3 12 s $ & s s % I'& s s %$I "4 ! "13 ! "4 ! "13 ! I I # # # B # '¦§s s¥¤ $ ¦Xs s¥ 5 14 5XX 14 XX XXX ''s s $$ Xs Xs 6 15 6XXX 15 XX ' s s $II s XXXs "7 ! "16 ! "7 ! "16 ! B B # I # # B # &¦s s¥ % &Xs s % 8 17 8XX 17 XXX XXX &s s % &Xs X s % 9 18 9 XX 18 XX &&s s %% &s XXXs % "10 ! "19 ! "10 ! "19 !
Figure 1 An example of G4(n, n +2)in(a)andG5(n, n +2)in(b)withn =1.
We want to diagonalize the blocks of above matrix. So we can apply following transfor- mation
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ T ⎜ O1 000⎟ ⎜ 0 Jn×(n+2) 00⎟ ⎜ O1 000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ T ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 O 00⎟ ⎜ J(n+2)×n ±I(n+2) I(n+2) 0 ⎟ ⎜ 0 O2 00⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (25) ⎜ 00OT 0 ⎟ ⎜ 0 I 0 B ⎟ ⎜ 00O 0 ⎟ ⎝ 3 ⎠ ⎝ (n+2) (n+2) ⎠ ⎝ 3 ⎠ T ± 000O4 00B(n+2) I(n+2) 000O4
⎛ ⎞ T ⎜ 0 O1 Jn×(n+2)O2 00⎟ ⎜ ⎟ ⎜ T T T ⎟ ⎜ O J(n+2)×nO1 ±O O2 O O3 0 ⎟ ⎜ 2 2 2 ⎟ = ⎜ ⎟ ⎜ 0 OT O 0 OT B O ⎟ ⎝ 3 2 3 (n+2) 4 ⎠ T ± T 00O4 B(n+2)O3 O4 O4 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 99
Then by choosing O2 = O3 = O4, the transformed matrix will be
⎛ ⎞ ⎜ 0 SV D(Jn×(n+2))0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ SV D(J(n+2)×n) ±I(n+2) I(n+2) 0 ⎟ ⎜ ⎟ ⎜ ⎟ (26) ⎜ 0 I 0 D ⎟ ⎝ (n+2) B ⎠
00DB ±I(n+2)
Therefore the eigenvalues of G4 will be the eigenvalues of these matrices:
⎛ ⎞ ⎛ ⎞ ⎜ 0 n(n +2) 0 0 ⎟ ⎜ ⎟ ±11 0 ⎜ ⎟ ⎜ ⎟ ⎜ n(n +2) ±110⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ , ⎜ 101⎟ (27) ⎜ 010n +1⎟ ⎝ ⎠ ⎝ ⎠ 01±1 00n +1 ±1
The eigenvalues will be
± ± ± ± ± (n +2), (n +1) , n, 2 , 1 2times (n+1)times 2(n+1)times
The same process can be applied to graph G5, So the eigenvalues of adjacency matrix of graph G5 will be the eigenvalues of these matrices:
⎛ ⎞ ⎛ ⎞ ⎜ 0 n(n +2) 0 0 ⎟ ⎜ ⎟ 01 1 ⎜ ⎟ ⎜ ⎟ ⎜ n(n +2) 0 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ , ⎜ 1 ±10⎟ (28) ⎜ 01±(n +1) 0 ⎟ ⎝ ⎠ ⎝ ⎠ 10±1 010±(n +1)
The eigenvalues will be
± ± ± ± ± (n +2), (n +1) , n, 2 , 1 2times (n+1)times 2(n+1)times
So these two graphs for all n are cospectral. 100 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
3.1 Investigation of GI problem via quantum walk in the antisymmetric fermionic basis
Now we want to use quantum walk for investigating graph isomorphism problem in these
two graphs. The total adjacency matrix for G4(n, (n + 2)) can be written as ⎛ ⎞
⎜ 0 Jn×(n+2) 000000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J × 0 I 0 I 00 0⎟ ⎜ (n+2) n (n+2) (n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 I(n+2) 0 B(n+2) 0000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00B(n+2) 000I(n+2) 0 ⎟ ⎜ ⎟ A(G4)=⎜ ⎟ (29) ⎜ 0 I 00 0I 0 J × ⎟ ⎜ (n+2) (n+2) (n+2) n ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000I(n+2) 0 B(n+2) 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000I 0 B 00⎟ ⎝ (n+2) (n+2) ⎠
0000Jn×(n+2) 00 0
And the adjacency matrix of G5(n, n + 2) can be written as
⎛ ⎞
⎜ 0 Jn×(n+2) 00 0 00 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ J × 0 I I 0000⎟ ⎜ (n+2) n (n+2) (n+2) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 I(n+2) 00 0Bb 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 I(n+2) 00 0 0B(n+2) 0 ⎟ ⎜ ⎟ A(G5)=⎜ ⎟ (30) ⎜ 00000I I J × ⎟ ⎜ (n+2) (n+2) (n+2) n ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00B(n+2) 0 I(n+2) 00 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000B I 00 0⎟ ⎝ (n+2) b ⎠
0000Jn×(n+2) 00 0
We want to rewrite the adjacency matrices of these two graphs in the new basis.
The strata of G4(n, n+2) and G5(n, n+2) are obtained by fermionization as following form | √1 | ⊗| ⊗ ⊗| φ0 = εi1,i2,...,i i1 i2 ... in n! n i1,i2,...,in n+2 | √ 1 | ⊗| ⊗ ⊗| | − ⊗| ⊗| φl = εi1,i2,...,i i1 i2 ... ik−1 ( n+(l 1)(n+2)+j ) ik+1 ... in n! n(n +2) n i1,i2,...,in j=1 (31) (l =1, ..., 6) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 101
| √1 | ⊗| ⊗ ⊗| φ7 = εi1,i2,...,i 1,2,...,n n+6(n+2)+i1 n+6(n+2)+i2 ... n+6(n+2)+in n! n i1,i2,...,in ⎛ ⎞ (32) ⎜ N ⎟ The dimension of this fermionic space is ⎝ ⎠ But we choose the above antisymmetric n
n-particle fermionic basis for graphs G4(n, n +2)and G5(n, n + 2). We want to apply the following adjacency matrices of two graphs on the defined basis. ⊗ ⊗ ⊗ ⊗ ⊗ A = I I ... A1 I... I (33) i i where I is identity matrix. Now, by applying adjacency matrices of G4(n, n +2)and
G5(n, n + 2) on the new basis, we have | | AG4(n,n+2) φ0 = n(n +2)φ1 | | | | AG4(n,n+2) φ1 = n(n +2)φ0 + φ2 + φ4 | | | AG4(n,n+2) φ2 = φ1 +(n +1)φ3 | | | AG4(n,n+2) φ3 =(n +1)φ2 + φ6 | | AG4(n,n+2) φ4 = n(n +2)φ7 | | | AG4(n,n+2) φ5 =(n +1)φ6 + φ4 | | | AG4(n,n+2) φ6 = φ3 +(n +1)φ5 | | AG4(n,n+2) φ7 = n(n +2)φ4 (34) So, the adjacency matrix in the stratification basis is ⎛ ⎞ ⎜ 0 n(n +2) 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ n(n +2) 0 1 0 1 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010n +1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00n +1 0 0 0 1 0 ⎟ A = ⎜ ⎟ G4(n,n+2) ⎜ ⎟ ⎜ 0100010n(n +2)⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000010n +1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00010n +1 0 0 ⎟ ⎝ ⎠ 0000n(n +2) 0 0 0 (35) 102 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
−iHt −iAG t The amplitudes of quantum walk (i.e. φi|e |φ0 = φi|e 4 |φ0 )forG4 are
− iAG4 t φ0|e |φ0 =
n2 +2n n2 n2 +2n cos (n +1)t + cos (n +2)t + cos nt 2(n +1)2 2(2n2 +3n) 4n2 +2n − iAG4 t φ1|e |φ0 = √ √ √ n2 +2n n n2 +2n n n2 +2n −i sin (n +1)t − i sin (n +2)t − i sin nt 2(n +1) 2(2n2 +3n) 4n2 +2n − iAG4 t φ2|e |φ0 = √ √ √ n2 +2n n n2 +2n n n2 +2n cos (n +1)t + cos (n +2)t + cos nt 2(n +1)2 2(2n2 +3n) 4n2 +2n √ √ 2 2 − n n +2n n n +2n φ |e iAG4 t|φ = −i sin (n +2)t + i sin nt 3 0 2(2n2 +3n) 4n2 +2n √ √ 2 2 − n n +2n n n +2n φ |e iAG4 t|φ = cos (n +2)t − cos nt 4 0 2(2n2 +3n) 4n2 +2n − iAG4 t φ5|e |φ0 = √ √ √ n2 +2n n n2 +2n n n2 +2n i sin (n +1)t − i sin (n +2)t − i sin nt 2(n +1)2 2(2n2 +3n) 4n2 +2n − iAG4 t φ6|e |φ0 = √ √ √ n2 +2n n n2 +2n n n2 +2n − cos (n +1)t + cos (n +2)t + cos nt 2(n +1)2 2(2n2 +3n) 4n2 +2n
2 2 − n n +2n φ |e iAG4 t|φ = −i sin (n +2)t + i sin nt 7 0 2(2n2 +3n) 4n2 +2n
The effect of adjacency matrix of G5 on the stratification basis are | | AG5(n,n+2) φ0 = n(n +2)φ1 | | | | AG5(n,n+2) φ1 = n(n +2)φ0 + φ2 + φ3 | | | AG5(n,n+2) φ2 = φ1 +(n +1)φ5 | | | AG5(n,n+2) φ3 = φ1 +(n +1)φ6 | | | | AG5(n,n+2) φ4 = φ5 + φ6 + n(n +2)φ7 | | | AG5(n,n+2) φ5 =(n +1)φ2 + φ4 | | | AG5(n,n+2) φ6 =(n +1)φ3 + φ4 | | AG5(n,n+2) φ7 = n(n +2)φ4 (36) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 103
So, the adjacency matrix in the stratification basis is ⎛ ⎞ ⎜ 0 n(n +2) 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ n(n +2) 0 1 1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 01000n +1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010000n +1 0 ⎟ A = ⎜ ⎟ G5(n,n+2) ⎜ ⎟ ⎜ 0000011n(n +2)⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00n +1 0 1 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000n +1 1 0 0 0 ⎟ ⎝ ⎠ 0000n(n +2) 0 0 0 (37)
The difference between amplitudes of quantum walk for two non-isomorph graphs G4(n, n+
2) and G5(n, n +2)are
− − iAG4 t iAG5 t φ0|e |φ0 − φ0|e |φ0 =0 − − iAG4 t iAG5 t φ1|e |φ0 − φ1|e |φ0 =0 − − iAG4 t iAG5 t φ2|e |φ0 − φ2|e |φ0 =0 − − iAG4 t iAG5 t φ3|e |φ0 − φ3|e |φ0 = enit e−(n+1)it e(n+2)it n(n +2)( − − ) 2(2n +1) (2n + 1)(2n +3) 2(2n +3) − − iAG4 t iAG5 t φ4|e |φ0 − φ4|e |φ0 = e−nit enit e−(n+1)it e(n+1)it n(n +2)(− − + + ) 2(2n +1) 2(2n +1) 2(2n +1) 2(2n +1) − − iAG4 t iAG5 t φ5|e |φ0 − φ5|e |φ0 = e−nit enit e−(n+1)it e(n+1)it n(n +2)( − − + ) 2(2n +1) 2(2n +1) 2(2n +1) 2(2n +1) − − iAG4 t iAG5 t φ6|e |φ0 − φ6|e |φ0 = e−nit e(n+1)it e−(n+1)it e(n+2)it n(n +2)( − − + ) 2(2n +1) 2(2n +3) 2(2n +1) 2(2n +1) − − iAG4 t iAG5 t φ7|e |φ0 − φ7|e |φ0 = e−nit enit e−(n+1)it e(n+1)it (− + + − ) (n +1) 2(2n +1) (2n +1) (2n +1) So from the difference between the amplitudes of n-particle quantum walk, we con- clude that the multiparticle quantum walk can distinguish two non-isomorph graphs. The complexity of calculations is reduced by fermionization of standard basis, and rewriting 104 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
the adjacency matrices of graphs in these basis. By this method, the process of finding the amplitudes of quantum walk is done, by using the 8 × 8-dimensional adjacency ma- trix for all amount of n. But if we didn’t use the new basis, then we had to work with (8n + 12) × (8n + 12)-dimensional adjacency matrices.
3.2 Investigation of GI problem via quantum walk in T4(n, n +2)and T5(n, n +2)
We can construct two nonisomorph graphs similar to G4(n, n +2)and G5(n, n +2) by replacing the A0 and A1 in adjacency matrices. The new graphs T4(n, n +2)and T (n, n + 2) are cospectral and non-isomorph. 5 ⎛ ⎞
⎜ A1 A0 ⎟ A = ⎝ ⎠ (38) A0 A1
Where A0,A1 are the same as (16), (17) for T4 and (21), (22) for T5.Weusetheanti- symmetric fermionic basis of (32) .Then, by applying adjacency matrix of T4(n, n +2) and T (n, n + 2) on these basis, we have 5 A |φ = n(n +2)|φ T4(n,n+2) 0 4 | | | | AT4(n,n+2) φ1 = n(n +2)φ7 + φ1 + φ5 | | | AT4(n,n+2) φ2 = φ4 +(n +1)φ6 A |φ =(n +1)|φ + |φ T4(n,n+2) 3 5 3 | | | | AT4(n,n+2) φ4 = n(n +2)φ0 + φ2 + φ4 | | | AT4(n,n+2) φ5 =(n +1)φ3 + φ1 A |φ = |φ +(n +1)|φ T4(n,n+2) 6 6 2 | | AT4(n,n+2) φ7 = n(n +2)φ1 (39) So, the adjacency matrix of T (n, n + 2) in the antisymmetric fermionic basis is ⎛ 4 ⎞ ⎜ 0000n(n +2) 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0100010n(n +2)⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000010n +1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00010n +1 0 0 ⎟ A = ⎜ ⎟ T4(n,n+2) ⎜ ⎟ ⎜ n(n +2) 0 1 0 1 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010n +1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00n +1 0 0 0 1 0 ⎟ ⎝ ⎠ 0 n(n +2) 0 0 0 0 0 0 (40) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 105
And | | AT5(n,n+2) φ0 = n(n +2)φ4 | | | | AT5(n,n+2) φ1 = n(n +2)φ7 + φ5 + φ6
| | | AT5(n,n+2) φ2 = φ4 +(n +1)φ2
| | | AT5(n,n+2) φ3 = φ4 +(n +1)φ3 | | | | AT5(n,n+2) φ4 = φ2 + φ3 + n(n +2)φ0
| | | AT5(n,n+2) φ5 =(n +1)φ5 + φ1
| | | AT5(n,n+2) φ6 =(n +1)φ6 + φ1 | | AT5(n,n+2) φ7 = n(n +2)φ1 (41)
So, the adjacency matrix of T5(n, n + 2) in the antisymmetric fermionic basis is ⎛ ⎞ ⎜ 0000n(n +2) 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000011n(n +2)⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00n +1 0 1 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000n +1 1 0 0 0 ⎟ A = ⎜ ⎟ T5(n,n+2) ⎜ ⎟ ⎜ n(n +2) 0 1 1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 01000n +1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010000n +1 0 ⎟ ⎝ ⎠ 0 n(n +2) 0 0 0 0 0 0 (42) Non-isomorphism of two cospectral graphs can be determined by n-particle quantum walk
as the same as G4(n, n +2)andG5(n, n + 2) by calculating 8 amplitudes of continuous
time quantum walks. Similar to G4(n, n +2)and G5(n, n + 2) there is no difference
between 3 amplitudes of T4(n, n +2)andT5(n, n + 2). But 5 amplitudes are different. for example one of them is:
− − iAT4 t iAT5 t φ2|e |φ0 − φ2|e |φ0 =
1 1 n(n +2)( (e−nit − enit) − (e−2it − e2it)) 2(2n +1) 2(n + 1)(2n +1) So the n-particle quantum walk is able to distinguish non-isomorph graphs. 106 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
(a)G1 (b)G2 65ss ss65 H C £J C H £J C £ J C HH £ J £ H £ C J C HH J C 2 £ J C 2 H£ J Xs £ Xs £ H C XX J C XX H J ¡¡A £XXX £XXXH sss C A XXXJJ sss C XHXXHJJ 7 C ¡ £ 10 7 C £ ¨ 10 J ¡ 1@A £ J 1@ £ ¨ C A C ¨ J Css ¡ @@A£ J 34Css @@£ ¨¨ J Q J Q ¨ 34Q Q ¨ J Q J ¨Q¨ J Q J ¨ Q Q ¨ Q JJs QQ s JJs¨¨ QQ s 89 89
Figure 2 A pair of nonisomorphic cospectral graphs: (a):G1 and (b):G2. Single particle quantum walk can distinguish these two graphs.
4 Investigation of graph isomorphism via quantum walk in some cospectral graphs
4.1 Example I
: Two cospectral nonisomorph graphs G1 and G2 are shown in Fig (II). They have ten ver- tices and eighteen edges. The degree distribution of two graphs is 5, 5, 5, 3, 3, 3, 3, 3, 3, 3.
The stratification basis are defined in two graph G1 and G2 as following
|φ0 = |1 1 |φ1 = √ (|2 + |3 + |4 ) 3 1 |φ2 = √ (|5 + |7 + |9 ) 3 1 |φ3 = √ (|6 + |8 + |10 ) (43) 3 So √ | | AG1 φ0 = 3 φ1 √ | | | | | AG1 φ1 = 3 φ0 +2φ1 + φ2 + φ3 | | | AG1 φ2 = φ1 +2φ3 | | | AG1 φ3 = φ1 +2φ2 (44)
And √ | | AG2 φ0 = 3 φ1 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 107
√ | | | | AG2 φ1 = 3 φ0 + φ2 + φ3 | | | AG2 φ2 = φ1 +2φ3 | | | | AG2 φ3 = φ1 +2φ2 +2φ3 (45) So, the adjacency matrix on strata basis is ⎛ √ ⎞ ⎜ 0 300⎟ ⎜ √ ⎟ ⎜ ⎟ ⎜ 3211⎟ A = ⎜ ⎟ (46) G1 ⎜ ⎟ ⎜ 0102⎟ ⎝ ⎠ 0120 ⎛ √ ⎞ ⎜ 0 300⎟ ⎜ √ ⎟ ⎜ ⎟ ⎜ 3011⎟ A = ⎜ ⎟ (47) G2 ⎜ ⎟ ⎜ 0102⎟ ⎝ ⎠ 0122
− − iAG1 t iAG2 t φ0|e − e |φ0 = −0.375e2it +0.5799e1.1774it − 0.2852e−1.3216it +0.0804e−3.8558it − − iAG1 t iAG2 t φ1|e − e |φ0 = 0.5e2it +0.268e1.1774it − 0.1661e−1.3216it +0.3981e−3.8558it − − iAG1 t iAG2 t φ2|e − e |φ0 = 0.375e2it − 0.5799e1.1774it +0.2852e−1.3216it − 0.0804e−3.8558it − − iAG1 t iAG2 t φ3|e − e |φ0 = 0.4999e2it − 0.268e1.1774it +0.1661e−1.3216it − 0.3981e−3.8558it We see that there are difference between the amplitudes of single particle quantum walk. then graph isomorphism can be distinguished from the single particle quantum walk.
4.2 Example II:
Two cospectral nonisomorph graphs H1 and H2 are shown in Fig (3). They have 12 vertices and 33 edges. The degree distribution of two graphs are
8, 8, 8, 8, 8, 8, 3, 3, 3, 3, 3, 3
The stratification basis are defined in the graph H1 as following
|φ0 = |6 108 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
(a)H1 (b)H2 ''''s s ''''s s 1 H ¨¨ 7 1 H ¨¨ 7 @HH¨ @HH¨ ¨s @¨ HHs ¨s @¨ HHs '' 2 H 8 '' 2 ¨ 8 H@ @ @¨ H ¨ ¨ &3 s H@H@s 9 &3¨ s @ @@s 9 H @¨¨ @ HH @ ¨ s @ HHs ¨s ¨@ @@s & 4§H ¨ 10 & 4§H ¨ 10 @H@¨ @H@¨ ¨ H¨ ¨H s ¨@ H@H@s ¨ s ¨@ H@H@s &&5H ¨ 11 &&5H ¨ 11 H¨@¨ H@¨ ¨H ¨ ¨H &&6¦s ¨ H@H@s 12 &&6¦¨ s H@H@s 12
Figure 3 A pair of nonisomorphic cospectral graphs:(a):H1 in the left and the (b):H2 in the right. Single particle quantum walk can distinguish these two graphs.
1 |φ1 = √ (|10 + |11 + |12 ) 3 1 |φ2 = √ (|3 + |4 + |5 ) 3 1 |φ3 = √ (|7 + |8 + |9 ) 3 1 |φ4 = √ (|1 + |2 ) (48) 2 So √ √ √ | | | | AH1 φ0 = 3 φ1 + 2 φ4 + 3 φ2 √ | | | AH1 φ1 = 3 φ0 +2φ2 √ √ | | | | | | AH1 φ2 = 3 φ0 +2φ1 +2φ2 + φ3 + 6 φ4 √ | | | AH1 φ3 = 6 φ4 + φ2 √ √ √ | | | | | AH1 φ4 = 2 φ0 + 6 φ2 + 6 φ3 + φ4 (49) ⎛ ⎞ √ √ √ ⎜ 0 3 30 2 ⎟ ⎜ ⎟ ⎜ √ ⎟ ⎜ 30 2 0 0⎟ ⎜ ⎟ ⎜ √ √ ⎟ A = ⎜ ⎟ (50) H1 ⎜ 32 2 1 6 ⎟ ⎜ √ ⎟ ⎜ ⎟ ⎜ 0010 6 ⎟ ⎝ √ √ √ ⎠ 20 6 61
The stratification basis are defined in the graph H2 as following
|φ0 = |12 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 109
1 |φ1 = √ (|4 + |5 + |6 ) 3 1 |φ2 = √ (|9 + |10 + |11 ) 3 1 |φ3 = √ (|1 + |2 + |3 ) 3 1 |φ4 = √ (|7 + |8 ) (51) 2 So √ | | AH2 φ0 = 3 φ1 √ | | | | | AH2 φ1 = 3 φ0 +2φ1 +2φ2 +3φ3 | | | AH2 φ2 =2φ1 + φ3 √ | | | | | AH2 φ3 =3φ1 + φ2 +2φ3 + 6 φ4 √ | | AH2 φ4 = 6 φ3 (52) ⎛ ⎞ √ ⎜ 0 30 0 0 ⎟ ⎜ ⎟ ⎜ √ ⎟ ⎜ 3223 0⎟ ⎜ ⎟ ⎜ ⎟ A = ⎜ ⎟ (53) H2 ⎜ 0 201 0⎟ ⎜ √ ⎟ ⎜ ⎟ ⎜ 0 312 6 ⎟ ⎝ √ ⎠ 00060 By some calculations, we see that
− − iAH1 t iAH2 t φ0|e − e |φ0 =
0.0655e2.7913it − 0.1067e1.4051it − 0.0655e−1.7913it +0.1067e−6.4051it − − iAH1 t iAH2 t φ1|e − e |φ0 = −0.1091e2.7913it +0.32e1.4051it +0.1091e−1.7913it − 0.32e−6.4051it − − iAH1 t iAH2 t φ2|e − e |φ0 = 0.0259e2.7913it − 0.32e1.4051it − 0.0218e−1.7913it +0.32e−6.4051it − − iAH1 t iAH2 t φ3|e − e |φ0 = −0.1091e2.7913it +0.32e1.4051it +0.1091e−1.7913it − 0.32e−6.4051it − − iAH1 t iAH2 t φ4|e − e |φ0 = 0.1309e2.7913it − 0.2133e1.4051it − 0.1309e−1.7913it +0.2133e−6.4051it The amplitudes of single particle quantum walk are different for two nonisomorph graphs. Therefore the single particle quantum walk can distinguish graph nonisomorphism. 110 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
(a)M1 (b)M2 s s ¨ 5 5 ¨ ¡ ¨ ¡ s¨¨ s s s 1 H 6 1 ¨¡¨ 6 HH J@A ¨ ¡ s HHs s¨AJ@¨ s 2 H 7 2 ¡ ¨ 7 @ H J AJ@¨ H ¨¡¨ 3 s @ HHs 8 s 13 3 s ¨J¡ AJ @@s 8 s 13 @ @ HHJA J¨¨ ¡ ¨¨H 4 s @ @@s 9 s 12 4 s ¨¡ JAHHJJs 9 s 12 HH@ JA H A H@H@s 10 s 11 JJAs 10 s 11
Figure 4 A pair of nonisomorphic cospectral graphs.(a):M1 and (b):M2. Single particle quantum walk can distinguish these two graphs.
Example III: Two graphs M1 and M2 in the Fig (4) are cospectral and nonisomorph. They have 13 vertices and 15 edges. The degree distribution of two graphs are
3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 1, 1, 1
The stratification basis are defined in the graph M1 as following
|φ0 = |1
1 |φ1 = √ (|5 + |6 + |7 ) 3
1 |φ2 = √ (|2 + |3 + |4 ) 3
1 |φ3 = √ (|8 + |9 + |10 ) 3
1 |φ4 = √ (|11 + |12 + |13 ) (54) 3 So √ | | AM1 φ0 = 3 φ1 √ | | | AM1 φ1 = 3 φ0 + φ2
| | | AM1 φ2 = φ1 +2φ3
| | | AM1 φ3 = φ4 +2φ2
| | AM1 φ4 = φ3 (55) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114 111
⎛ ⎞ √ ⎜ 0 3000⎟ ⎜ ⎟ ⎜ √ ⎟ ⎜ 30100⎟ ⎜ ⎟ ⎜ ⎟ A = ⎜ ⎟ (56) M1 ⎜ 0 1 020⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 201⎟ ⎝ ⎠ 0 0 010
The stratification basis are defined in the graph M2 as following
|φ0 = |1 1 |φ1 = √ (|8 + |9 + |10 ) 3 1 |φ2 = √ (|2 + |3 + |4 ) 3 1 |φ3 = √ (|5 + |6 + |7 ) 3 1 |φ4 = √ (|11 + |12 + |13 ) (57) 3 So √ | | AM2 φ0 = 3 φ1 √ | | | | AM2 φ1 = 3 φ0 + φ2 + φ4 | | | AM2 φ2 = φ1 +2φ3 | | AM2 φ3 =2φ2 | | AM2 φ4 = φ1 (58) ⎛ ⎞ √ ⎜ 0 3000⎟ ⎜ ⎟ ⎜ √ ⎟ ⎜ 30101⎟ ⎜ ⎟ ⎜ ⎟ A = ⎜ ⎟ (59) M2 ⎜ 0 1 020⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 200⎟ ⎝ ⎠ 0 1 000 The difference between amplitudes of single particle quantum walk for two graphs are
− − iAM1 t iAM2 t φ0|e − e |φ0 =
0.6554e2.5616it +0.1492e1.5616it − 0.1875 + 0.1492e−1.5616it − 0.0555e−2.5616it − − iAM1 t iAM2 t φ1|e − e |φ0 = 112 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 91–114
−0.1212e2.5616it +0.1212e1.5616it +0.1212e−1.5616it − 0.1212e−2.5616it − − iAM1 t iAM2 t φ2|e − e |φ0 = 0.0554e2.5616it − 0.1492e1.5616it +0.1875 − 0.1492e−1.5616it − 0.1875e−2.5616it − − iAM1 t iAM2 t φ3|e − e |φ0 = 0.1212e2.5616it − 0.1212e1.5616it − 0.1212e−1.5616it +0.1212e−2.5616it − − iAM1 t iAM2 t φ4|e − e |φ0 =0 The amplitudes of single particle quantum walk are different for two nonisomorph graphs. Therefore the single particle quantum walk can distinguish graph nonisomor- phism.
5 Conclusion
We investigated the graph isomorphism problem, in which one wishes to determine
whether two graphs are isomorphic. In two non-isomorph cospectral graphs G4(n, n +2)
and G5(n, n+2), we used n-particle quantum walk to distinguish these two graphs. It was performed by using the antisymmetric fermionic basis. The adjacency matrices of graphs was written in these new basis. The amplitudes of n-particle quantum walk, were differ- ent for two graphs, so the multiparticle quantum walk could detect non-isomorph pairs. In the process of fermionization of basis, the complexity has been reduced. Also in two
other similar cases T4(n, n+2) and T5(n, n+2), the n-particle quantum walk could detect these graphs. It was shown that n-particle quantum walk can detect non-isomorph pairs of G4(n, n +2)andG5(n, n + 2). In some examples of non-isomorph cospectral graphs, we show that the single particle quantum walk can detect non-isomorphism. One expect that the quantum walk in antisymmetric basis be able to distinguish some other kinds of graphs. Also it seems that the entanglement entropy is a powerful tool for detecting non-isomorph graphs.
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Thermodynamics of hot quantum scalar field in a (D +1) dimensional curved spacetime
W. A. Rojas C.∗ andJ.R.ArenasS.†
Received 16 August 2017, Accepted 25 December 2017, Published 20 April 2018
Abstract: We use the brick wall model to calculate the free energy of quantum scalar field in a curved spacetime (D +1) dimensions. We find the thermodynamics properties of quantum scalar field in several scenaries: Minkowski spacetime, Schwarzschild spacetime and BTZ spacetime. For the cases analyzed, the thermodynamical properties of quantum scalar field is exactly with the reported. It was found that the entropy of the gas is proportional to the horizon area in a gravity field strong, which is consistent with the holographic principle. c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Brick wall, black holes, holographic principle PACS (2010): 04.,04.20-q,04.70Bw,05.20.Gg
1 Introduction
In the early 70’s, a connection between gravity and thermodynamics is established. Where the horizon for a black hole has an associated temperature κ T = 0 (1) H 2π and entropy 1 SBH = 2 A. (2) 4lpl This entropy is proportional to black hole surface. The Bekenstein-Hawking entropy is considered a true thermodynamic entropy of the black holes. In a typical thermodynamic system, the thermal properties must reflect the microscopic physics. The temperature is a measure of average energy of particles and entropy counts the number of microstates of the system.
∗ Departamento de F´ısica , Universidad Nacional de Colombia, Email: [email protected] † Observatorio Astron´omico Nacional, Universidad Nacional de Colombia, Email: [email protected] 116 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124
The first models emerged to explain the origin of entropy arising from Gibbons/Hawking studies and model t’Hooft brick wall. For the former model (Euclidean approach) pro-
vides no insight into the dynamical origin of SBH. The latter (brick wall model) studies behavior of scalar fields near to black hole
horizon [2]. Under the last model, the SBH is related with the vacuum fluctuations in strong gravitational fields [1]. Because, an observer it at rest with black hole horizon sees a thermal bath of particles on the horizon [2]. In this point, we must clarify the concept of brick wall. Following to Israel [3]: The brick wall is treated here as a real physical barrier. To prevent misunderstanding this conception must be distinguished from a quite different one, according to which the wall is fictitious, merely a mathematical cutoff used to regularize the (Δr)−1 and logΔr terms in the expression for the thermal entropy . Our main purpose in this paper is study of hot quantum scalar field in a (D +1) dimensional curved spacetime. Under the brick wall model, we calculate the free energy
FD of hot quantum scalar field in (D + 1) dimensions. For this study D are n dimensional space, where n =2, 3... This method allows find the black hole entropy SD and this is proportional to area. In the section two, we calculate the free energy for hot quantum scalar field in a (D+1) dimensional curved spacetime. Under standard prescription we derive the entropy for this scenery. In the section three, we revised the thermodynamics of a hot quantum field in (3 + 1) dimensional flat spacetime. We study the particular case of D = 3 and we find that the thermodynamics properties are same reported for [1] and [4]. In the section four, we study the BTZ black hole, we find that entropy under method explained en the section two is proportional al area. In the end, we present discussions, we conclude that this model corresponds to a generalization of Brick wall model in (D + 1) dimensions.
2 Thermodynamics of ideal gas in (D +1) dimensions
We consider a spacetime with metric 1 ds2 = −f(r)dt2 + dr2 + r2dΩ2 (3) f(r) this includes the spacetimes Schwarzschild, Reissner-Nordstronm and (anti-) de Sitter in spacetime (3+1) dimensions [2].
For this spacetime Fursaev showed that free energy F3 for a bosonic field in three spatial dimensions near to black hole horizon is [1] 1 √ F = − ζ(4) T 4 −gd3x, (4) 3 π2 where ζ(4) is Riemann function zeta, T is temperature of the bosonic gas, g is the determinant of metric and d3x is differential element of volume spatial. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124 117
Free energy in (2 + 1) for a gas of bosons is 1 √ F = − ζ(3) T 3 −gd2x. (5) 2 π
If we consider a (D + 1)-dimensional spacetime with a metric [5]
1 ds2 = −f(r)dt2 + dr2 + rD−1dΩD−1. (6) f(r) and metric tensor gμν given by ⎛ ⎞ ⎜ −f(r) ... 0 ⎟ ⎜ ⎟ ⎜ . 1 ⎟ gμν = ⎜ . ... ⎟ , (7) ⎝ f(r) ⎠ 0 ... rD−1Ω
Then, the free energy is 1 √ F = − ζ(D +1) T D+1 −gdDx, (8) D πD−1 where T is temperature of ideal gas and obey the Tolman’s law
T∞ T (r)= (9) f(r) and g is the determinant of metric tensor
g = rD−1Ω. (10)
Their square root is √ √ D−1 −g = rD−1Ω=r 2 Ω1/2, (11) where r is radial part and Ω is solid angle in D dimensions. The differential element of volume spatial dDx, is the product of a radial part (dr) and an angular part (dD−1Ω)
dDx = drdD−1Ω. (12)
Thus, the free energy is reduced to D+1 1 T∞ D−1 − F = − ζ(D +1) r 2 drΩ1/2dD 1Ω. (13) D D−1 D+1 π f(r) 2 Separating radial part and angular part in (13), we can write free energy as
D−1 1 r 2 F = − ζ(D +1)T D+1 dr Ω1/2dD−1Ω. (14) D D−1 ∞ D+1 π f(r) 2 118 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124
The first integral in (14) is associated to radial part and second one to angular part. Following S. Kolelar and T. Padmanabhan [5], and can be written as
D−1 1 r 2 F = − ζ(D +1)T D+1Ω dr. (15) D D−1 ∞ D+1 π f(r) 2
Where Ω is solid angle in D dimensions. Consider the following conditions [5]:
f(rH )=0,
which defines the horizon 2κ = f (rH ) It is a finite temperature in the vicinity of the black hole. Near to horizon, we write f(r) as [2, 5, 10] f(r)=f (rH )(r − rH ). (16) According to the above, the integral of (15) reduces to
D−1 D−1 D−1 r 2 r 2 1 r 2 D+1 dr = D+1 dr = D+1 D+1 dr. (17) 2 2 2 − 2 f(r) (f (rH )(r − rH )) f (rH ) (r rH )
Defining u = r − rH
D−1 D−1 1 r 2 1 (u + r ) 2 dr = H du. (18) D+1 D+1 D+1 D+1 f (rH ) 2 (r − rH ) 2 f (rH ) 2 u 2 And using a binomial expansion −1 −3 −3 D−1 D 1 D − 1 D 1 D − 1 D − 3 D D−1 (r + u) 2 = r 2 + r 2 u + r 2 u2 + ...+ u 2 . H H 1! 2 H 2! 2 2 H (19) So, (18) rewrites D−1 1 (u + r ) 2 H du = D+1 D+1 f (rH ) 2 u 2 −1 −3 −3 1 1 D 1 D − 1 D 1 D − 1 D − 3 D D−1 r 2 + r 2 u + r 2 u2 + ...+ u 2 du. D+1 D+1 H H H f (rH ) 2 u 2 1! 2 2! 2 2 (20) Then the integral in (18) is rewritten as (20) [5]. S. Kolelar and T. Padmanabhan says: the main contribution to this integral comes from the lower limit of the integral D−1 r 2 H h D+1 . Because near the horizon u = h and r lP [5]. u 2 H
D−1 D−1 1 r 2 2r 2 H du = − H (21) D+1 D+1 D+1 D−1 f (rH ) 2 u 2 f (r) 2 (D − 1)h 2 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124 119
where lp is the Planck length. Near the horizon, we can replace h to lp as a distance own H dr ≈ h lp = 2 , (22) rH f(r) f (rh) solvig h and replace in (21)
−1 D−1 D r 1 2Dr 2 dr = − H (23) D+1 D − 2 − D 1 f(r) (D 1) [f (rH )] lp Finally we write the free energy for ideal gas in (D + 1) dimensions in the following form ζ(D +1) A D−1 T D+1 F = − 2 ∞ (24) D − D−1 D−1 D (D 1)π lp κ
D−1 2 where the product A D−1 =Ωr is the horizon area in (D + 1) dimensions. The free 2 H energy is an off-shell prescription and expresses in four independent variables: • The temperature T∞, • geometrical characteristics (D + 1) spacetime dimensions, horizon’s area A and sur- face gravity κ. The entropy is recoverable from the free energy by the standard prescription ∂F S = − (25) D ∂T ζ(D +1) A D−1 (D +1)T D S = 2 ∞ . (26) D − D−1 D−1 D (D 1)π lP κ At this point, we observe that this is an off-shell prescription [2, 6]. Because, geometrical quantities (A y κ) are kept fixed when the temperature is varied [2].
3 Particular case (3+1) dimensional flat spacetime
If we take the equation (8) and we do gμν = ημν where ημν is tensor of Minkowski for a flat spacetime. So the equation (8) become in 1 √ F = − ζ(D +1) T D+1 −ηdDx, (27) D πD−1 √ D where −η =1,T = T∞ and VD = d x. Then, we have 1 F = − ζ(D +1)T D+1V . (28) D πD−1 ∞ D We find the entropy of the hot quantum scalar field in the Minkowski spacetime like ∂F SD = − ∂T∞ 120 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124
(D +1) S = ζ(D +1)T DV . (29) D πD−1 ∞ D In particular the case when D = 3 for a flat spacetime. The free energy is π2 F = − T 4 V (30) 3 90 ∞ 3 and the entropy is 2 S = π2T 3 V (31) 3 45 ∞ 3 other properties of hot quantum scalar field in the minkowski spacetime are well known [19].
4 Particular case (3+1) dimensional curved spacetime
κ ∞ | If the temperature of quantum field is T = TH = 2π , so the entropy is S3 T∞=TH using (26) 1 S | = A (32) 3 T∞=TH 2 360πlp it is agree with Fursaev’s result!! The quantum field is supposed to be in thermal equi- librium with the horizon [6]. Another hand, if D = 3, (24) and (26) reduced to
π2 T 4 A F = − ∞ (33) 3 2 3 180 lpκ π2 T 3 A S = ∞ (34) 3 2 3 45 lpκ such that (26) is entropy of bosonic field in (3+1). For a photon gas in Schwarzschild spacetime, their entropy is [4] 1 π2k4 c3 A S = B T 3 . (35) 3 3 2 3 ∞ 45 lP κ Also we calculate other thermodynamics properties with standard recipes: • The internal energy of quantum field, dE = dF + T∞dS,is ζ(D +1) A D−1 DT D+1 E = 2 ∞ . (36) D − D−1 D−1 D (D 1)π lp κ If D = 3 the internal energy π2 AT 4 = ∞ (37) E3 2 3 , 60 lpκ that is the internal energy of photon gas in Schwarzschild spacetime [4]. • The specific heat is ∂E CV = ∂T∞ V Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124 121 ζ(D +1) A D−1 D(D +1)T D C = 2 ∞ , (38) V − D−1 D−1 D (D 1)π lp κ with D =3 2 3 π AT∞ − = (39) CV 3 2 3 . 15 lpκ • And the pressure of quantum field near to horizon is ∂F 1 ∂F P = − = ∂V T∞ lP ∂A T∞ ζ(D +1) 1 T D+1 P = ∞ (40) D − D−1 D D (D 1)π lp κ with D =3 π2 T 4 P = ∞ . (41) 3 3 3 180 lpκ Is interesting to note that the thermodynamic properties described by (33), (34), (37), (39) and (41) have already been studied by the authors in the case of a photon gas in the Schwarzschild spacetime [4] under the approximation of brick wall model by Mukohyama and Israel for a hot quantum scalar field near the horizon in the Boulware state [2].
5 BTZ spacetime
This spacetime is interesting because the BTZ black holes are asymptotically anti de Sitter. In gravity (2+1) the curvature is constant (R = −6/l2). This black holes don’t have points and regions in which the curvature is divergent. However, the BTZ black holes have a horizon, an ergosphere and thermodynamic properties to the classical solutions of General Relativity [9]. Their metric is r2 − r2 1 2 − + 2 2 2 ds = dt + 2− 2 dr + r dφ, (42) l2 r r+ l2
− 1 with a negative cosmological constant λ = l2 . The area of horizon of BTZ black hole is the length 2πr+ [1, 8]. We calculate the black hole entropy for this spacetime with method of the section II. The first step is to use (8) in which D = 2 spatial dimensions, then we have 1 √ F = − ζ(3) T 3 −gd2x, (43) 2 π √ in agree with (9); (11), −g = r1/2Ω1/2 and (12) d2x = drdΩ. The free energy in the BTZ spacetime is rewritten as 3 1/2 1 T∞ 1 r F = − ζ(3) r1/2Ω1/2drdΩ=− ζ(3)T 3 Ω1/2dΩ dr, (44) 2 π f(r)1/2 π ∞ f(r)3/2 122 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124 2− 2 1/2 r r+ where the solid angle is Ω = Ω dΩ=2π and f(r)= l2 for the BTZ spacetime. We consider the last integral is agree with the condition (16), the integral reduces to r1/2 1 r1/2 dr = dr, (45) 3/2 3/2 − 3/2 f(r) [f (r+)] (r r+) this integral is similar to (17) y it can solve the same form. With u = r − r+, thus (45) become in r1/2 1 (r + u)1/2 dr = + du. (46) 3/2 3/2 3/2 f(r) [f (r+)] u And using a binomial expansion
1 − (r + u)1/2 = r1/2 + r 1/2u + ...+ u1/2. (47) + + 2 + Then the integral (46) is rewritten as 1 (r + u)1/2 + du = 3/2 3/2 [f (r+)] u 1/2 −1/2 1 1/2 1/2 1/2 1 r+ + 2 r+ u + ...+ u 1 r 2r 1 du ≈ + du = − + .√ . 3/2 3/2 3/2 3/2 3/2 [f (r+)] u [f (r+) ] u f (r+) u (48) Under the condition of S. Kolekar and T. Padmanabhan [5]. Again, near to horizon u = h
and we replace h to lp as distance own in agree with (22) r1/2 22r1/2 = − + . (49) 3/2 2 f(r) [f (r)] lp We write the free energy for hot quantum scalar field in BTZ spacetime as 2 3 A 2−1 1 2 1 T 2 = − (3) 3 Ω 1/2 = − (3) ∞ (50) F2 ζ T∞ r+ 2 2 ζ 2 π 2 κ lp π κ lp
1/2 where the area of horizon is A 2−1 =Ωr =2πr+. We can obtain the entropy as 2 + 3 T 2 2πr = (3) ∞ + (51) S2 ζ 2 . π κ lp
κ Again, if the temperature of quantum field is T∞ = TH = 2π and the entropy to the temperatura Hawking is 3 1 κ2 2πr S | = ζ(3) + 2 T∞=TH 2 2 π κ 4π lp
| 2πr+a S2 T∞=TH = . (52) lp Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124 123
The mass M of the BTZ black hole is defined as
r2 M = + , (53) 2 8l G3 where G3 is the 3D gravitational coupling and has the dimension of length [1]. | Thus, S2 T∞=TH is 2πa | 2 S2 T∞=TH = 8l MG3 (54) lp
6 Summary and Discussion
The last result shows that entropy of ideal gas is proportional to area in (D + 1) dimen- sions, when this is in equilibrium state with the horizon at temperature T∞. Under the brick wall model for (D + 1) dimensional curved spacetime, the entropy can be determined by response of the free energy of the system to change of temperature given by (25) [1, 2, 4, 6]. Also the distinction between thermodynamical and statistical entropies disappears in this model, because the geometrical and thermal variables are kept independent. Observe that this is an off-shell prescription [2, 6]. The free energy was made from the brick wall model, which studies quantum fields close to the horizon [2]. Following this model, thus affirms Fursaev and Mukohyama: entropy defined in (26) and corresponds to the Bekenstein-Hawking entropy SBH for quantum fields near the horizon in spacetime (D+1). And also depends on the geometrical and thermal quantities as expressed Mukohyama [1, 2] A D−1 (D +1)T D S ∝ 2 ∞ (55) D − D−1 D (D 1)lP κ if T = TH ,thenSD = SBH, that are defined only for geometrical quantities in (D +1) dimensions [2]. We find that the entropy to the Hawking temperature in BTZ black hole is propor- tional to √ | ∝ 2 S2 T∞=TH 2π l M, (56) This result is consistent with [1, 8]. From the above we can consider that the present model is a generalization of the study of hot quantum scalar field in a (D + 1) dimensional curved spacetime of brick wall model. In the case when D = 3 the usual space, the entropy is reduced to the results reported by [1] and [4].
Acknowledgements
This work was supported by the Departamento de Administrativo de Ciencia, Tec- nolog´ıa e Innovaci´on, Colciencias. 124 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 115–124
References
[1] D. V. Fursaev. Phys. Part. Nucl. 36 (2005):81. [2] Mukohyama, Shinji et al. Phys.Rev. D58 (1998) 104005 [3] W. Israel, Phys. Lett. A57, 107 (1976). [4] W. A. Rojas C. and R. Arenas arXiv:[gr-qc] 1110.4058v2 [5] S. Kolekar and T. Padmanabhan, Phys.Rev. D83 (2011) 064034. [6] V. P. Frolov, D. V. Fursaev, and A. I. Zelnikov, Phys. Rev. D 54, 2711 (1996). [7] G. t’Hooft. Nucl. Phys. B256.(1985):727. [8] M. Ba˜nados, C. Tietelboim and J. Zanelli. Phys. Rev. Lett. 69 (1992) 1849. [9] E. A. Larra˜naga R. Tesis de Doctorado Termodin´amica de agujeros negros en BTZ (2+1) dimensiones. Departamento de F´ısica. Universidad Nacional de Colombia. Director: Juan Manuel Tejeiro Sarmiento (2009). [10] L. Susskind and J. Lindesay. An Introduction to Black Holes, Information, and String Theory Revolution. World Scientific Publising Co. Pte. Ltd., London, (2005). [11] A. Corichi and D. Sudarky. Mod. Phys. Lett A17 (2002):1431. [12] S. W. Hawking. Comm. Math. Phys. 43 (1975):199. [13] J. D. Benkenstein. Phys. Rev. D 7 (1973):2333. [14] Bardeen, J. M.; Carter, B.; Hawking, S. W. Com. in Math. Phys. 31 (2): 16170 (1973). [15] R. C. Tolman. Relativity Thermodynamics and Cosmology. Dover Publications Inc., New York (1987). [16] T. Padmanabhan. Gravitation foundations and frontiers. Cambrigde University Press., New York (2010). [17] R. M. Wald. General Relativity. The Chicago University Press. USA (1984). [18] R. P. Feynman. Statistical Mechanics: A set of lectures. The Benjamin/Cummings Publishing Company, Inc, Massachusetts, (1961). [19] L. Landau and E. Lifshitz. Curso de F´ısica Te´orica. Teor´ıa Cl´asica de Campos, volumen 3. Editorial Revert´e S.A., Barcelona, 1973. [20] K. S. Thorne, C. W. Misner and Wheeler. Gravitation. W.H. Freedman and Company. San Francisco. (1973). EJTP 14, No. 37 (2018) 125–146 Electronic Journal of Theoretical Physics
Spin and Zitterbewegung in a Field Theory of the Electron.(‡)
Erasmo Recami 1 ∗†and Giovanni Salesi 2 1Facolt`a di Ingegneria, Universit`a Statale di Bergamo, 24044–Dalmine (BG), Italy; INFN–Sezione di Milano, Milan, Italy; and 1DECOM, Faculty of Electrical Engineering (FEEC), State University at Campinas (UNICAMP), Campinas, Brazil. 2Facolt`a di Ingegneria, Universit`a Statale di Bergamo, 24044–Dalmine (BG), Italy; INFN–Sezione di Milano, Milan, Italy.
Received 6 January 2018, Accepted 20 March 2018, Published 20 April 2018
Abstract: In previous papers we investigated the classical theory of the electron spin, and its ”quantum” version, by using Clifford algebras; and we ended with a new non-linear Dirac-like equation (NDE). We re-address in this review the whole subject, and extend it, by adopting however the ordinary tensorial language, within the first quantization formalism. In particular, we re-derive the NDE for the electron field, showing it to be associated with a new conserved probability current. Incidentally, the Dirac equation results from the former by averaging over a Zbw cycle. Afterward, we derive an equation of motion for the 4-velocity field, which allows regarding the electron as an extended-like object with a classically intelligible internal structure. We carefully study the solutions of the NDE; with special attention to those implying (at the classical limit) light-like helical motions, since they appear to be the most adequate solutions for the electron description. At last we introduce a natural generalization of our approach, for the case in which an external electromagnetic potential is present.
c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Spin and Zitterbewegung; Structure of Spinning Particles; QFT of the Electron; Extended-like Particles; Barut-Zangh Theory; New non-linear Dirac-like Equation PACS (2010): 75.10.Hk, 75.10.Jm; 21.10.Hw, 03.70.+k; 04.20.Gz; 11.10.-z; 14.60.Cd.
(‡) Work supported by CAPES (Brazil), and by INFN and MIUR (Italy) ∗ Email: [email protected] † Bolsista CAPES/BRASIL: Supported as a Vising Professor by a PVE Fellowship of CAPES/BRAZIL. 126 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146
1 Premise
In previous work, we investigated the classical Barut–Zanghi (BZ) theory[1] for the electron spin, which involves internal Zitterbewegung (zbw) motions along cylindrical helices, by using the powerful language of Clifford algebras[2]. The “quantum” version of such approach lead us[2], in particular, to a new, non-linear, Dirac-like equation (NDE). In this review article we implement a first quantization of the BZ theory for the free electron, by adopting however the ordinary tensorial language. In so doing, we re- derive the NDE for the electron quantum field. Such equation involves a new probability current J μ, which is shown to be a conserved quantity, endowed (except in the “trivial” case corresponding to the standard Dirac theory) with a zbw motion having the typical frequency ω =2m,wherem is the electron mass. Then, we derive a new equation of motion for the field velocity vμ quite different from that of Dirac’s theory. Indeed, it allows us referring to realistic internal motions endowed with an intuitive kinematical meaning. The consequence is that our electron results to be an extended-like particle, with a classically intelligible internal structure. Afterwards, we write down the general solution of the NDE, which appears to be a superposition of plane waves with positive and negative frequencies; in the present theory such superposition results to entail always a positive field energy, both for particles and for antiparticles. We analyse the kinematical (zbw) structure of vμ in the center-of-mass frame as a function of the initial spinor ψ(0). After having shown that, for extended-like particles 2 μ like our electron, quantity v ≡ vμv in general is not constant in time (quite differently from the scalar particle case, in which v2 = 1 is always constant), we look for the par- ticular NDE solutions that on the contrary imply a constant v2; and we find this case to correspond to a circular, uniform zbw motion with ω =2m and orbital radius R = |v|/2m (where |v|, and therefore R, depend on the particular NDE solution considered). Even more, the simple requirement of a uniform (classical) motion seems to play the role of the ordinary quantization conditions for the z-component of the field spin s;namely,v2 = ± 1 constant does imply sz = 2 , that is to say, the electron polarization. We also examine, then, the oscillating linear motions associated with unpolarized electrons. Special attention is devoted to the light-like (v2 = 0) helical motions, either clock- wise or anti-clockwise, for which the orbital radius results to be equal to half a Compton wavelength. In fact, as already mentioned, such motions appear to be the most adequate for the electron description, both kinematically and physically, and correspond to the electromagnetic properties of the electron, such as Coulomb field and intrinsic magnetic moment. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 127
2 Introduction
Attempts to put forth classical theories for the electron are known since almost a century.[3−6] For instance, Schr¨odinger’s suggestion[4] that the electron spin was related to the Zitterbewegung (internal) motion did originate a large amount of subsequent work, including Pauli’s. In ref.[7] one can find, for instance, even the proposal of models with clockwise and anti-clockwise “inner motions” as classical analogues of quantum relativistic spinning particles and antiparticles, respectively. Most of the cited works did not pay attention, however, to the transition from the classical theory to its quantum version, i.e., to finding out a wave equation whose classical limit be consistent with the assumed model. Among the approaches in which on the contrary a passage to the quantum version was performed, let us quote the papers by Barut and Pavsic.[1,8] Namely, those authors first considered a classical lagrangian[1] L describing (internal) helical motions; and second —after having passed to the hamiltonian formalism— they constructed a quantum version of their model and thus derived (in their opinion) the Dirac equation. Such results are really quite interesting and stimulating. We show below, however, that from the BZ theory one does naturally derive a non-linear Dirac-like equation,[2] rather than the Dirac (eigenvalue) equation itself, whose solutions are only a subset of the former’s. Many further results have been met,[2,9], as we were saying, by using the powerful Clifford algebra language.[10] Due to their general interest and importance, we reformu- late here the results appeared in refs.[2] by the ordinary tensorial language: This will allow us to show more plainly and clearly, and more easily outline, their geometrical and kinematical significance, as well as their field theoretical implications. In so doing, also we shall further develop such a theory; for instance: (i) by showing the strict correlation existing between electron polarization and zbw kinematics (in the case of both time-like and –with particular attention, as we know— light-like internal helical motions); and (ii) by forwarding a probabilistic interpretation of the NDE current (after having shown it to be always conserved).
3 The classical Barut–Zanghi (BZ) theory
The existence of an “internal” motion (inside the electron) is denounced, besides by the presence of spin, by the remarkable fact that, according to the standard Dirac theory, the electron four-impulse pμ is not parallel to the four-velocity: vμ = pμ/m; moreover, while [pμ,H]=0 sothatpμ is a conserved quantity, on the contrary vμ is not a constant of the motion: [vμ,H] = 0. Let us recall that indeed, if ψ is a solution of Dirac equation, only the “quantum average” (or probability current)
pendent canonically conjugate variables are necessary; for example:
xμ ,pμ ; vμ , Pμ , or alternatively (as we are going to see): xμ ,pμ ; z, z
where z and z ≡ z†γ0 are ordinaryC I 4–bispinors. In the BZ theory,[1] the classical electron was actually characterized, besides by the usual pair of conjugate variables (xμ,pμ), by a second pair of conjugate classical spinorial variables (z,z), representing internal degrees of freedom, which are functions of the proper time τ of the electron global center-of-mass (CM) system. The CM frame is the one in which at every instant of time it is p = 0 (but in which, for spinning particles, v ≡ x˙ is not zero, in general!). Barut and Zanghi, then, introduced a classical lagrangian that in
the free case (Aμ = 0) writes [c =1] 1 L = iλ(zz˙ − zz˙)+p (˙xμ − zγμz)(1) 2 μ where λ has the dimension of an action. [The extension of this lagrangian to the case with external fields, Aμ = 0, has been treated elsewhere]. One of the consequent (four) motion equations yields directly the particle four-velocity:
x˙ μ ≡ vμ = zγμz. (1)
We are not writing down explicitly the spinorial indices of z and z. Let us explicitly notice that, in the classical theories for elementary spinning par- ticles, it is convenient[11] to split the motion variables as follows
xμ ≡ ξμ + Xμ ; vμ = wμ + V μ , (2) where ξμ and wμ ≡ ξ˙μ describe the translational, external or drift motion, i.e. the motion of the CM, whilst Xμ and V μ ≡ X˙ μ describe the internal or spin motion. From eq.(1) one can see[1] that also
μ μ H ≡ pμv = pμzγ z (3) is a constant of the motion (and precisely it is the CMF energy); being H, as one may easily check, the BZ hamiltonian in the CMF, we can suitably set H = m, quantity m being the particle rest-mass. In this way, incidentally, we obtain, even for spinning particles, the ordinary relativistic constraint (usually met for scalar particles):
μ pμv = m. (3 ) The general solution of the equations of motion corresponding to lagrangian (1), with λ = −1[and = 1], is: p γμ z(τ)=[cos(mτ) − i μ sin(mτ)]z(0) , (4a) m Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 129
p γμ z(τ)=z(0)[cos(mτ)+i μ sin(mτ)] , (4b) m with pμ = constant; p2 = m2; and finally: pμ pμ x¨μ x˙ μ ≡ vμ = +[˙xμ(0) − ]cos(2mτ)+ (0) sin(2mτ) . (4c) m m 2m This general solution exhibits the classical analogue of the phenomenon known as Zitterbewegung: in fact, the velocity vμ contains the (expected) term pμ/m plus a term describing an oscillating motion with the characteristic zbw frequency ω =2m.The velocity of the CM will is given of course by wμ = pμ/m. Notice that, instead of adopting the variables z and z, we can work in terms of the spin variables, i.e., in terms of the set of dynamical variables
xμ ,pμ ; vμ ,Sμν
where i Sμν ≡ z[γμ,γν]z ; (5a) 4 then, we would get the following motion equations:
μ μ μ μ μν ˙ μν ν μ μ ν p˙ =0; v =˙x ;˙v =4S pν ; S = v p − v p . (5b)
By varying the action corresponding to L, one finds as generator of space-time rotations the conserved quantity J μν = Lμν + Sμν,whereLμν ≡ xμpν − xνpμ is the orbital angular momentum tensor, and Sμν is just the particle spin tensor: so that J˙μν = 0 implies L˙ μν = −S˙ μν. Let us explicitly observe that the general solution (4c) represents a helical motion in the ordinary 3-space: a result that has been met also by means of other,#1 alternative approaches.[12,13]
4 Field theory of the extended–like electron
The natural way of “quantizing” lagrangian (1) is that of reinterpreting the classical spinors z and z as Dirac field spinors, say ψ and ψ [in the following the Dirac spinors will be merely called “spinors”, instead of bi-spinors, for simplicity’s sake]:
z → ψ ; z → ψ ;
which will lead us below to the conserved probability current J μ = m ψγμψ/p0. Recall that here the operators (xμ,pμ; ψ, ψ) are field variables; for instance,[2] ψ = ψ(xμ); ψ =
1#1Alternative approaches to the kinematical description of the electron spin have been proposed, e.g., [12] by Pavsic and Barut in refs.[12,13]. In connection with Pavsic’s approach, we√ would like here to mention that the classical angular momentum was defined therein as s ≡ 2βv ∧ a/ 1 − v2, whilst in the BZ theory it is s ≡ r ∧ mw,wherea ≡ v˙ . Both quantities s result to be parallel to p. 130 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146
ψ(xμ). Thus, the quantum version of eq.(1) is the field lagrangian i L = λ(ψψ˙ − ψψ˙)+p (˙xμ − ψγμψ)(6) 2 μ that refers (so as in the classical case) to free electrons with fixed impulse pμ. The four Euler–Lagrange equations, with −λ = = 1, yield the following motion equations: ⎧ ⎪ ⎪ ψ˙ + ip γμψ =0 (7a) ⎨⎪ μ x˙ μ = ψγμψ (7b) ⎪ ⎪ ⎩ p˙ =0, (7c)
besides the hermitian adjoint of eq.(7a), holding for ψ = ψ+γ0. In eqs.(7), the μ invariant time τ is still the CMF proper time, and it is often put pμγ ≡ p/. We can pass to the hamiltonian formalism by introducing the field hamiltonian corresponding to the energy in the CMF:
μ H = pμψγ ψ, (8) which easily yields the noticeable equation#2
μ pμψγ ψ = m (9)
This non-linear equation, satisfied by all the solutions of eqs.(7), is very probably the simplest non-linear Dirac–like equation.[14] Another non-linear Dirac–like equation, quite equivalent to eq.(7a) but employing the generic coordinates xμ and no longer the CMF proper time, was anticipated in the first one of refs.[2]; it is obtained by inserting the identity d dxμ ∂ ≡ ≡ x˙ μ∂ (10) dτ dτ ∂xμ μ μ μ μ μ into eq.(7a). In fact, one gets ix˙ ∂μψ = pμγ ψ, and, sincex ˙ = ψγ ψ because of eq.(7b), one arrives at the important equation:
μ μ iψγ ψ∂μψ = pμγ ψ. (11)
μ A more general equation, in which pμγ is replaced by m,
μ iψγ ψ∂μψ = mψ , can be easily obtained (cf. the last one of refs.[2]) by releasing the fixed–pμ condition. Let us notice that, differently from eqs.(6)–(7), equation (11) can be valid a priori 1 even for massless spin 2 particles, since the CMF proper time does not enter it any longer.
1#2 μ In refs.[12] it was moreover assumed that pμx˙ = m, which actually does imply the very general μ μ relation pμψγ ψ = m, but not the Dirac equation pμγ ψ = mψ, as claimed therein; these two equations in general are not equivalent. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 131
The remarkable non-linear equation (11) corresponds to the whole system of eqs.(7): quantizing the BZ theory, therefore, does not lead to the Dirac equation, but rather to the non-linear, Dirac-like equation (11), that we call NDE. The eq.(11) might be even adopted in substitution for the free Dirac (eigenvalue) equation, since it apparently admits a sensible classical limit [which describes an inter- nal periodic motion with frequency 2m, implying the existence of an intrinsic angular momentum tensor: the “spin tensor” Sμν of eq.(5a)]. Moreover, eq.(11) admit assigning a natural, simple physical meaning to the negative frequency waves.[15] In other words, eqs.(6)–(11) seem to allow us overcoming the wellknown problems related with physical meaning and time evolution of the position operator xμ and of the velocity operatorx ˙ μ: we shall come back to this point. [We always indicate a variable and the corresponding operator by the same simbol]. In terms of field quantities (and no longer of operators), eq.(11) corresponds[2] to the four motion equations
μ μ μ μ μν ˙ μν ν μ μ ν p˙ =0; v =˙x ;˙v =4S pν; S = v p − v p , (12a) in which now vμ and the spin tensor Sμν are the field quantities vμ ≡ ψ(x)γμψ(x)and
i Sμν = ψ(x)[γμ,γν]ψ(x) . 4 By deriving the third one of eqs.(12a), and using the first one of them, we obtain
μ ˙ μν v¨ =4S pν ; (12b) by substituting now the fourth one of eqs.(12a) into eq.(12b), and imposing the previous μ 2 μ constraints pμp = m ,pμv = m, we get the time evolution of the field four-velocity:
pμ v¨μ vμ = − . (13) m 4m2 Let us recall, for comparison, that the corresponding equation for the standard Dirac case[1−7] was devoid of a classical, realistic meaning because of the known appearance of an imaginary unit i in front of the acceleration:[16]
pμ i vμ = − v˙ μ . (13) m 2m
μ One can observe, incidentally, that by differentiating the relation pμv = m = constant, one immediately gets that the (internal) accelerationv ˙ μ ≡ x¨μ is orthogonal to μ μ the electron impulse p since pμv˙ = 0 at any instant. To conclude, let us recall that the Dirac electron has no classically meaningful internal structure; on the contrary, our elec- tron, an extended–like particle, does possess an internal structure, and internal motions which are all kinematically, geometrically acceptable and meaningful: As we are going to see. 132 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146
5 General solution of the new non-linear, Dirac–like equation (NDE), and conservation of the probability current
In a generic frame, the general solution of eq.(11) can be easily shown to be the μ following [p/ ≡ pμγ ]:
m − p/ μ m + p/ − μ ψ(x)=[ eipμx + e ipμx ] ψ(0) ; (14a) 2m 2m which, in the CMF, reduces to
1 − γ0 1+γ0 ψ(τ)=[ eimτ + e−imτ ] ψ(0) , (14b) 2 2 or, in a simpler form, to:
ψ(τ)=[cos(mτ) − iγ0 sin(mτ)] ψ(0) , (14c)
quantity τ being the particle proper time, as above. Let us explicitly observe that, by introducing eq.(14a), or eq.(14b), into eq.(9), one obtains that every solution of eq.(11) does correspond to the CMF field hamiltonian H = m>0, even if it appears (as expected in any theories with zbw) to be a suitable superposition of plane waves endowed with positive and negative frequencies.[15] Notice that superposition (14a) is a solution of eq.(11), due to the non-linearity of such an equation, only for suitable pairs of plane waves, with weights
m ± p/ =Λ± , 2m
respectively, which are nothing but the usual projectors Λ+ (Λ−)overthepositive (negative) energy states of the standard Dirac equation. In other words, the plane wave solution (for a fixed value of p) of the Dirac eigenvalue equation pψ/ = mψ is a particular case of the general solution of eq.(11): namely, for either
Λ+ψ(0) = 0 or Λ−ψ(0) = 0 . (15)
Therefore, the solutions of the Dirac eigenvalue equation are a subset of the set of solutions of our NDE. It is worthwhile to repeat that, for each fixed p, the wave function ψ(x) describes both particles and antiparticles: all corresponding however to positive energies, in agreement with the reinterpretation forwarded in refs.[15] (as well as with the already mentioned fact that we can always choose H = m>0).
We want now to study, eventually, the probability current J μ corresponding to the wave functions (14a,b,c). Let us define it as follows: m J μ ≡ ψγμψ (16) p0 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 133
where the normalization factor m/p0 (the 3-volume V being assumed to be equal to 1, as usual; so that p0V ≡ p0) is required to be such that the classical limit of J μ,that is (m/p0) vμ, equals (1; v), like for the ordinary probability currents. Notice also that sometimes in the literature p0 is replaced by E;andthatJ 0 ≡ 1, which means that we have one particle inside the unitary 3-volume V = 1. This normalization allows us μ μ 0 to recover, in particular, the Dirac current JD = p /p when considering the (trivial) solutions, without zbw, corresponding to relations (15). Actually, if we substitute quantity ψ(x) given by eq.(14a) into eq.(16), we get pμ J μ = + Eμ cos(2p xμ)+Hμ sin(2p xμ) , (16) p0 μ μ where Eμ ≡ J μ(0) − pμ/p0 ; Hμ ≡ J˙(0)/2m. (16) If we now impose conditions (15), we have Eμ = Hμ = 0 and get therefore the Dirac μ μ μ 0 current J = JD = constant = p /p . Let us notice too that the normalization factor m/p0 cannot be inserted into ψ and ψ, as it would seem convenient, because of the non-linearity of eq.(11) and/or of constraint (9). μ μ μ 0 2 0 2 0 Since pμE ≡ pμJ (0) − pμp /p = m /p − m /p = 0 (where we used eq.(9) for μ ˙μ x = 0) and since pμH ≡ pμJ (0)/2m = 0 obtained deriving both members of eq.(9) —note incidentally that both Eμ and Hμ are orthogonal to pμ— it follows that
μ μ μ ∂μJ =2pμH cos(2px) − 2pμE sin(2px)=0. (16 ) We may conclude, with reference to equation (11), that our current J μ is conserved: We are therefore allowed to adopt the usual probabilistic interpretation of fields ψ, ψ. Equation (16’) does clearly show that the conserved current J μ, as well as its classical limit (m/p0)vμ [see eq.(4c)], are endowed with a Zitterbewegung–type motion: precisely, with an oscillating motion having the CMF frequency Ω = 2m 1021s−1 and period T = π/m 10−20s (we may call Ω and T the zbw frequency and period, respectively). From eq.(16’) one can immediately verify that in general
J μ = pμ/p0 ,Jμ ≡ J μ(x);
μ whilst the Dirac current JD for the free electron with fixed p, as already mentioned, is constant:
μ μ 0 JD = p /p = constant , corresponding to no zbw. In other words, our current behaves differently from Dirac’s, even if both of them obey#3 the constraint [cf. eq.(9)]
μ μ 2 0 pμJ = pμJD = m /p . 1#3 μ In the Dirac case, this is obtained by getting from the ordinary Dirac equation, pμγ ψD = mψD, μ 0 the non-linear constraint pμψDγ ψD = mψDψD, and therefore by replacing ψDψD by m/p , consistently −ipx 0 with the ordinary normalization ψD = e up/ 2p ,withupup =2m. 134 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146
It’s noticeable, moreover, that our current J μ goes into the Dirac one, not only in the no-zbw case of eq.(15), but also when averaging it (in time) over a zbw period:
pμ
In the next section we study the kinematical zbw structure of J μ, relative to some given solutions of the NDE: Let us stress that this structure is the same for both J μ and its classical limit mvμ/p0 (and that in the CMF it is mvμ/p0 ≡ vμ). That is to say, the probability current stream-lines correspond just to the classical world-lines of a pointlike particle, in agreement with the Correspondence principle. Consequently, we can study the kinematical features of our J μ by means of the analysis of the time evolution of the four-velocity vμ.
6 Uniform motion solutions of the NDE
Before examining the solutions, let us stress that we do use the term electron#4 to indicate the whole spinning system associated with the geometrical center of the helical trajectories (which corresponds to the center of the electron Coulomb field[17]). let us repeat that this geometrical center is always at rest in the CMF, i.e., in the frame where p and w (but not v ≡ V ) vanish [cf. eqs.(2)]. On the contrary, we shall call Q the pointlike object moving along the helix; we shall refer to it as to the electron “constituent”, and to its (internal) movement as to a “sub-microscopic” motion. We need first of all to make explicit the kinematical definition of vμ, rather different from the common (scalar particle) one. In fact, from the very definition of vμ, one obtains
dt dx dt vμ ≡ dxμ/dτ ≡ (dt/dτ;dx/dτ) ≡ ( ; ) dτ dt dτ
√ √ =(1/ 1 − w2; u/ 1 − w2) , [u ≡ dx/dt] (18) [where, let us recall, w = p/m is the velocity of the CM in the chosen reference frame (that is, in the frame in which the quantities xμ are measured)]. Below, it will be 2 μ convenient to choose as reference frame the CMF (even if quantities as v ≡ vμv are frame invariant); so that [cf. definition (2)]:
μ μ ≡ vCM = V (1; V ) , (19)
wherefrom one deduces for the speed |V | of the internal motion (i.e., for the zbw speed) the new conditions:
0 1#4Usually we speak about electrons, but this theory could be applied to all leptons. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 135 V 2(τ)=0 ⇔ V 2(τ) = 1 (light-like) (19) V 2(τ) < 0 ⇔ V 2(τ) > 1 (space-like) , where V 2 = v2. Notice that, in general, the value of V 2 does vary with τ; except in special cases (e.g., the case of polarized particles: as we shall see). Our NDE in the CMF (where, let us remember, J μ ≡ vμ) can be written as μ 0 (ψγ ψ)i∂μψ = mγ ψ (20) whose general solution is eq.(14b) or, equivalently, eq.(14c). In correspondence to it, we also have [due to eq.(16’)] that J μ = pμ/m + Eμ cos(2mτ)+Hμ sin(2mτ) ≡ V μ (21a) 2 2 2 2 2 2 μ V ≡ J =1+E cos (2mτ)+H sin (2mτ)+2EμH sin(2mτ)cos(2mτ) . (21b) Let us select the solutions ψ of eq.(20) corresponding to constant V 2 and A2,where Aμ ≡ dVμ/dτ ≡ (0; A), quantity V μ ≡ (1; V ) being the zbw velocity. Therefore, we shall suppose in the present frame that quantities V 2 =1− V 2 ; A2 = −A2 are constant in time: V 2 = constant ; A2 = constant , (22) so that V 2 and A2 are constant in time too. (Notice, incidentally, that we are dealing exclusively with the internal motion, in the CMF; thus, our results are quite independent of the global 3-impulse p and hold both in the relativistic and in the non- relativistic case). The requirements (22), inserted⎧ into eq.(21b), yield the following constraint ⎪ ⎨ E2 = H2 (23a) ⎩⎪ μ EμH =0. (23b) Constraints (23) are necessary and sufficient (initial) conditions to get a circular uniform motion (the only uniform and finite motion conceivable in the CMF). Since both E and H do not depend on the time τ, also eqs.(23) hold for any time instant. In the euclidean 3-dimensional space, and at any instant of time, constraints (23) read: ⎧ ⎪ ⎨ A2 =4m2V 2 (24a) ⎪ ⎩ V · A =0 (24b) which correspond to a uniform circular motion with radius R = |V |/2m. (24c) 136 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 Quantity R is the “Zitterbewegung radius”; the zbw frequency was already found to be always Ω = 2m. Quantities E,H (with p0 = m) given in eqs.(16) are functions of the initial spinors ψ(0), ψ(0). Bearing in mind that in the CMF v0 = 1 [cf. eq.(19)] and therefore ψγ0ψ = 1 (which, incidentally, implies the CMF normalization ψ†ψ =1),one gets pμ Eμ ≡− + ψ(0)γμψ(0) = (0; ψ(0)γψ(0)) . (24a) m By substituting into the second one of eqs.(16) the expression (14c) for the general solution of the NDE, we finally have Hμ = iψ(0)[γ0γμ − g0μ]ψ(0) = (0; iψ(0)αψ (0)) , (24b) where γ ≡ (γ1,γ2,γ3), while α ≡ γ0γ and gμν is the metric. Therefore, conditions (23) or (24) can be written in spinorial form, for any time τ, as follows ⎧ ⎪ ⎨ (ψγψ)2 = −(ψαψ)2 (25a) ⎪ ⎩ (ψγψ) · (ψαψ)=0. (25b) At this point, let us show that this classical uniform circular motion, occurring around the z-axis (which in the CMF can be chosen arbitrarily, while in a generic frame is parallel to the global three-impulse p, as we shall see), does just correspond to the ± 1 case of polarized particles with sz = 2 . It may be interesting to notice, once more, that in this case the classical requirements (23) or (24) —namely, the uniform motion ± 1 conditions— play the role of the ordinary quantization conditions sz = 2 .Now,let us first observe [cf. eq.(5a)] that in the CMF the z-component of the spin vector i s = S12 ≡ ψ(τ)[γ1,γ2]ψ(τ) z 4 can actually be shown to be a constant of the motion. In fact, by easy calculations on eq.(14c), one finds sz to be independent of τ: i 1 s (τ)= ψ(0)γ1γ2ψ(0) = ψ(0)Σ ψ(0) ; (26) z 2 2 z where Σ is the spin operator, that in the standard representation reads ⎛ ⎞ ⎜ σ 0 ⎟ ⎜ ⎟ ⎜ ⎟ Σ ≡ ⎜ ⎟ ⎝ ⎠ 0 σ quantities σx,σy,σz being the well-known Pauli 2 × 2 matrices. Then, it is straightforward to realize that the most general spinors ψ(0) satisfying the conditions sx = sy = 0 (27a) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 137 1 1 s = ψ(0)Σ ψ(0) = ± (27b) z 2 z 2 are (in the standard representation) of the form T | ψ(+)(0) = ( a 0 0 d ) (28a) T | ψ(−)(0) = ( 0 b c 0), (28b) respectively; and obey in the CMF the normalization constraint ψ†ψ = 1. [It can − 1 ≤ ≤ 1 be easily shown that, for generic initial conditions, it is 2 sz 2 ]. In eqs.(28) we separated the first two from the second two components, bearing in mind that in the standard Dirac theory (and in the CMF) they correspond to the positive and negative fre- quencies, respectively. With regard to this, let us observe that the “negative-frequency” components c and d do not vanish at the non-relativistic limit [since, let us repeat, it is p = 0 in the CMF]; but the field hamiltonian H is nevertheless positive and equal to m, as already stressed. Now, from relation (28a) we are able to deduce that (with ∗≡complex conjugation) <γ>≡ ψγψ =2(Re[a∗d], +Im[a∗d], 0) <α>≡ ψαψ =2i(Im[a∗d], −Re[a∗d], 0) and analogously, from eq.(28b), that <γ>≡ ψγψ =2(Re[b∗c], −Im[b∗c], 0) <α>≡ ψαψ =2i(Im[b∗c], +Re[b∗c], 0) , which just imply relations (25): ⎧ ⎪ ⎪ <γ>2= − <α>2 ⎨⎪ ⎪ ⎪ ⎩ <γ>· <α>=0. In conclusion, the (circular) polarization conditions eqs.(27) imply the internal zbw motion to be uniform and circular (V 2 = constant; A2 = constant); eqs.(27), in other words, do imply that sz is conserved and quantized, at the same time. Notice that, when passing from the CMF to a generic frame, eq.(27) transforms into 1 Σ · p 1 λ ≡ ψ ψ = ± = constant . (29) 2 |p| 2 Therefore, to get a uniform motion around the p-direction [cf. equations (4c) or (16’)], we have to request that the helicity λ be constant (over space and time), and quantized 1 in the ordinary way, i.e., λ = 2 . We shall come back to the question of the double sign ± 1 2 in the case of the light-like helical trajectories; here, for simplicity, let us confine to the + sign. 138 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 It may be interesting, now, to calculate |V | as a function of the spinor components a and d. With reference to eq.(28a), since ψ†ψ ≡|a|2 + |d|2 = 1, we obtain (for the 1 sz =+2 case): V 2 ≡<γ>2=4|a∗d|2 =4|a|2 (1 −|a|2) (30a) A2 ≡ (2im <α> )2 =4m2V 2 =16m2|a|2 (1 −|a|2) , (30b) and therefore the normalization (valid now in any frame, at any time) ψψ = 1 − V 2 , (30c) which show that to the same speed and acceleration there correspond two spinors ψ(0), related by an exchange of a with d. From eq.(30a) we derive also that, as 0 ≤|a|≤1, it is: 0 ≤ V 2 ≤ 1; 0≤ ψψ ≤ 1 . (30d) ≤ ≤ 1 Correspondingly, from eq.(24c) we obtain for the zbw radius 0 R 2 m. The second one of eqs.(30d) is a new, rather interesting (normalization) boundary condition. From eq.(30c) one can easily see that: (i) for V 2 =0(nozbw)wehave ψψ =1andψ is a “Dirac spinor”; (ii) for V 2 = 1 (light-like zbw) we have ψψ =0and ψ is a “Majorana spinor”; (iii) for 0 < V 2 < 1 we meet, instead, spinors with properties “intermediate” between the Dirac and the Majorana ones. [18] As an example,√ let us write down the “Caldirola√ solution” ψ(0), corresponding to the zbw speed 3/2 and to the zbw radius 3/4m, and yielding correct values for the zero and first order contributions to the electron magnetic moment (for simplicity we chose a and d real): √ 1 3 ψT(0) = ( 0 | 0 ); 2 2 √ as well as that got by interchanging 1/2and 3/2. The “Dirac” case,[2,9] corresponding to V 2 = A2 = 0, that is to say, corresponding to no zbw internal motion, is merely represented (apart from phase factors) by the spinors ψT(0) ≡ (1 0 | 0 0) (31) and (interchanging a and d) ψT(0) ≡ (0 0 | 01). (31) − 1 The spinorial quantities (31), (31’), together with the analogous ones for sz = 2 , satisfy eq.(15): i.e., they are also solutions (in the CMF) of the Dirac eigenvalue equation. This is the unique case in which the zbw disappears, while the field spin is still present! 1 1 In fact, even in terms of eqs.(31)–(31’), one still gets that 2 ψΣzψ =+2 . Since we have been discussing the classical limit (vμ) of a quantum quantity (J μ), let us add that even the well-known change in sign of the fermion wave function, under Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 139 360o-rotations around the z-axis, gets in our theory a natural classical interpretation. In fact, a 360o-rotation of the coordinate frame around the z-axis (passive point of view) is indeed equivalent to a 360o-rotation of the constituent Q around the z-axis (active point of view). On the other hand, as a consequence of the latter transformation, the zbw angle 2mτ does vary of 360o, the proper time τ does increase of a zbw period T = π/m,andthe pointlike constituent does describe a complete circular orbit around z-axis. At this point it is straightforward to notice that, since the period T =2π/m of the ψ(τ) in eqs.(14b)– (14c) is twice as large as the zbw orbital period, the wave function of eqs.(14b)–(14c) does suffer a phase–variation of 180o only, and then does change its sign: as it occurs in the standard theory. To conclude this Section, let us shortly consider the interesting case obtained when releasing the conditions (22)–(25) (and therefore abandoning the assumption of circular uniform motion), requiring instead that: |a| = |c| and |b| = |d| , (32) so to obtain an internal oscillating motion along a constant straight line; where we un- derstood ψ(0) to be written, as usual, ψT(0) ≡ (ab| cd) . For instance, one may choose either 1 ψT(0) ≡ √ (1 0 | 10), (32) 2 or ψT(0) ≡ √1 (1 0 | i 0), or ψT(0) ≡ 1 (1 −1 |−11),orψT(0) ≡ √1 (0 1 | 01), 2 2 2 and so on. In case (32’), for example, one actually gets <γ>=(0, 0, 1) ; <α>=(0, 0, 0) which, inserted into eqs.(24’a), (24’b), yield Eμ =(0;0, 0, 1) ; Hμ =(0;0, 0, 0) . Therefore, because of eq.(21a), we have now a linear, oscillating motion [for which equa- tions (22), (23), (24) and (25) do not hold: here V 2(τ) does vary from 0 to 1!] along the z-axis: Vx(τ)=0; Vy(τ)=0; Vz(τ) = cos(2mτ) . This new case could describe an unpolarized, mixed state, since it is 1 s ≡ ψΣ ψ =(0, 0, 0) , 2 in agreement with the existence of a linear oscillating motion. 140 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 7 The light-like helical case. Let us go back to the case of circular uniform motions (in the CMF), for which con- ditions (22)-(25) hold, with ψ(0) given by eqs.(28). Let us fix our attention, however, on the special case of light-like motion[2]. The spinor fields ψ(0), corresponding to 2 2 | | | | 1 V =0;V = 1, are given by eqs.(28) with a = d for the sz =+2 case, or | | | | − 1 1 b = c for the sz = 2 case; as it follows from eqs.(30) for sz =+2 andfromthe analogous equations V 2 =4|b∗c| =4|b|2(1 −|b|2)(30a) A2 =4m2V 2 =16m2|b|2(1 −|b|2) , (30b) − 1 holding for the case sz = 2 . It can be easily shown that a difference in the phase factors of a and d (or of b and c, respectively) does not change the kinematics, nor the rotation direction, of the motion; but it does merely shift the zbw phase angle at τ =0. Thus, we may choose purely real spinor components (as we did above). As a consequence, the simplest spinors may be written as follows 1 ψT ≡ √ (1 0 | 0 1) (33a) (+) 2 T 1 ψ − ≡ √ (0 1 | 1 0) ; (33b) ( ) 2 and then <γ>(+)=(1, 0, 0) ; <α>(+)=(0, −i, 0) <γ>(−)=(1, 0, 0) ; <α>(−)=(0,i,0) which, inserted into eqs.(24’), yield μ μ E (+) =(0;1, 0, 0) ; H (+) =(0;0, 1, 0) . μ μ E (−) =(0;1, 0, 0) ; H (−) =(0;0, −1, 0) . 1 With regard to eqs.(33), let us observe that eq.(21a) implies for sz =+2 an anti- clockwise[6,2] internal motion, with respect to the chosen z-axis: Vx =cos(2mτ); Vy = sin(2mτ); Vz =0, (34) that is to say ⎧ ⎪ −1 ⎪ X =(2m) sin(2mτ)+X0 ⎨⎪ − −1 Y = (2m) cos(2mτ)+Y0 (34 ) ⎪ ⎪ ⎩⎪ Z = Z0 ; − 1 and a clockwise internal motion for sz = 2 : Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 141 Vx =cos(2mτ); Vy = − sin(2mτ); Vz =0, (35) that is to say ⎧ ⎪ −1 ⎪ X =(2m) sin(2mτ)+X0 ⎨⎪ −1 Y =(2m) cos(2mτ)+Y0 ⎪ ⎪ (35 ) ⎩⎪ Z = Z0 . 1 Let us explicitly observe that spinor (33a), related to sz =+2 (i.e., to an anti- clockwise internal rotation), gets equal weight contributions from the positive–frequency spin-up component and from the negative–frequency spin-down component, in full agree- ment with our “reinterpretation” in terms of particles and antiparticles, given in refs.[15]. − 1 Analogously, spinor (33b), related to sz = 2 (i.e., to a clockwise internal rotation), gets equal weight contributions from the positive-frequency spin-down component and the negative-frequency spin-up component. As we have seen above [cf. eq.(29)], in a generic reference frame the polarized states are characterized by a helical uniform motion around the p-direction; thus, the 1 − 1 λ =+2 [λ = 2 ] spinor will correspond to an anti-clockwise [a clockwise] helical motion with respect to the p-direction. Going back to the CMF, we have to remark that in this case eq.(24c) yields for the zbw radius R the traditional result: |V | 1 λ R = ≡ ≡ , (36) 2m 2m 2 where λ is the Compton wave-length. Of course, R =1/2m represents the maxi- mum (CMF) size of the electron, among all the uniform motion solutions (A2 =const; V 2 =const); the minimum, R = 0, corresponding to the Dirac case with no zbw (V = A = 0), represented by eqs.(31), (31’): that is to say, the Dirac free electron is a pointlike, extensionless object. Finally, let us to underline that the present light-like solutions, among all the uniform motion, polarized state solutions, are likely the most suitable solutions for a complete and “realistic” (i.e., classically meaningful) picture of the free electron. Really, the uniform circular zbw motion with speed c seems to be the sole that allows us —if we think the electric charge of the whole electron to be carried around by the internal con- stituent Q— to obtain the electron Coulomb field and magnetic dipole (with the correct strength μ = e/2m), simply by averaging over a zbw period[17,19 the electromagnetic field generated by the zbw circular motion of Q (and therefore oscillating with the zbw fre- quency). Moreover, only in the light-like case the electron spin can be actually regarded as totally arising from the internal zbw motion, since the intrinsic term Δμν entering the BZ theory[1] does vanish when |V | tends to c. 142 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 8 Generalization of the NDE for the non-free cases. Let us eventually pass to consider the presence of external electromagnetic fields: Aμ = 0. For the non-free case, Barut and Zanghi[1] proposed the following lagrangian 1 L = i(ψψ˙ − ψψ˙)+p (˙xμ − ψγμψ)+eA ψγμψ (37) 2 μ μ which in our opinion should be better rewritten in the following form, obtained directly from the free lagrangian via the minimal prescription procedure: 1 L = i(ψψ˙ − ψψ˙)+(p − eA )(x ˙ μ − ψγμψ), (38) 2 μ μ all quantities being expressed as functions of the (CMF) proper time τ, and the general- ized impulse being now pμ − eAμ. We shall call as usual F μν ≡ ∂μAν − ∂νAμ. Lagrangian (38) does yield, in this case, the following system of differential equations: ⎧ ⎪ ⎪ ψ˙ + i(p − eA )γμψ =0 (39a) ⎨⎪ μ μ x˙ μ = ψγμψ (39b) ⎪ ⎪ ⎩ p˙μ − eA˙ μ = eF μνx˙ μ . (39c) As performed in Sect.4, we can insert the identity (10) into eqs.(39a), (39b), and exploit the definition of the velocity field, eq.(39c). We easily get the following five first– order differential equations (one scalar plus one vector equation) in the five independent variables ψ and pμ: ⎧ ⎪ ⎨ μ μ i(ψγ ψ)∂μψ =(pμ − eAμ)γ ψ (40a) ⎩⎪ ν μ μ μν (ψγ ψ)∂ν(p − eA )=eF ψγνψ, (40b) which are now field equations (quantities ψ, ψ, p and A being all functions of xμ). The solutions ψ(x) of system (40) may be now regarded as the classical spinorial 1 μ fields for relativistic spin- 2 fermions, in presence of an electromagnetic potential A =0. We can obtain from eqs.(40) well-defined time evolutions, both for the CMF velocity pμ/m and for the particle velocity vμ. A priori, by imposing the condition of finite mo- tions, i.e., v(τ)andp(τ) periodic in time (and ψ vanishing at spatial infinity), one will be able to find a discrete spectrum, out from the continuum set of solutions of eqs.(40). Therefore, without solving any eigenvalue equation, within our field theory we can indi- viduate discrete spectra of energy levels for the stationary states, in analogy with what we already found in the free case (in which the uniform motion condition implied the z-componens sz of spin s to be discrete). The case of a uniform, external magnetic field has been treated in ref.[16], where —incidentally— also the formal resolvent of system (39) is given. We shall expand on this point elsewhere: having in mind, especially, the applications to classical problems, so as the hydrogen atom, the Stern and Gerlach Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 143 experiment, the Zeeman effect, and tunnelling through a barrier. 9 Conclusions. In this paper we have analyzed and studied the electron internal motions (predicted by the BZ theory), as functions of the initial conditions, for both time-like and space- like speeds. In so doing, we revealed the noticeable new kinematical properties of the 2 μ velocity field for the spinning electron. For example, we found that quantity v ≡ vμv may assume any real value and be variable in time, differently from the ordinary (scalar particle) case in which it is always either v2 =1orv2 = 0. By requiring v2 to be constant in time (uniform motions), we found however that [since the CMF does not coincide with the rest frame!] quantity v2 can assume all values in the interval 0 to 1, depending on the value of ψ(0). Moreover, this paper shows clearly the correlation existing between electron polar- ization and Zitterbewegung. In particular, in Sect.6 we show that the requirement ± 1 s =(0, 0, 2 ) [i.e., that the classical spin magnitude, corresponding to the average quan- 1 Q tum spin magnitude, be 2 ] corresponds to the requirement of uniform motions for ,and vice-versa. We found the zbw oscillation to be a uniform circular motion, with frequency ω =2m, and with an orbital radius R = |v|/2m that in the case of light-like orbits equals half the Compton wave-length; the clockwise (anti-clockwise) helical motions cor- responding to the spin-up (spin-down) case. We introduced also an equation for the 4-velocity field (different from the correspond- ing equation of Dirac theory), which allows an intelligible description of the electron internal motions: thus overcoming the well-known problems about the physical mean- ing of the Dirac position operator and its time evolution.[20] Indeed, the equation for the standard Dirac case was devoid of any intuitive kinematical meaning, because of the appearance of the imaginary unit i in front of the acceleration (which is related to the non-hermiticity of the velocity operator in that theory). Finally, we have considered a natural generalization of the NDE for the non-free case, which allows a classical description of the interaction between a relativistic fermion and an external electromagnetic field (a description we shall deepen elsewhere). Acknowledgements The authors acknowledge continuous stimulating discussions with P. Cardieri, P.L. Dias Peres, W. de Freitas Filho, A. Pagano, M. Pavˇsiˇc, M. Zamboni-Rached, and particularly Hugo E. Hern/’andez-Figueroa. Thanks for helpful collaboration and interest are also due to C. Giardini, I. Licata, P. Riva, C. Rizzi, and particularly to S. Paleari and (for his quite generous editorial help) to A. Sakaji. At last, we would like to recall the work performed by Waldyr A. Rodrigues Jr. (recently passed away), who contributed to the creation of research groups in theoretical physics, which paid particular attention to the 144 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 125–146 structure of spinning particles, to magnetic poles, and to the use of Clifford algebras in space-time physics. .. References [1] A.O. Barut and N. Zanghi: Phys. Rev. Lett. 52 (1984) 2009. See also A.O. Barut andA.J.Bracken:Phys.Rev.D23 (1981) 2454; D24 (1981) 3333. [2] M. Pavˇsiˇc, E. Recami, W.A. Rodrigues, G.D. Maccarrone, F. Raciti and G. Salesi: Phys. Lett. B318 (1993) 481. See also W.A. Rodrigues, J. Vaz, E. Recami and G. Salesi: Phys. Lett. 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EJTP 14, No. 37 (2018) 145–160 Electronic Journal of Theoretical Physics Solutions to the Gravitational Field Equations in Curved Phase-Spaces Carlos Castro∗ Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia. 30314, USA Received 4 June 2017, Accepted 20 August 2017, Published 20 April 2018 Abstract: After reviewing the basics of the geometry of the cotangent bundle of spacetime, via the introduction of nonlinear connections, we build an action and derive the generalized gravitational field equations in phase spaces. A nontrivial solution generalizing the Hilbert- Schwarzschild black hole metric in spacetime is found. The most relevant physical consequence is that the metric becomes momentum-dependent (observer dependent) which is what one should aim for in trying to quantize geometry (gravity) : the observer must play an important role in any measurement (observation) process of the spacetime he/she lives in. To finalize, some comments about modifications of the Weyl-Heisenberg algebra [xi, pj]=i gij(x, p)andits implications are made. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Gravity; Finsler Geometry; Born Reciprocity; Phase Space PACS (2010): 11.10.-z; 04.40.Nr; 11.25.-w; 04.20.-q 1 Introduction : Quantum Gravity and Curved Phase Space In the first introduction to Quantum Mechanics we are exposed to the Weyl-Heisenberg i i algebra given by the commutators [x , pj]=i δj of the coordinate and momentum oper- ators, and which hold the key behind Heisenberg’s uncertaintity principle via the relation i ≥ 1 | i | Δx Δpj 2 < [x , pj] > , after taking expectation values. Inspired from the results obtained in the very high energy limit of string scattering amplitudes [1], a lot of work has been devoted in the past two decades to deformations of the Weyl-Heisenberg algebra [17], and which is associated to a generalized uncertaintity principle (GUP) leading to the notion of a minimal length scale (of the order of the Planck length). The strings begin ∗ Email: [email protected] 146 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 to grow in size when trans-Planckian energies are reached, rather than probing smaller and smaller distances. Most of the work devoted to Quantum Gravity has been focused on the geometry of spacetime rather than phase space per se. The first indication that phase space should play a role in Quantum Gravity was raised by [2]. The principle of Born’s reciprocal rela- tivity [2] was proposed long ago based on the idea that coordinates and momenta should be unified on the same footing, and consequently, if there is a limiting speed (temporal derivative of the position coordinates) in Nature there should be a maximal force as well, since force is the temporal derivative of the momentum. A maximal speed limit (speed of light) must be accompanied with a maximal proper force (which is also compatible with a maximal and minimal length duality). The generalized velocity and acceleration i boosts/rotations transformations (where x ,pi; i =1, 2, 3, 4 are all boosted/rotated into each-other) and which leave invariant the infinitesimal line interval of the 8D flat phase space, were given by [3] based on the group U(1, 3) and which is the Born version of the Lorentz group SO(1, 3). The extension of these transformations to Noncommutative phase spaces was analyzed in [12] We explored in [7] some novel consequences of Born’s reciprocal Relativity theory in flat phase-space and generalized the theory to the curved spacetime scenario. We provided, in particular, six specific results resulting from Born’s reciprocal Relativity and which are not present in Special Relativity. These are : momentum-dependent time delay in the emission and detection of photons; energy-dependent notion of locality; superluminal behavior; relative rotation of photon trajectories due to the aberration of light; invariance of areas-cells in phase-space and modified dispersion relations. A discussion of Mach’s principle within the context of Born Reciprocal Gravity in Phase Spaces was described in [14]. The Machian postulate states that the rest mass of a particle is determined via the gravitational potential energy due to the other masses in the 2 universe. It is also consistent with equating the maximal proper force mPlanck(c /LPlanck) 2 to MUniverse(c /RHubble) and reflecting a maximal/minimal acceleration duality. By in- voking Born’s reciprocity between coordinates and momenta, a minimal Planck scale should correspond to a minimum momentum, and consequently to an upper scale given by the Hubble radius. Further details can be found in [14]. It is better understood now that the Planck-scale modifications of the particle dis- persion relations can be encoded in the nontrivial geometrical properties of momentum space [16]. When both spacetime curvature and Planck-scale deformations of momentum space are present, it is expected that the nontrivial geometry of momentum space and spacetime get intertwined. The interplay between spacetime curvature and non-trivial momentum space effects was essential in the notion of “relative locality” and in the deep- ening of the relativity principle [16]. Recently the authors [18] described the Hamilton geometry of the phase space of particles whose motion is characterized by general dis- persion relations. Explicit examples of two models for Planck-scale modified dispersion relations, inspired from the q-de Sitter and κ-Poincare quantum groups, were considered. In the first case they found the expressions for the momentum and position dependent Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 147 curvature of spacetime and momentum space, while for the second case the manifold is flat and only the momentum space possesses a nonzero, momentum dependent curvature. The purpose of this work is to study deeper the geometry of the cotangent bundle in order to derive the analog of Einstein’s field equations in curved phase spaces, and construct specific solutions. The curved phase-space geometry of the cotangent bundle of spacetime can be explored via the introduction of nonlinear connections which are associated with certain nonholonomic modifications of Riemann–Cartan gravity within the context of Finsler geometry. The geometry of the 8D tangent bundle of 4D space- time and the physics of a limiting value of the proper acceleration in spacetime [6] has been studied by Brandt [4]. Generalized 8D gravitational equations reduce to ordinary Einstein-Riemannian gravitational equations in the infinite acceleration limit. We must emphasize that the results found in this work are very different than those obtained earlier by us in [13] and by [4] [11], [9], among others. In the next section we review the geometry of the cotangent bundle of spacetime, build an action and derive the generalized field equations. In the final section we find some nontrivial solutions generalizing the Hilbert-Schwarzschild black hole solution in space- time. The most relevant consequence is that the metric becomes momentum-dependent (observer dependent) which is what one should aim for in trying to quantize geometry (gravity) : the observer must play an important role in any measurement (observation) process of the spacetime he/she lives in. 2 Field Equations in Curved Phase Spaces In the first part of this section we shall review the geometry of the cotangent bundle case T ∗M (phase space) following the monographs by [11]. Readers familiar with this material can proceed to the second part of this section where we derive the field equations. A classical treatise on the Geometry of Phase Spaces can be found in [8]. In the case of ∗ the cotangent space of a d-dim manifold T Md the metric can be equivalently rewritten in the block diagonal form [11] as 2 k i j ab k (ds) = gij(x ,pa) dx dx + h (x ,pc) δpa δpb = k i j k a b gij(x ,pa) dx dx + hab(x ,pc) δp δp (2.1) i, j, k =1, 2, 3, .....d, a, b, c =1, 2, 3, .....d, if instead of the standard coordinate basis one introduces the following anholonomic frames (non-coordinate basis) ∂ i a a ≡ δi = δ/δx = ∂xi + Nia ∂ = ∂xi + Nia ∂pa ; ∂ ∂pa = (2.2) ∂pa One should note the key position of the indices that allows us to distinguish between i derivatives with respect to x and those with respect to pa. The dual basis of (δi = i a δ/δx ; ∂ = ∂/∂pa)is i − j a a − a j dx ,δpa = dpa Nja dx ,δp = dp Nj dx (2.3) 148 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 where the N–coefficients define a nonlinear connection, N–connection structure. An N-linear connection D on T ∗M can be uniquely represented in the adapted basis in the following form k a − a b Dδj (δi)=Hij δk; Dδj (∂ )= Hbj ∂ ;(2.4a) ka b − ba c D∂a (δi)=Ci δk; D∂a (∂ )= Cc ∂ (2.4b) k a ka ba where Hij(x, p),Hbj(x, p),Ci (x, p),Cc (x, p) are the connection coefficients. Our nota- tion for the derivatives is a i a ∂ = ∂/∂pa,∂i = ∂xi ,δi = δ/δx = ∂xi + Nia ∂ (2.4c) The N–connection structures can be naturally defined on (pseudo) Riemannian space- times and one can relate them with some anholonomic frame fields (vielbeins) satisfying − γ the relations δαδβ δβδα = Wαβδγ. The only nontrivial (nonvanishing) nonholonomy coefficients are a a − a Wija = Rija; W jb = ∂ Njb = Wjb (2.5a) and Rija = δjNia − δiNja (2.5b) is the nonlinear connection curvature (N–curvature). ab Imposing a zero nonmetricity condition of gij(x, p),h (x, p) along the horizontal and vertical directions, respectively, gives − l − l Digjk = δi ggk Hij glk Hik gjl =0, (2.6a) a bc a bc ab dc ac bd D h = ∂ h + Cd h + Cd h =0 (2.6b) Performig a cyclic permutation of the indices in eqs-(2.6), followed by linear combination of the equations obtained yields the irreducible (horizontal, vertical) h-v-components for the connection coefficients 1 Hi = gin (δ g + δ g − δ g )(2.7) jk 2 k nj j nk n jk 1 Cab = − h ∂bhad + ∂ahbd − ∂dhab (2.8) c 2 cd ab a The additional conditions Dih =0,D gij = 0, yield the mixed components of the connection coefficients 1 Ha = hac δ h − hac h ∂dN + ∂aN (2.9) bj 2 j bc bd jc jb ac ac (after using h δjhbc = −(δjh ) hbc) 1 Cja = gjk ∂ag (2.10) i 2 ik Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 149 For any N-linear connection D with the above coefficients the torsion 2-forms are 1 Ωi = T i dxj ∧ dxk + Cia dxj ∧ δp (2.11a) 2 jk j a 1 1 Ω = R dxj ∧ dxk + P b dxj ∧ δp + Sbc δp ∧ δp (2.11b) a 2 jka aj b 2 a b c and the curvature 2-forms are 1 1 Ωi = Ri dxk ∧ dxm + P ia dxk ∧ δp + Siab δp ∧ δp (2.12) j 2 jkm jk a 2 j a b 1 1 Ωa = Ra dxk ∧ dxm + P ac dxk ∧ δp + Sacd δp ∧ δp (2.13) b 2 bkm bk c 2 b c d i a i where one must recall that the dual basis of δi = δ/δx ,∂ = ∂/∂pa is given by dx ,δpa = j dpa − Njadx . The distinguished torsion tensors are i i − i ab ab − ba ia ia − ia Tjk = Hjk Hkj; Sc = Cc Cc ; Tj = Cj = T j a a − a a − a Pbj = Hbj ∂ Njb,Pbj = Pbj δN δN R = ja − ia (2.14) ija δxi δxj The distinguished tensors of the curvature are i i − i l i − l i − ia Rkjh = δhHkj δjHkh + Hkj Hlh Hkh Hlj Ck Rjha (2.15) ab a b ad b − ab b da a bd − d ab Pcj = ∂ Hcj + Cc Pdj δj Cc + Hdj Cc + Hdj Cc Hcj Cd (2.16) ak a k al k − ak a bk k al − l ak Pij = ∂ Hij + Ci Tlj δj Ci + Hbj Ci + Hlj Ci Hij Cl (2.17) abc c ab − b ac eb ac − ec ab Sd = ∂ Cd ∂ Cd + Cd Ce Cd Ce ;(2.18) ibc c bi − b ci bh ci − ch bi Sj = ∂ Cj ∂ Cj + Cj Ch Cj Ch (2.19) a a − a c a − c a − ca Rbjk = δkHbj δjHbk + Hbj Hck Hbk Hcj Cb Rjkc (2.20) Equipped with these curvature tensors one can perform suitable contractions involving ij g ,hab to obtain two curvature scalars of the R, S type R j i kl S d abc = δi Rkjl g ; = δb Sd hac (2.21) 150 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 and construct a 2d-dim gravitational phase space action involving a linear combination of the curvature scalars 1 S = ddxddp |det g| |det h| ( c R + c S ) (2.22) 2κ2 1 2 where c1,c2 are real-valued numerical coefficients (with the appropriate physical units) and κ2 is the analog of gravitational coupling constant in phase space. We shall fix c1 = c2 = 1. The metric in the nonholonomic (non-coordinate basis) is block diagonal as described by eq-(2-1). As a result the determinant factorizes into a product giving for measure |det g| |det h|. Other measures of integration are possible as well as many more actions besides (2.22). For instance, one may add curvature and torsion squared terms. In natural units = c = 1, the physical units are fixed by ensuring the action is dimensionless and such that c1R has the same physical units as c2S. The reason one is not adding to the action the other curvature contractions involving the remaining components c ab i ak b a i jbc δa Pcj ,δk Pij ,δa Rbjk,δj Si (2.23) is that the latter two curvature contractions are antisymmetric in the jk,bc indices, kl respectively. Thus a further contraction with g ,hbc will be identically zero, so one will not be able to include these curvature components into an action linear in the curva- ture unless there are antisymmetric components to the metrics. One could introduce antisymmetric metrics but for the moment this would be the subject of a future investi- gation. The first two curvature contractions in (2.23) could be contracted further if one had at our disposal a second rank tensor with mixed upper and lower indices, but such ab tensor is not available. One cannot use Niah because the nonlinear connection does not transform as a tensor. Given the action linear in curvatures, the vacuum field equations associated with the geometry of the cotangent bundle are δS δS δS =0, =0, =0 (2.24) ab δgij δh δNia When i, j =1, 2,...,d,anda, b =1, 2,...,d the number of field equations is 1 1 2d(2d +1) d(d +1) + d(d +1) + d2 = (2.25) 2 2 2 which match the number of independent degrees of freedom of a metric gMN in 2d- dimensions. One should emphasize that there is no mathematical equivalence of the above eqs-(2.24) with the ordinary Einstein vacuum field equations in a Riemannian spacetime of 2d-dimensions 1 R (X) − g (X) R(X)=0; M,N =1, 2, 3, ...... , 2d (2.26) MN 2 MN one of the reasons being is the nontrivial presence of the nonlinear connection Nia(x, p). Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 151 The variation of the action δS is more complicated than usual due to the fact that the δ ··· ∼ variation does not commute with the elongated derivatives : the commutator [δ, δxi ]( ) ∂ (δNia) (···) =0. ∂pa d d M Furthermore, the integral d xdp |g||h| DM (J )isno longer a total derivative leading to boundary terms which can be dropped when the fields vanish fast enough at infinity. The reason being that the covariant horizontal derivative operator Di is defined δ ∂ in terms of the elongated noncommuting derivatives i = ∂ i +N . For these reasons δx x ia ∂pa we shall bypass the more complicated variational procedure of eqs-(2.24) and instead recur to the Bianchi identities in order to derive the field equations. The Bianchi identities in the absence of torsion for the horizontal and vertical curva- ture tensors are [11] R m R m R m (Di ) jkl +(Dk ) jli +(Dl ) jik =0 (2.27) S m S m S m (Da ) bcd +(Dc ) bda +(Dd ) bac =0 (2.28) In the presence of torsion the Bianchi identities are modified by the inclusion of torsion- curvature terms in the right hand side, and the field equations are more complicated. From the Bianchi identities in the absence of torsion (and when the nonmetricity is zero) one can retrieve Einstein’s tensor, as usual, by performing two successive contractions of the indices in eqs-(2.27,2.28) giving 1 Di(2R − g R )=0⇒ Di( R − g R )=0 (2.29) ij ij ij 2 ij 1 Da(2S − h S )=0⇒ Da( S − h S )=0 (2.30) ab ab ab 2 ab Therefore, the field equations consistent with the Bianchi identities in the absence of torsion (and for zero nonmetricity) are given by 1 1 R − g R = T (H), S − h S = T (V ) (2.31) ij 2 ij ij ab 2 ab ab (H) (V ) where Tij ,Tab are the conserved energy-momentum tensors in the horizontal and ver- tical space, respectively. The vacuum field equations are then 1 1 R − g R =0, S − h S =0 (2.32) ij 2 ij ab 2 ab and which are equivalent to the Ricci flat conditions obtained after taking the trace of eqs-(2.32) Rij =0, Sab =0 (2.33) We must supplement the above equations with the vanishing torsion conditions i i − i ab ab − ba Tjk = Hjk Hkj =0; Sc = Cc Cc =0, ia ia − ia Tj = Cj = T j =0 152 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 a a − a − a Pbj = Hbj ∂ Njb = Pbj =0 δN δN R = ja − ia =0 (2.34) ija δxi δxj 3 Some Solutions to Gravity in Curved Phase Spaces Next we shall find some solutions to the above eqs-(2.33) for the metric fields gij(x, p),hab(x, p) (inverse of hab) in the absence of torsion (2.34) (and zero nonmetricity). In the partic- ular case when the nonlinear connection is Nia = 0, and the metric components depend solely on x and p, respectively, gij(x),hab(p), while the connection coefficients are given in eqs-(2.7-2.10), one can verify that all the torsion components (2.34) are zero. Thus, a static spherically symmetric metric, consistent with Born’s reciprocity principle (x ↔ p), ∗ and which is a solution of the field equations (2.33) in the 8D cotangent bundle T M4 associated with a 4D spacetime is 2 2GM 2 2GM −1 2 2 2 (ds) = −(1 − )(dt) +(1− ) (dr) + r (dΩ(x)) + r r 2M 2M κ−4 −(1 − )(dE)2 +(1− )−1 (dpr)2 +(pr)2 (dΩ )2 (3.1) pr pr (p) the spacetime and momentum space infinitesimal solid angle elements are respectively 2 2 2 2 2 (dΩ(x)) = r ( sin θ(x) (dφ(x)) +(dθ(x)) ), 2 r 2 2 2 2 (dΩ(p)) =(p ) ( sin θ(p) (dφ(p)) +(dθ(p)) )(3.2) care must be taken not to confuse the angles. In particular, for the momentum variables in spherical coordinates one has pr = (px)2 +(py)2 +(pz)2, pt = E,and z r x r y r p = p cos(θ(p)),p = p sin(θ(p)) cos(φ(p)),p = p sin(θ(p)) sin(φ(p))(3.3) κ has mass units and can be equated to the Planck’s mass ; i.e. inverse of the Planck scale L. The Newtonian gravitational coupling G = L2. The above metric is the phase space counterpart of the Hilbert-Schwarzschild metric. If we were to set κ = ∞,the pre-factor in front of the second line in eq-(3.1) collapses to zero and the momentum- space metric components would degenerate to zero. To avoid a degenerate metric in the momentum space would require a cutoff κ = ∞, and consequently L = 0. This requirement is compatible with the minimal Planck length postulate of the literature [15], [16]. The metric solution (3.3) is in a sense trivial since there is no entanglement among x and p. The solution is simply a “diagonal” sum of a spacetime and momentum metric. As expected, one finds (i) when pr = 0, eq-(3-3) yields a singularity in momentum space, while r = ∞ leads to an asymptotically flat spacetime metric. The interesting feature is that low values of Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 153 the momentum correspond to the interior region (inside the momentum horizon) of the momentum space. Meaning that the location of the momentum horizon signals a natural infrared cutoff in the values of the momentum . (ii) when r = 0, eq-(3-3) yields a black hole singularity in the underlying spacetime, while pr = ∞ leads to an asymptotically flat momentum space metric. One can construct a nontrivial phase space metric solution to the vacuum field equa- tions, and which is given in the block diagonal form described by the second line of eq-(2.1), as follows 2 2GM 2 2GM −1 2 2 2 (ds) = −(1 − )(dt) +(1− ) (dr) + r (dΩ(x)) + r r r 2 2M (∂ r ρ(r, p )) r κ−4 −(1− )(dE)2 + p ( dpr − N p (r, pr) dr )2 + ρ(r, pr) − 2M r (1 ρ(r,pr) ) −4 r 2 2 κ ρ(r, p ) (dΩ(p)) (3.4) where one has modified the phase space metric by introducing the nonlinear connec- pr r a tion Nr (r, p ) = 0 (after setting all the other components of Ni = 0) and inserting the function ρ(r, pr) which plays the role of the area radial-momentum function. The area radial-momentum function ρ(r, pr) can be determined in addition to the nonlinear pr r connection Nr (r, p ) by solving eqs- (2.34) as follows. Firstly, an entire Appendix is devoted to show explicitly that the momentum space metric components hab(x, p) solve the Ricci flat Sab = 0 flat eqs-(2.33). The gij(x) components (Hilbert-Schwarzschild solution) solve the Ricci flat Rij = 0 eqs-(2.33). The next step is to impose the zero torsion conditions which will allow to determine ρ(r, pr) a r pr r a and Ni (r, p ). Setting Nr (r, p ) =0,andall the other components Ni = 0, simplifies dramatically the zero torsion conditions (2.34). The nontrivial equations turn out to be pr pr 1 prpr ∂ pr ∂ ∂Nr P r = h ( + N ) h r r − =0 (3.5) p r 2 ∂r r ∂pr p p ∂pr pt 1 ptpt ∂ pr ∂ P = h ( + N ) h t t =0 (3.6) pt r 2 ∂r r ∂pr p p Therefore one ends up with the above two differential equations for the sought-after two r pr r functions ρ(r, p )andNr (r, p ) after substituting −4 2M ptpt 1 h t t = − κ (1 − ),h = (3.7) p p r ρ(r, p ) hptpt r 2 −4 (∂pr ρ(r, p )) prpr 1 hprpr = κ 2M ,h = (3.8) − h r r (1 ρ(r,pr) ) p p r r pr r r into eqs-(3.5,3.6). The factorization condition ρ(r, p )=Ψ(r)Ξ(p ); Nr (r, p )=p F (r) allows us to integrate the second equation (3.6) giving r r r − pr r r ρ(r, p )=p exp dr F (r ) ,Nr (r, p )=p F (r)(3.9) 0 154 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 Inserting these solutions into the first equation (3.5) yields, after some algebra, that pr r r F (r) = 0 and one ends up with the trivial solution Nr =0andρ(r, p )=p . For this reason we should discard this trivial factorization and look for non factorizable solutions of eqs-(3.5, 3.6). Nontrivial solutions to the field equations in the four-dim tangent bundle of a two-dim spacetime have been found by [10]. They require arbitrary integrating functions and generating functions. In general, when the nonlinear connection is Nia(x, p) = 0, but still obeys the zero ab torsion conditions (2.34), the metric components gij(x, p),h (x, p) obeying the field equa- i tions will have more flexibility and freedom to depend on both the x ,pa coordinates of phase space. A special class of solutions consistent with all the zero torsion conditions ab (2.34) when Nia(x, p) =0areoftheform gij(x),h (x, p). The physical relevance of the solutions (3.4) is that when one works with holonomic coordinates, the metric is no longer block diagonal as in eq-(2.1), but instead is given by 2 ab i j (ds) = gij(x)+h (x, p) Nia(x, p) Njb(x, p) dx dx + ab ab i ab j h (x, p) dpa dpb − Nib(x, p) h (x, p) dx dpa − Nja(x, p) h (x, p) dx dpb (3.10) Consequently the effective spacetime metric is now momentum-dependent; i.e. observer dependent eff ab gij (x, p)=gij(x)+h (x, p) Nia(x, p) Njb(x, p)= a b gij(x)+hab(x, p) Ni (x, p) Nj (x, p)(3.11) To sum up, the metric (3.4) leads to momentum-dependent (observer dependent) modi- fications of the radial components of the Hilbert-Schwarzschild solution eff r pr r pr r grr (r, pr)= grr(r)+hprpr (r, p ) Nr (r, p ) Nr (r, p )(3.12) r r pr r where hprpr (r, p ) is given by eq-(3.8) and ρ(r, p ),Nr (r, p ) are nontrivial solutions of eqs-(3.5, 3.6). The possibility that the underlying spacetime geometry might become observer dependent was envision also by Gibbons and Hawking long ago. We should note that this curved phase-space procedure is not the same as the Rainbow gravity approach proposed in the literature after inserting, by hand, extra momentum- dependent scalar factors f(E,p) into the spacetime metric components in order to modify the energy-momentum dispersion relations. The immediate extension of this work is to introduce matter. Brandt [5] has studied the wave equations of scalar fields Φ(xμ,vμ)in the spacetime tangent bundle and found that by complexifying the coordinates zμ = xμ + iL vμ a natural UV (ultraviolet) regulator L in the space of solutions of the wave equations exists at the Planck scale. The possibility that a fundamental physical theory might provide a physical cutoff for field theory was speculated long ago by Landau and others. The regularization of quantum fields in complex spacetimes have been studied in particular by [19]. String and p-branes propagating in spacetime tangent bundle backgrounds were briefly studied as well by Brandt [5]. Accelerated strings in tangent bundle backgrounds were Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 155 studied in further detail by [20]. The worldsheet associated with those accelerated open strings envisages a continuum family of worldlines of accelerated points. It is when one embeds the two-dim string worldsheet into the tangent bundle TM background (associ- ated with a uniformly accelerated observer in spacetime) that the effects of the maximal acceleration are manifested. The induced worldsheet metric as a result of this embedding has a null horizon. It is the presence of this null horizon that limits the acceleration values of the points inside string. If the string crosses the null horizon some of its points will exceed the maximal acceleration value c2/L and that portion of the string will be- come causally disconnected from the rest of string outside the horizon. We also found a modified Rindler metric which has a true curvature singularity at the location of the null horizon due to a finite maximal acceleration c2/L. This might have important physical consequences in the construction of generalized QFT in accelerated frames and the black hole information paradox. To finalize, let us go back to the beginning of the introduction and postulate the fol- lowing modification of the Weyl-Heisenberg algebra [xi, pj]=i gij(x, p), in addition to [xi, xj]=[pi, pj] = 0, with the provision that the above commutators obey the Jacobi identities [17] . There is a vast literature devoted to deformations of the Weyl-Heisenberg algebra, deformed Quantum Mechanics, generalized uncertainty principle, Noncommuta- tive geometry, discretization of space and time, ···, see references in [21]. The idea behind the above modified commutators is that in the classical limit one will obtain c-number expressions for the metric gij(x, p). The next question would be to verify whether or not such metrics gij(x, p) obey the generalized field equations for gravity in curved phase spaces. APPENDIX : Momentum Space Schwarzschild-like solutions in D>3 We will show below that the momentum space metric components of eq-(3.4) obey the momentum space Ricci flat conditions Sab = 0. In this Appendix we shall find the most general static spherically symmetric vacuum solutions to the momentum space Ricci flat Sab = 0 equations ( ) in any momentum space of dimension D>3. The phase space is 2D-dim. Let us start with the momentum space line element with signature (−, +, +, +, ...., +) 2 − μ(r,pr) 2 ν(r,pr) r 2 r 2 ˜ a b (ds)(p) = e (dE) + e (dp ) + ρ(r, p ) hab dξ dξ . (A.1) r ˜ where the area radial-momentum function is ρ(r, p ). The metric components hab corre- ˜ spond to a D − 2-dim homogeneous space. The indices of hab span a, b =3, 4, ..., D − 2, whereas the temporal and radial indices are denoted by 1, 2 respectively. The only non-vanishing Christoffel symbols in momentum space are given in terms of the fol- lowing partial derivatives with respect to the radial momentum pr variable, and are denoted with a prime. These derivatives must not be confused with derivatives with respect to the radial variable r in spacetime. Therefore our notation here is such that r r r ρ (r, p ) ≡ ∂pr ρ(r, p ); μ ≡ ∂pr μ(r, p ), etc ... Eq-(2.8) yields 156 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 1 1 2 1 2 1 μ−ν C21 = 2 μ ,C22 = 2 ν ,C11 = 2 μ e , (A.2) 2 − −ν ˜ a ρ a a ˜a ˜ Cab = e ρρ hab,C2b = ρ δb ,Cbc = Cbc(hab), and the only nonvanishing momentum space curvature tensor components are S1 − 1 − 1 2 1 S1 − 1 −ν ˜ 212 = 2 μ 4 μ + 4 ν μ , a1b = 2 μ e ρρ hab, S2 μ−ν 1 1 2 − 1 S2 −ν 1 − ˜ (A.3) 121 = e ( 2 μ + 4 μ 4 ν μ ), a2b = e ( 2 ν ρρ ρρ )hab, Sa ˜a − 2 −ν a˜ − a˜ bcd = Rbcd ρ e (δc hbd δd hbc). The vacuum field equations are 1 1 1 (D − 2) ρ S = eμ−ν( μ + μ2 − μν + μ )=0, (A.4) 11 2 4 4 2 ρ 1 1 1 1 ρ ρ S = − μ − μ2 + μν +(D − 2)( ν − )=0, (A.5) 22 2 4 4 2 ρ ρ and e−ν 1 k S = (ν − μ)ρρ − ρρ − (D − 3)ρ2 h˜ + (D − 3)h˜ =0, (A.6) ab ρ2 2 ab ρ2 ab ˜ where k = ±1, depending if hab refers to positive or negative curvature. From the −μ+ν combination e S11 + S22 =0weget 2ρ μ + ν = . (A.7) ρ The solution of this equation is μ + ν =lnρ2 + C, (A.8) where C is an integration constant that one can set to zero if one wishes to recover the flat momentum space metric in the asymptotic region pr → ∞. Substituting (A.7) into the equation (A.6) we find e−ν ( νρρ − 2ρρ − (D − 3)ρ2 )=−k(D − 3) (A.9) or γρρ +2γρρ +(D − 3)γρ2 = k(D − 3), (A.10) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 145–160 157 where γ = e−ν. (A.11) The solution of (A.10) corresponding to a D − 2-dim homogeneous momentum space of positive curvature (k = 1) can be written as β M ∂ρ γ =(1− D )( )−2 ⇒ r D−3 r (D − 2) ΩD−2 ρ(r, p ) ∂p ν βDM −1 ∂ρ 2 h r r = e =(1− ) ( ) . (A.12) p p r D−3 r (D − 2) ΩD−2 ρ(r, p ) ∂p where ΩD−2 is the appropriate momentum space solid angle in D − 2-dim and βD is a suitable constant. For the most general D − 2-dim homogeneous momentum space we may write β M −ν = ln(k − D ) − 2lnρ (A.13) D−3 (D − 2) ΩD−2 ρ Thus, according to (A.8) we get β M μ = ln(k − D )+constant. (A.14) D−3 (D − 2) ΩD−2 ρ we can set the constant to zero, and this means the momentum space line element (A.1) can be written as 2 β M (∂ r ρ) 2 − − D 2 p r 2 (ds)(p) = (k )(dE) + (dp ) + − D−3 βDM (D 2) ΩD−2 ρ (k − D−3 ) (D−2) ΩD−2 ρ 2 r ˜ a b ρ (r, p ) hab dξ dξ (A.15) InthecaseofaD − 2-dim sphere in momentum space we have k = 1, and the angular r 2 2 part of (A.15) is simply ρ(r, p ) (dΩ(p)) .WhenD = 4, one has βD =16π so that r D−3 r βDM/(D − 2)ΩD−2 ρ(r, p ) ⇒ 2M/ρ(r, p ). It is interesting to observe that the only ˜ effect of the homogeneous metric hab is reflected in the k = ±1 parameter, associated with a positive (negative) constant scalar curvature of the homogeneous D−2-dim momentum space. k = 0 corresponds to a spatially flat D−2-dim section. Concluding, we have shown in the static spherically symmetric case, that the momentum space metric components in eq-(3-4) obey the momentum space Ricci flat conditions Sab =0. Acknowledgements We thank M. 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Khalil and S. Das, “Time crystals from minimum time uncertainty” Eur. Phys. J. C (2016) 76 : 30. EJTP 14, No. 37 (2018) 161–178 Electronic Journal of Theoretical Physics Electromagnetic Media in pp-wave Spacetime Mohsen Fathi∗ Department of Physics, Payame Noor University (PNU), PO BOX 19395-3697 Tehran, Iran Received 4 June 2017, Accepted 20 December 2017, Published 20 April 2018 Abstract: The physical configuration of a dielectric transformation media is investigated, whilst it is subjected in a spacetime formed by gravitational waves, namely the pp-wave. Furthermore, such media is also manipulated to resemble the spacetime itself, and we deal with peculiar cases of transformations, exploited by the media. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Transformation Optics; Dielectric Analog; Gravitational waves PACS (2010): 04.20.Jb; 04.30.Nk; 42.25.Dd; 42.15.Eq 1 Introduction The connection between theory of transformation optics (TO) and engineered metamate- rials [1, 2, 3] stems from the capability of these materials to be designed flexibly, to have variety of applications which are proposed by the theory. These may include cloaking devices [4], invisibility devices [5, 6] and perfect lenses [7]. It may also become important to develop a joint between general theory of relativity and man-made metamaterials. The importance of general relativistic modifications has been noted in engineering [8], for example in Global Positioning System (GPS). However, when fabrication of an optical device is desired which has to undergo general relativistic effects (for example, an orbit- ing telescope containing a superlens [9], or a satellite antenna based on TO [10]), these modifications has to be applied in interpreting the behavior of electromagnetic fields in materials, when for instance it is considered in curved spacetime [11]. Here it should be noted that despite the fact that general relativity and TO basically share a same mathematical language, it is a crucial task to fully understand an optical device in the context of general relativity, since TO is discussed on a fixed background, whereas general relativity is a dynamic theory of spacetime. ∗ Email: [email protected]; [email protected] 162 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 This is the original idea of transformation optics. The underlying physics of TO, firstly proposed by Eddington [12], is that the trajectory of light in an arbitrary vacuum curved spacetime, could be regenerated in an appropriate dielectric media, residing in Minkowski spacetime. Later, Gordon [13] discussed the way of finding the analog curved vacuum spacetime, out of a definite dielectric, using an effective optical metric. However it was Plebanski [14] who found out that the constitutive equations for electromagnetic fields in an arbitrary vacuum curved spacetime, are equivalent to those in an appropriate dielectric media in Minkowski spacetime. Further, De Felice [15] and afterwards Reznik [16], used this equivalence in studying gravitational systems and De Felice generalized it to Friedmann-Robertson-walker spacetime in a universe with no spatial curvature. Note that, TO is based on transformation media performing coordinate transformations, which were initially considered to be purely spatial transformations [6, 17, 18, 19]. However since De Felice’s approach is linked to differential geometry, one can perform both space and time transformations [9]. As stated above, the most important feature of the Plebanski’s equivalence, is provid- ing the possibility of studying an analog system in Minkowski spacetime that mimics some aspects of gravitational systems, like trajectories of light. Because of complexity of such systems and also lack of a unified theory of gravitation, some remarkably distinguishable phenomena are not still completely understood, even if they have been observationally approved. For example, there are some experimental evidences for stimulated emissions from an analog system [see for example [20]], that may provide some kind of Hawking- like radiation [21]. However since this appears to be a quantum phenomena, we have yet not been able to properly explain it within available gravitational theories. Therefore it seems that if we could resemble spacetime properties using a definite dielectric with certain configuration, it might be possible to facilitate the investigation of gravitational systems, experiencing that spacetime. In this paper we consider such simplification to study the some features of a spacetime constructed by gravitational waves, by identifying a pp-wave geometry and a dielectric in Minkowski spacetime. However the method we exploit is not the Plebanski- De Felice approach, because of its limitations, specially lack of covariance which prompts us to take only a certain class of transformations. Note that, in accordance with what Plebanski-De Felice TO relies on, the appearance of magnetoelectric coupling can be regarded as the velocity of an isotropic media at low speeds. However instead, we will apply a covariant theory of TO, introduced in [22] , developed in [23] and studied in the context of vacuum solutions of general relativity in [24] and [11]. This method, because of its covariance, provides a greater freedom in considering diversified types of motions and transformations (even for time-varying dielectric [25]), while designing materials. The paper is organized as follows: in section 2, a general survey on classical electro- dynamics is made. In section 3, we provide a constructive introduction to the mentioned covariant approach of TO, and this method will be used for a moving dielectric in a vacuum pp-wave spacetime, in order to specifying some characteristics of such media. In Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 163 section 4, we construct a dielectric analog of pp-wave spacetime and assuming a peculiar transformation, we state that some sort of coordinate singularity can be obtained, which may provide a electromagnetically cloaked region. The concluding remarks are given in section 5, and also some relations which are in use in this paper, are brought in the appendix. 2 Classical Electrodynamics The covariant formulation of classical electrodynamics is based on its tensorial representa- tion on a Riemannian manifold. The reader may find detailed discussions in [26, 27, 28]. For a manifold with spacetime metric g, the field strength 2-form F ,intermsofthe potential co-vector A, is defined by the exterior derivative F =dA. (1) −→ −→ The co-vector A = Aμ itself consists of the electric field E and the magnetic flux B . Consequently, the tensorial form of the field strength tensor in Minkowski spacetime and local Cartesian frame reads as ⎛ ⎞ − − − ⎜ 0 Ex Ey Ez ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Ex 0 Bz −By ⎟ ⎜ ⎟ Fμν = ⎜ ⎟ . (2) ⎜ E −B 0 B ⎟ ⎝ y z x ⎠ Ez By −Bx 0 −→ −→ Moreover, in terms of the electric flux D and the magnetic field H , the excitation 2-form G may be written in the following tensorial form: ⎛ ⎞ ⎜ 0 Hx Hy Hz ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −Hx 0 Dz −Dy ⎟ ⎜ ⎟ Gμν = ⎜ ⎟ . (3) ⎜ −H −D 0 D ⎟ ⎝ y z x ⎠ −Hz Dy −Dx 0 Together with (2), this provides the Maxwell’s equations. dF =0, dG = J, (4) where J is the current 3-form [27]. On the other hand, linear dielectric materials admit the following relation between F and G G = χ( F ), (5) 164 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 or in component form αβ Gμν = χμν ( F )αβ. (6) Here, the Hodge dual is a map on a manifold (M,g), which for 2-forms is defined as 1 μν = |g| gσμgρν, (7) αβ 2 αβσρ with αβσρ as the Levi-Civita tensor and g the determinant of metric. In equation (5), F maps F to another 2-form on the same manifold. Furthermore, the susceptibility χ contains the material characteristics. Being also a linear material, the vacuum configured by [22] αβ (χvac)μν = ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎞ ⎜ 0000⎟ ⎜ 0100⎟ ⎜ 0 010⎟ ⎜ 0001⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 0000⎟ ⎜ −1000⎟ ⎜ 0 000⎟ ⎜ 0000⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ −1000⎟ ⎜ 0000⎟ ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎟ ⎜ 0000 0000 0 000 −1000 ⎟ ⎜ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ ⎜ 0 −100⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ 0010⎟ ⎜ 0001⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ 0 −100⎟ ⎜ 0000⎟ ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎟ 1 ⎜ 1000 0000 0000 0 −100 ⎟ × ⎜ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ , (8) 2 ⎜ 00−10⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 00 0 0⎟ ⎜ 00−10⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ 10 0 0⎟ ⎜ 0100⎟ ⎜ 0000⎟ ⎜ 0001⎟ ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎟ ⎜ 00 0 0 0000 0000 00−10 ⎟ ⎜ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ ⎜ ⎜ 000−1 ⎟ ⎜ 000 0 ⎟ ⎜ 000 0 ⎟ ⎜ 0000⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 000 0 ⎟ ⎜ 000−1 ⎟ ⎜ 000 0 ⎟ ⎜ 0000⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 000 0 ⎟ ⎜ 000 0 ⎟ ⎜ 000−1 ⎟ ⎜ 0000⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 100 0 010 0 010 0 0000 may perform χvac( F )= F . Maintaining above definitions, the classical theory of electrodynamics becomes structured by means of the constitutive equation (5), which results in the following collected form of six independent relations in Minkowski spacetime: −→ →− −→ −→ →− −→ −1 ∗ ∗ ∗ H =ˇμ B +ˇγ1 E, D =ˇε E +ˇγ2 B. (9) Equation (9) may be rearranged to −→ −→ −→ −→ −→ −→ B =ˇμ H +ˇγ1 E, D =ˇε E +ˇγ2 H. (10) The 3-dimensional matricesε ˇ, the permittivity,μ ˇ, the permeability andγ ˇ, the magneto- electric coupling are viable to provide an equivalence between (9) and (10) by relating −1 −1 ∗ ∗ ∗ μˇ = μˇ , εˇ =ˇε − γˇ2 μˇ γˇ1 , ∗ ∗ γˇ1 = −μˇγˇ1 , γˇ2 =ˇγ2 μ.ˇ (11) The way in which these 3 × 3 matrices constitute the components of χ, has been given in the appendix. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 165 3 Covariant Formulation of Transformation Optics A brief survey on a covariant formulation of TO is given here, which has been demon- strated in [22] and developed in [23]. Consider an initial manifold (M,g, )withfield configuration (F , G, J) and material distribution χ on which the Maxwell’s equations dF =0anddG = J and constitutive equations G = χ( F ) are valid, is mapped to its sub-manifold M ⊆ M by T : M −→ M (see Figure 1). The Maxwell’s and constitutive equations also hold on M ;dF =0anddG = J . Parallel to geometric transformations however, the physical fields are transformed by T , the inverse of T , and therefore F and G are mapped to F and G by T ∗, the pullback of T , i.e. [22] Figure 1 Manifold M and its sub-manifold M , which is obtained using a map like T from M to M . M contains a material χ . The map could be defined, in a way to cerate a hole in M (cloaked region). G = T ∗(G)=T ∗ (χ( F )) = χ ( T ∗(F )) . (12) For every point x ∈ M , we can relate the material distributions in M and M, as follows: τη α β μν σρ −1 π −1 θ τη − |T |T | χλκ (x)= Λ λΛ κχαβ (x) μν (x) Λ σ Λ ρ πθ x, (13) where Λ is the Jacobian of the transformation T (x). In (13) the first is calculated on M and the other one on M . Note that, if the initial media is supposed to be a vacuum, then the first χ has to be the one defined in (8), however non-vacuum initial media in this covariant formulation has also been considered in [29]. Now using (8) and the fact that χvac = , the material distribution in M is characterized by τη α β σρ −1 π −1 θ τη − |T | χλκ (x)= Λ λΛ κ αβ (x) Λ σ Λ ρ πθ x, (14) by means of which, one can relate a vacuum initial manifold with an arbitrary metric, to its sub-manifold containing a material χ . 3.1 Moving Dielectric in pp-Wave Spacetime In this subsection, it is assumed that a transformation media is moving in a region, con- structed by gravitational waves and determined by the pp-wave spacetime metric. We discuss that how will be the dielectric’s configuration, when special coordinate transfor- mations are performed. Also some examples are given. 166 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 The pp-wave line element in (u, v, x, y) coordinate system reads as [30, 31] ds2 = −c2dτ 2 =Φ(u, x, y)du2 + dudv + dx2 + dy2, (15) implying ⎛ ⎞ 1 ⎜ Φ(u, x, y) 2 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 000⎟ ⎜ 2 ⎟ gμν = ⎜ ⎟ . (16) ⎜ 0010⎟ ⎝ ⎠ 0001 Keeping c = 1, the 4-velocity vectors are derived from the following equation μ ν gμνU U = −1, (17) μ dxμ for time-like geodesics, with U = dτ . According to (15), this results in −1=Φ˙u2 +˙uv˙ +˙x2 +˙y2, (18) d with dot standing for dτ . This is equivalent to μ e0 e0μ = η00, (19) μ ∂xμ where eA = ∂xA is the set of basis vectors, transforming a 1-form in Minkowskian manifold (parameterized by xA) to a 1-form in pp-wave manifold (parameterized by xμ). The coordinates in pp-wave formalism, consist of two light-cone elements, namely u and v; u = t − z, v = t + z. (20) Also, the only non-zero components of Christoffel symbols in this geometry, are ∂Φ Γv = , uu ∂u ∂Φ Γv = , iu ∂xi 1 ∂Φ Γi = , (21) uu 2 ∂xi where i =2, 3. Therefore, the geodesic equations imply that u¨ =0 → u˙ =const.=α. (22) Thereforeu ˙ could be considered as a constant of motion. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 167 3.2 Transformation Optics in pp-Wave Spacetime Now we are at a stage of exploiting the covariant formulation of TO introduced in section 3. Let us consider a transformation media moving in pp-wave spacetime. The covariant relation (14) for a definite coordinate transformation T (x), results in the configuration of the considered dielectric in same spacetime. However, what a local observer examines is the same configuration obtained in Minkowski spacetime. Therefore we are in need of a transformation operator, to transform χ in pp-wave spacetime, to its Minkowskian correspondent, χˆ. To obtain the proper transformation matrix we pursue the following μ procedure. The matrix consists of the basis vectors eA, exhibiting the following traits: • Each vector, respectively must give the time-like and space-like results; i.e. μ ν gμνeAeB = ηAB. (23) • Orthogonality condition: μ ν gμνeAeB =0, (24) for A = B. We consider the dielectric to move along the z direction with coordinate velocity dz dt = vz. Therefore the x and y coordinates are left unchanged. The zero components of μ the set of basis vectors, namely e0 , is equivalent to the velocity 4-vector. We have μ e0 =(˙u, v,˙ 0, 0). (25) According to the geodesic result (22), and the orthogonality condition, we obtain 1+α2Φ eμ =(α, − , 0, 0). (26) 0 α For the 1st and 2nd basis vectors, we could use the pure coordinate differentiation, since they are exactly revealing the Minkowskian representation. Therefore ∂xμ eμ = =(0, 0, 1, 0), 1 ∂x ∂xμ eμ = =(0, 0, 0, 1). (27) 2 ∂y μ The 3rd vector however, is dependent to e0 which due to the orthogonality condition (23), becomes 1 − α2Φ eμ =(α, , 0, 0). (28) 3 α All these vectors, satisfy both mentioned conditions. Using them, one can conclude the transformation matrix as ⎛ ⎞ − 1+α2Φ(u,x,y) ⎜ α α 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0010⎟ μ ⎜ ⎟ SA = ⎜ ⎟ . (29) ⎜ 0001⎟ ⎝ ⎠ 1−α2Φ(u,x,y) α α 00 168 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 It is straightforward to see that μ ν SA SB gμν = ηAB. (30) μ SA in (29) transforms a 1-form θμ in pp-wave spacetime, to θA in local frame. To A μ transform vectors, we must use the matrix S μ; the transpose of the inverse of SA . Finally, one can relate [32] CD λ κ C D τη χˆAB = SA SB S τ S η χ λκ , (31) to obtain the locally examined χˆ form its pp-wave correspondent χ . 3.3 Examples Having the transformation matrix obtained, we still need to consider a coordinate trans- formation to be performed by the transformation media, otherwise, the media becomes unnoticeable. Let T (u, v, x, y)=(u,v,x,y), to transform the components from the media (unprimed) to the vacuum space (primed). 3.3.1 Transformation on the null coordinates As a first step, let us take (u,v,x,y)=T (u, v, x, y)=(f(u),v,x,y). (32) The corresponding Jacobian reads as ⎛ ⎞ ⎜ ∂uf(u)000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0100⎟ μ ⎜ ⎟ Λ ν = ⎜ ⎟ . (33) ⎜ 0010⎟ ⎝ ⎠ 0001 Using (16), (7), (33) and (14) together with (31), and comparing with (A.1), one obtains the dielectric 3×3 vector, defined in the constitutive equations (9) and (10) in Minkowski spacetime. ⎛ ⎞ 1 ⎜ − 2 2 00⎟ ⎜ Φ(u,x,y)α +Φ(f(u),x,y)f (u)α +1 ⎟ ⎜ 1 ⎟ εˆ =ˆμ = ⎜ 0 2 2 0 ⎟ , ⎝ −Φ(u,x,y)α +Φ(f(u),x,y)f (u)α +1 ⎠ 1 00f (u) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 169 ⎛ ⎞ ⎜ 0 −10⎟ 2 − ⎜ ⎟ T α (Φ(u, x, y) f (u)Φ(f(u),x,y)) ⎜ ⎟ γˆ1 =ˆγ = ⎜ 100⎟ . (34) 2 α2f (u)Φ(f(u),x,y)+α2(−Φ(u, x, y)) + 1 ⎝ ⎠ 000 Note that f ≡ ∂uf(u). These results show that, the potential Φ appears explicitly in the dielectric characteristics. In the localized limit f(u) ∼ u, we get the Newtonian result ⎛ ⎞ ⎜ 100⎟ ⎜ ⎟ ⎜ ⎟ εˆ =ˆμ = ⎜ 010⎟ , (35) ⎝ ⎠ 001 which is the identity matrix. Here, Φ ≡ Φ(u, v, x, y). This result also corresponds to an isotropic dielectric. Moreover, the limit, when applied for the magnetoelectric coupling termγ ˆ, yields ⎛ ⎞ ⎜ 0 −10⎟ ⎜ ⎟ 2 ⎜ ⎟ γˆ1 =(α − 1)Φ ⎜ 100⎟ . (36) ⎝ ⎠ 000 So one can see that the magnetoelectric terms do not vanish in the Newtonian limit. Traditionally, non-vanishing coupling terms are in accord with moving materials [33], which expose direct impacts on the permittivity values, i.e.ε ˆ. In this case, the non- vanishingγ ˆ implies the initial motion of the dielectric, in the pp-wave spacetime. However, for α2 = 1, i.e. for du (˙u)2 =( )2 =1, dτ these terms will vanish. In other words we have du d(t − z) ≡ = t˙ − z˙ = γ(1 − v )=1. dτ dτ z Since for low speeds, γ ≈ 1, then the above relation results inz ˙ =0,orvz =0;nomotion along z direction, or in our case, no motion at all. 3.3.2 y-dependent transformation of x In this case, the transformation is supposed to be preformed on the x coordinate, orthog- onal to the dielectric’s direction of motion, namely thez ˆ direction. Let us investigate, what the contribution of media’s velocity, would be in the transformation media charac- teristics. The coordinate transformation is supposed to be T : T (u, v, x, y) −→ (u,v,x,y)=(u, v, f(x, y),y), (37) 170 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 where the function f(x, y)actsonthex coordinate, with a minor coupling to the y coordinate. The corresponding Jacobian reads as ⎛ ⎞ ⎜ 10 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 01 0 0 ⎟ μ ⎜ ⎟ Λ ν = ⎜ ⎟ . (38) ⎜ 00f f ⎟ ⎝ ,x ,y ⎠ 00 0 1 Same procedure as in the previous example, if pursued for the transformation (38), yields ⎛ ⎞ 2 − 1+f,y f 0 ⎜ f,x ,y ⎟ 1 ⎜ ⎟ εˆ =ˆμ = ⎜ − ⎟ , 2 2 ⎜ f,y f,x 0 ⎟ −1+α Φ − α Φf ⎝ ⎠ 00f,x ⎛ ⎞ ⎜ 010⎟ α2 ⎜ ⎟ ˆ =ˆT = [Φ − Φ ] ⎜ − ⎟ (39) γ1 γ2 2 2 f ⎜ 100⎟ , −1+α Φ − α Φf ⎝ ⎠ 000 with Φf ≡ Φ(u, f(x, y),y)andα =˙u. In this case, the vanishing magnetoelectric coupling terms, are followed by the following items: • α = 0: therefore we have du u˙ ≡ =0 ⇒ u = δ =const. dτ Or one can write d(t − z) = t˙ − z˙ = γ − v γ =0 ⇒ v =1. dτ z z • Φ=Φf : this implies that f(x, y) ≡ f(x)=x. Thereforeγ ˆ1 =ˆγ2 =0,and ⎛ ⎞ ⎜ 10 0 ⎟ ⎜ ⎟ ⎜ ⎟ εˆ =ˆμ = ⎜ 01 0 ⎟ . ⎝ ⎠ 00−1 None of the above items refers to a stationary situation, for which vz = 0. Therefore, one may infer that for perpendicular transformations, the magnetoelectric coupling terms do not explicitly retain the motion of the transformation media. Likewise, the vanishing magnetoelectric coupling terms here, do not lead to vanishing medium velocity. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 171 4 Dielectric Analog of pp-wave Spacetime Having introduced the covariant formulation of TO, here we mention that it is also possible to establish the concept of dielectric analogs. This important implementation of a dielectric analog is exploited here to investigate the characteristics of pp-wave spacetime, in a locally flat Minkowskian frame. The covariant method which is applied here, has been presented in [24], however since the initial vacuum manifold is supposed to be the pp-wave spacetime, therefore the light waves (rays) in this case are those which are supposed to have interactions with the gravitational waves (to obtain insights into this concept, please see Ref. [31]). We are concerning with the propagation of light waves in a dielectric, analogous to the pp-wave spacetim (see Figure 2). We begin with pp-wave Figure 2 The map T transforms the Minkowskian manifold M , containing a material χ , to the vacuum curved manifold Mˆ . Pullback of this map, T ∗, performs an associate map, taking differential forms in the cotangent bundle of Mˆ ,intothe cotangent bundle of M and here, appears to be the reverse map. metric, written in Cartesian coordinates ds2 =[1+Φ(u, x, y)]dt2 +dx2 +dy2 +[Φ(u, x, y) − 1]dz2 − 2Φ(u, x, y)dtdz. (40) One can note that, in contrast with the previous section, we wrote the metric in Cartesian coordinates. This is because of the fact that, the resultant dielectric analog is also characterized in the same coordinate system and as it is seen in Figure 2, the coordinate transformation T (t, x, y, z)=(t,x,y,z), maps Minkowski spacetime to vacuum pp- wave. However, as we will seen below, this map is not always in need of a peculiar expression other than the identity map. In contrast with the previous discussion where the dielectric had to perform a coordinate transformation ( otherwise it became unnoticeable ), the dielectric analog, because of different geometric background, could be valid even when the identity transformation is applied. Turning back to our discussion, the susceptibility in a local frame is obtained by the following tensor representation [24]: τη α β σρ −1 π −1 θ τη − ˆ |T | χλκ (x)= Λ λΛ κ αβ (x) Λ σ Λ ρ πθ x, (41) τη where χλk (x) is obtained in Minkowski spacetime. Putting T = T0, the identity map, μ results in Λ ν = 14 and equation (41) becomes τη σρ τη χλκ (x)=− ˆλκ |x σρ |x, (42) 172 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 in which the first Hodge dual is calculated in pp-wave spacetime, without any coordinate transformation. According to this identity transformation, using (40) and (42) and com- paring with (A.1), our impedance matched dielectric analog in Minkowski spacetime, is characterized by the following vectors: ⎛ ⎞ 1 ⎜ − 00⎟ ⎜ g00(x) ⎟ ⎜ ⎟ μˇ =ˇε = ⎜ 0 − 1 0 ⎟ , (43) ⎝ g00(x) ⎠ 001 ⎛ ⎞ 1 ⎜ 01− 0 ⎟ ⎜ g00(x) ⎟ T ⎜ 1 ⎟ γˇ1 =(ˇγ2) = ⎜ − 100⎟ , (44) ⎝ g00(x) ⎠ 000 where g00(x) = 1+Φ(u, x, y), according to the metric in (40). Note that the above magnetoelectric couplings are caused by the off-diagonal terms in the metric, which could be regarded as the time-dependency of the dielectric analog. 4.1 Square Cloak: electromagnetic invisibility In order to create a cloaked region, one should consider a transformation from the vacuum spacetime (pp-wave), to a square shell (with z =const.) in Minkowski spacetime. Such transformation has been derived in [34] as below: s2 − s1 s2 − s1 s1 T (t ,x,y,z)=(t, x, y, z)= t ,x + s1,y + ,z (45) s2 s2 x for 0 Figure 3 A square cloak. The transformation (45), maps to the shaded region I. The whole region is obtained by using a rotation matrix. Substitution of (46) and (47) in (41) results in the following vectors, corresponding to same parameters in constitutive equations (9) and (10), in region I of Figure 3: ⎛ ⎞ s1−x s1y ⎜ 2 0 ⎟ ⎜ x x ⎟ − 4 2 2 T 1 ⎜ x +s1 y ⎟ μˇI =ˇεI = g00( (x)) ⎜ s1y 3 0 ⎟ , (48) ⎝ (s1−x)x ⎠ 2 s2 (s1−x) 00− 2 g (T (x)) (s1−s2) x 00 ⎛ ⎞ T − ⎜ 0 g00( (x)) 10⎟ ⎜ ⎟ T −1 ⎜ ⎟ (ˇγ1)I =(ˇγ2)I = g00(T (x)) ⎜ 1 − g (T (x)) 0 0 ⎟ , (49) ⎝ 00 ⎠ 000 where g00(T (x))=1+Φ(u, f(x),g(x, y)) according to metric (15), maintaining the coor- dinate transformation (46). One can note that the form of the magnetoelectric couplings, in (44) and (49) are the same since the off-diagonal term of the pp-wave metric remains the same, consequently the dielectric analog does not experience any further time-dependent alternations. Also as it was noted above, relations (48) and (49) only cover the region I in Figure 3, therefore we shall apply the following rotations, to obtain the whole region: ⎛ ⎞ ⎛ ⎞ ⎜ 010⎟ ⎜ 010⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T ⎜ ⎟ μˇII =ˇεII = ⎜ −100⎟ · μˇI, (ˇγ1)II =(ˇγ2)II = ⎜ −100⎟ · (ˇγ1)I, (50) ⎝ ⎠ ⎝ ⎠ 001 001 174 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 ⎛ ⎞ ⎛ ⎞ ⎜ −100⎟ ⎜ −100⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T ⎜ ⎟ μˇIII =ˇεIII = ⎜ 0 −10⎟ · μˇI, (ˇγ1)III =(ˇγ2)III = ⎜ 0 −10⎟ · (ˇγ1)I, (51) ⎝ ⎠ ⎝ ⎠ 001 001 ⎛ ⎞ ⎛ ⎞ ⎜ 0 −10⎟ ⎜ 0 −10⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T ⎜ ⎟ μˇIV =ˇεIV = ⎜ 100⎟ · μˇI, (ˇγ1)IV =(ˇγ2)IV = ⎜ 100⎟ · (ˇγ1)I. (52) ⎝ ⎠ ⎝ ⎠ 001 001 The vectors in (48)-(52), characterize a region, in which no electromagnetic wave can enter. Therefore we created a cloaked region, through which no optical data can be exchanged and consequently has been electromagnetically isolated. In fact, dielectric configurations are not interacting gravitational waves, but just electromagnetic waves. However, we should bear in mind that the analog dielectric obtained above, is indeed replaced by the pp-wave spacetime and if any real-world opto-gravitational interactions do exist, the aforementioned cloaked region will be capable of exhibiting them. We should note also that, such cloaked region is indeed caused by a coordinate transformation. Therefore such a singularity is a coordinate singularity, note a real one. However, because of its optical characteristics, this region may be regarded as a trapped surface, which has several correspondences in regular solutions to general relativity. 5 Conclusion The configuration of a transformation media, when it is moving in a region constructed by gravitational waves, was discussed as the main aim. To fulfill this task, a covariant formulation of transformation optics was applied. It is essential for this transformation media to be applied to light waves. In this work however, one may consider some possi- ble opto-gravitational interactions in the region, and performing some definite coordinate transformations, may demonstrate specific behaviors. Moreover, we also constructed a dielectric analog of the mentioned region, and showed that this analog has to be mag- netoelectric. According to a peculiar coordinate transformation, we also dealt with a 2-dimensional electromagnetically isolated region; no light rays may pierce nor emanate from the region. This is traditionally related to electromagnetic [35] or acoustic [36] cloaks. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 175 A Susceptibility Reference Frame Susceptibility reference matrix in Cartesian coordinates [23]: ⎛ ⎞ ∗∗∗ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜ M 0 ∗∗⎟ αβ ⎜ ⎟ χμν = ⎜ ⎟ , (A.1) 2 ⎜ NO0 ∗ ⎟ ⎝ ⎠ PQR0 where ⎛ ⎞ − −1 − −1 − −1 ⎜ 0 μxx μxy μxz ⎟ ⎜ ⎟ ⎜ − ∗ ∗ ⎟ ⎜ μ 1 0 −γ γ ⎟ M ⎜ xx 1xz 1xy ⎟ = ⎜ ⎟ , ⎜ μ−1 γ∗ 0 −γ∗ ⎟ ⎝ xy 1xz 1xx ⎠ −1 − ∗ ∗ μxz γ1xy γ1xx 0 ⎛ ⎞ − −1 − −1 − −1 ⎜ 0 μyx μyy μyz ⎟ ⎜ ⎟ ⎜ − ∗ ∗ ⎟ ⎜ μ 1 0 −γ γ ⎟ N ⎜ yx 1yz 1yy ⎟ = ⎜ ⎟ , ⎜ μ−1 γ∗ 0 −γ∗ ⎟ ⎝ yy 1yz 1yx ⎠ −1 − ∗ ∗ μyz γ1yy γ1yx 0 ⎛ ⎞ − ∗ − ∗ − ∗ ⎜ 0 γ2zx γ2zy γ2zz ⎟ ⎜ ⎟ ⎜ ∗ ∗ ∗ ⎟ ⎜ γ 0 −ε ε ⎟ O ⎜ 2zx zz zy ⎟ = ⎜ ⎟ , ⎜ γ∗ ε∗ 0 −ε∗ ⎟ ⎝ 2zy zz zx ⎠ ∗ − ∗ ∗ γ2zz εzy εzx 0 ⎛ ⎞ − −1 − −1 − −1 ⎜ 0 μzx μzy μzz ⎟ ⎜ ⎟ ⎜ − ∗ ∗ ⎟ ⎜ μ 1 0 −γ γ ⎟ P ⎜ zx 1zz 1zy ⎟ = ⎜ ⎟ , ⎜ μ−1 γ∗ 0 −γ∗ ⎟ ⎝ zy 1zz 1zx ⎠ −1 − ∗ ∗ μzz γ1zy γ1zx 0 ⎛ ⎞ ∗ ∗ ∗ ⎜ 0 γ2yx γ2yy γ2yz ⎟ ⎜ ⎟ ⎜ ∗ ∗ ∗ ⎟ ⎜ −γ 0 ε −ε ⎟ O ⎜ 2yx yz yy ⎟ = ⎜ ⎟ , ⎜ −γ∗ −ε∗ 0 ε∗ ⎟ ⎝ 2yy yz yx ⎠ − ∗ ∗ − ∗ γ2yz εyy εyx 0 176 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 161–178 ⎛ ⎞ − ∗ − ∗ − ∗ ⎜ 0 γ2xx γ2xy γ2xz ⎟ ⎜ ⎟ ⎜ ∗ ∗ ∗ ⎟ ⎜ γ 0 −ε ε ⎟ R ⎜ 2xx xz xy ⎟ = ⎜ ⎟ . ⎜ γ∗ ε∗ 0 −ε∗ ⎟ ⎝ 2xy xz xx ⎠ ∗ − ∗ ∗ γ2xz εxy εxx 0 References [1] C. 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Phys., 43, 113001 (2010) EJTP 14, No. 37 (2018) 179–194 Electronic Journal of Theoretical Physics Validation of the Hadron Mass Quantization from Experimental Hadronic Regge Trajectories Navjot Hothi1∗,ShuchiBisht2† 1Department of Physics, University of Petroleum and Energy Studies, Dehradun - 248007, India 2Department of Physics, Kumaun University, Nainital-263002, India Received 4 June 2017, Accepted 20 August 2017, Published 20 April 2018 Abstract: This contribution provides a validation to the earlier proposal of hadron mass quantization in the units of 70 MeV mass quanta. The linear experimental Hadronic Regge Trajectories constructed from the recent 2014 Particle Data Group Listings serve as a prominent tool in solving the Hadronic mass spectrum mystery. Application of the Barut’s solution to relativistic Balmer formula helps in deriving quark masses for mesons and baryons. This astonishingly produces the quark masses very close the 70 MeV mass quanta, which turns out to be the mass quantum for buliding hadrons. The slight deviation from this mass quantization is also evidently explained. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Regge Trajectories; Hadrons; Quantization; Slope PACS (2010): 11.55Jy; 12.40Nn; 14.20-c; 14.40.-n 1 Introduction The hadronic mass spectrum is a mystery. Hadrons are not point particles like leptons and they do possess an internal structure. Mass quantization is evident at atomic scale. This led to the proposal that even hadronic masses ought to be quantized as nature loves symmetry. Thereby, the general expectancy is that hadrons have rotational and other excited states which become evident through the discovery of large number of hadronic resonances. Malcolm Mac Gregor [1-3] did an inventive work in this field of mass quantization and has shown that in case of hadrons, there exists a mass band structure in the units of Q= 70 MeV. This quanta lead to a common shell structure separating ∗ Email: [email protected] † Email: [email protected] 180 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 hadron states. Gregor observed [2,3] that the hadron spectrum shows regularity with energy separations in the units of 70 MeV [4, 5]. He proposed that quarks are light and have small binding energies. These small binding energies of quarks leash two types of quanta. The first one is a 70 MeV spinless quanta Q, which reproduces the pion and the second one is a 330 MeV spin quanta S, which reproduces the nucleon. The second quanta is basically a derivate of the original 70 MeV mass quanta. This 70 MeV spinless mass quanta of Gregor [1-3] can be undoubtedly derived from the electron mass and the fine structure constant as Q = me/α =70MeV (1) Also, the Nambu empirical mass formula: mn =(n/2)137me(n is a positive integer) (2) which can tabularize the particle spectrum, is candidly related to this particular quan- tum. In the present contribution, this famous 70 MeV mass quanta is derived from the experimental linear Regge trajectories derived from the Particle Data Group 2014 listings [6]. Regge trajectories basically classify dynamic hadrons of the same quantum numbers but different spins and masses. The idea behind the Regge theory for the purpose of classification of elementary particles is quite simple [7]. With the continuation of nuclear experiments, physicists began to find discover more and more hadrons. These hadrons projected out to be copies of the original hadrons, but with higher spins and masses. It was viable to plot some of them on a spin (J) versus (mass)2 (M 2) plot and an astonishing pattern was observed. Each set of particles with the same internal quantum numbers lied on straight lines and these lines were called Regge Trajectories. The Regge trajectories are parameterized by the equation: M 2 = J/α + β (3) where α is called the Regge slope and β is the Regge intercept. It turned out that the Regge trajectories were far more valuable than just another classification scheme of hadrons. They serve as a tool to study hadron dynamics. Evidently, with the availability of large hadronic data, new experimental Regge trajectories are being plotted. In the present contribution, application of the Barut’s solution to relativistic Balmer formula [8,9] helps in deriving quark masses for mesons and baryons which are plotted on their respective Regge trajectories. This astoundingly produces the quark masses very close the 70 MeV mass quanta, which turns out to be the ultimate hadronic building block. This provides a validation to concept of hadron mass quantization [10] from the experimental hadronic Regge trajectories [11]. 2 Hadron Spectrum and the 70 MeV mass quanta The spinless quantum Q=70 MeV and the spin quantum S=330 MeV, both act as the adequate basis set for light hadrons, where the quark-antiquark binding energies Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 181 specifically match the nucleon-antinucleon binding energies. Moreover, when the special relativity is employed to the rotating masses, it is revealed that S is itself composed of three quanta Q in a relativistically spinning configuration, so that Q emerges out as the ultimate hadronic mass quantum. The relativistically spinning mass is 3/2 times the 70 MeV mass quanta. Q =70MeV ⇒ Qs =(3/2)Q = 105MeV (4) where Qs is termed as the relativistically spinning mass quanta. Therefore, 3Qs ≈ 330MeV =(Squanta) (5) For the mesonic sector, the lowest mass particle in the pion, whose mass is approximately 140 MeV/c2, which can be viewed as a composite mass of two 70 MeV/c2 mass quanta. We know that π = QQ , where Q and Q are 70 MeV light mass quanta. In the similar fashion, K meson is 7Q that is seven times the mass quantum Q. The other mesonic resonances which are exact multiple of 70 are:- 1)ρ(770) 2)a0(980) 3)π2(2100) 4)a6(2450) 5)K1(1400) 6)K2(1820) etc. Also from the level spacing observed in the case of baryons [2], it appears that excitations occur in the units of Q=70 MeV, eg 3 3Q=210 MeV, 4 4Q=280 MeV etc. It may be so that the particles and resonances which lie on the bary- onic RT, may not be exact multiples of Q. For some baryons, following combinations of S and Q account for their masses. Σ = 330 ⊕ 330 ⊕ 330 ⊕ (4 ⊗ 70) Λ = 330 ⊕ 330 ⊕ 330 ⊕ (3 ⊗ 70) Ω = 330 ⊕ 330 ⊕ 330 ⊕ (3 ⊗ 70) ⊕ (4 ⊗ 70) ⊕ (4 ⊗ 70) Ξ = 330 ⊕ 330 ⊕ 330 ⊕ (3 ⊗ 70) ⊕ (3 ⊗ 70) N = 330 ⊕ 330 ⊕ 330 The subsistence of the 70 MeV boson was also stemmed from the mass of the classical Dirac Magnetic Monopole [12]. Further, the sequel [13] of the previous reference pro- vides testimony of this quanta from the findings of the Particle Data Group listings [14]. Furthermore, according to Gregor, all particles more massive than the electron can be constructed from a single mass quantum Q. 3 Hadron Spectrum and the 70 MeV mass quanta In case of mesons, the pion is the lowest mass meson and thus the starting member of the meson RT. The pion is composed of two 70 MeV/c2 mass quanta. From Barut’s solution to a relativistic Balmer mass formula [8,9], J M 2 =(m + m )2 +2m m (6) 1 2 1 2 α If m1=m2=m, then 2m2 slope = (7) α 182 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 and slopeα m = (8) 2 similarly slope(baryons)=[(m1m2m3)/(m1 + m2 + m3)]αs (9) where m1, m2 and m3 are individual quark masses and αs is the strength of the coupling constant. If m1=m2=m3=m, then m2 slope(baryons)= α (10) 3 s Taking the nucleon to be the starter of the series for the baryon RT, it is obvious that m1 = m2 = m3 = u = 315MeV (Since u and d are approximately equal in mass). The particle mass is taken to be 97 percent of the total mass, the rest being the contribution of the binding energy. Therefore, the quark mass will be 0.97 × 315 = 305.6MeV (11) αs = 137/4=34.25 (12) This value of coupling constant is taken from the work of Sawada [15], who proposed that coupling parameter takes this particular value when Coulombic interaction of the magnetic monopole [16] is taken into account. This analysis of slope is done with the hypotheses of idealized mass quanta in the units of spinless, 70 MeV and spin1/2, 315 MeV mass quantas. An accordance of these values with experimental data would provide validation to this concept of hadron mass quantization. In all, 9 Figures have been plotted for a series of mesonic and baryonic resonances. Figures I-V have been plotted for series of mesonic resonances, while Figures VI-IX correspond to series of baryonic resonances. The slopes have been calculated for these trajectories and their quark masses have been calculated subsequently. Table I displays the quark mass for each trajectory along with the deviation from standard value of 70 MeV and 305.6 MeV respectively for mesonic and baryonic RTs. From the table it is evident that the deviation from the standard values is very low and profoundly the experimental quark masses match very distinctly with the theoretical postulation of the 70 MeV mass quanta. The reason for slight deviation from the exact quantized masses has been explained in the next section. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 183 Table I: Calculated values of quark masses for several series of Hadronic RTs 61R +DGURQV 6ORSH 4XDUNPDVV 'HYLDWLRQIURP VWDQGDUGYDOXH )LJ, 8QIODYRUHGXGPHVRQV 0H9 0H9 0H9 0H9 0H9 0H9 0 0H9 0H9 ( )LJ,, 6 8QIODYRUHGXGPHVRQV 0H9 0H9 2 0H9 0H9 1 6 0H9 0H9 )LJ,,, 8QIODYRUHGXGPHVRQV 0H9 0H9 0H9 0H9 )LJ,9 8QIODYRUHGXGPHVRQV 0H9 0H9 0H9 0H9 )LJ9 .PHVRQV 0H9 0H9 0H9 0H9 0H9 0H9 )LJ 9, 1EDU\RQV 0H9 0H9 )LJ9,, % ǻEDU\RQV 0H9 0H9 $ 5 0H9 0H9 < 2 0H9 0H9 1 6 0H9 0H9 )LJ ȁEDU\RQV 0H9 0H9 9,,, 0H9 0H9 0H9 0H9 )LJ,; ȈEDU\RQV 0H9 0H9 0H9 0H9 4 Factors contributing to deviation from 70 MeV mass quanta Several features manifestly cause the departure of masses of hadrons from the integral multiples of 70 MeV mass quanta. Foremost, the Coulomb corrections and spins are protuberant contributory factors along with magnetic moment ratios and charge splitting. Another conscientious factor could be the strangeness quantum number. Constituent quark binding energies is also one of the factors steering to variation in mass. Additionally, the constituent quark binding energies also relegates the masses of hadrons by nearly about 3 to 4 percent [2,17]. The relativistically spinning configuration of rotating masses also leads to digression of masses of hadrons from the integral multiples of seventy. 5 Conclusions Quark mass quantization is a revolutionary notion and the present contribution provides an authentication to it. A restored estimation of the hadron dynamics is feasible once 184 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 quantization becomes evident. The derivation of quark masses from Barut’s solution to relativistic Balmer formula provides a foundation for calculating quark masses for mesons and baryons from the experimental hadronic RTs for Particle Data Group 2014 listings. Resplendently, the quark masses calculated from this approach are very intimate to the earlier proposed 70 MeV mass quanta. This in itself is validation to the proposal of quantized hadronic masses, which will ultimately prove to be very useful in unraveling the ambiguities of hadron dynamics. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 185 ϵ ϴ ϳ Ĩϲ;ϮϱϭϬͿ ϲ ͲͲ ʔϱ;ϮϮϵϬͿϱ ϱ нн ĨϮ;ϮϭϱϬͿϮ нн Ϯ Ĩ ;ϮϬϱϬͿϰ ϰ ϰ D нн ĨϮ;ϭϵϭϬͿϮ ͲͲ ϯ нн ʔϯ;ϭϳϳϬͿϯ ĨϬ;ϭϳϭϬͿϬ ͲͲ ͲͲ ʘ;ϭϲϱϬͿϭ ʘϯ;ϭϲϳϬͿϯ Ĩ ;ϭϰϯϬͿϮнн Ϯ Ĩ ;ϭϯϳϬͿϬнн Ϯ Ϭ нн ĨϮ;ϭϮϳϬͿϮ ϭ ʔ;ϭϬϮϬͿϭͲͲ ʘ;ϳϴϮͿϭͲͲ Ϭ ϬϭϮϯϰϱϲϳ Ͳϭ : M 2 =0.849J +2.924 M 2 =0.885J +1.863 M 2 =1.056J − 0.035 M 2 =1.088J − 0.506 Figures I:-Regge trajectories for four separate series of unflavored (u-d) mesons along with their straight line equations and the string tensions. 186 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 ϳ ϲ ͲͲ ʌϱ;ϮϯϱϬͿϱ ͲͲ ϱ ʌϯ;ϮϮϱϬͿϯ Ă ;ϮϬϰϬͿϰнн ϰ ͲͲ ϰ ʌϯ;ϭϵϵϬͿϯ ʌ;ϭϵϬϬͿϭͲͲ ϯ нн ͲͲ ĂϮ;ϭϳϬϬͿϮ ʌϯ;ϭϲϵϬͿϯ Ϯ D ͲͲ Ϯ ʌ;ϭϰϱϬͿϭ нн ĂϮ;ϭϯϮϬͿϮ нн ϭ ĂϬ;ϭϬϳϬͿϬ ʌ;ϳϳϬͿϭͲͲ Ϭ ϬϭϮϯϰϱϲ Ͳϭ ͲϮ : M 2 =0.726J +2.883 M 2 =0.928J +1.126 M 2 =1.227J − 0.708 Figure II :- RTs for three separate series of unflavored (u-d) mesons along with their straight line equations and the string tensions. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 187 ϴ ϳ ϲ нн ĨϮ;ϮϯϰϬͿϮ ϱ нн Ĩ:;ϮϮϮϬͿϰ ʔ;ϮϭϳϬͿϭͲͲ Ϯ нн ϰ ĨϬ;ϮϬϮϬͿϬ ͲͲ D ʘϯ;ϮϬϬϬͿϯ нн ϯ ĨϮ;ϭϳϯϱͿϮ Ϯ ʘ;ϭϰϮϬͿϭͲͲ ϭ нн ĨϬ;ϵϴϬͿϬ Ϭ ϬϭϮϯϰϱϲ : M 2 =0.697J +4.057 M 2 =0.987J +0.989 Figure III:- RTs for two separate series of unflavored (u-d) mesons along with their straight line equations and the string tensions. 188 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 ϰ Ͳн ϯ͘ϱ ɻϮ;ϭϴϳϬͿϮ ϯ Ͳн ʋϮ;ϭϲϳϬͿϮ нͲ Ϯ͘ϱ Śϭ;ϭϱϵϱͿϭ Ϯ D Ϯ нͲ Śϭ;ϭϯϴϬͿϭ ɻ;ϭϮϵϱͿϬͲн ϭ͘ϱ ϭ ɻΖ;ϵϱϴͿϬͲн Ϭ͘ϱ Ϭ Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ Ϯ Ϯ͘ϱ : M 2 =0.909J +1.662 M 2 =0.935J +0.934 Figure IV- RTs for two separate series of unflavored (u-d) mesons along with their straight line equations and the string tensions. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 189 ϵ ϴ ϳ < ;ϮϱϬϬͿ ϲ ϰ Ͳ <ϱΎ;ϮϯϴϬͿϱ < ;ϮϮϱϬͿϮͲ ϱ Ϯ <ϯ;ϮϮϲϱͿ н Ϯ < Ύ;ϮϬϰϱͿϰ ϰ ϰ D <Ϯ;ϭϴϮϬͿ <;ϭϴϯϬͿϬͲ < Ύ;ϭϳϴϬͿϯͲ ϯ ϯ <ϭ;ϭϰϬϬͿ н Ϯ <Ϭ;ϭϯϮϱͿ <ϮΎ;ϭϰϯϬͿϮ ϭ <Ύ;ϴϵϮͿϭͲ Ϭ ϬϭϮϯϰϱϲ Ͳϭ : M 2 =0.856J +3.348 M 2 =1.145J +1.657 M 2 =1.187J − 0.391 Figure V:- RTs for three separate series of K-mesons along with their straight line equations and string tensions. 190 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 ϴ E;ϮϳϬϬͿ ϳ E;ϮϰϴϱͿ ϲ ϱ E;ϮϮϱϬͿ Ϯ ϰ D E;ϭϵϵϬͿ ϯ E;ϭϲϳϱͿ Ϯ E;ϭϮϱϬͿ ϭ Ϭ ϬϭϮϯϰϱϲϳ : M 2 =1.118J +0.025 Figure VI:- RTs for one series of N-baryons along with their with straight line equations and string tensions. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 191 ϭϮ ϭϬ ȴ;ϮϵϱϬͿ ϴ ȴ;ϮϳϱϬͿ Ϯ ϲ D ȴ;ϮϯϱϬͿ ȴ;ϮϯϵϬͿ ȴ;ϮϰϬϬͿ ȴ;ϮϰϮϬͿ ȴ;ϮϭϴϬͿ ȴ;ϮϮϬϬͿ ȴ;ϭϵϭϬͿ ϰ ȴ;ϭϵϱϬͿ ȴ;ϭϵϮϬͿȴ;ϭϵϬϱͿ ȴ;ϭϲϮϬͿ ȴ;ϭϲϬϬͿ Ϯ ȴ;ϭϮϯϮͿ ȴ;ϭϭϰϱͿ Ϭ ϬϮϰϲϴ : M 2 =0.937J +3.179 M 2 =1.026J +2.125 M 2 =1.081J +0.954 M 2 =1.196J − 0.374 Figure VII :- RTs for four separate series of Δ baryons along with their straight line equations and string tensions. 192 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 ϳ ϲ ȿ;ϮϯϱϬͿ ϱ ȿ;ϮϭϭϬͿ ȿ;ϮϭϬϬͿ ϰ Ϯ ȿ;ϭϴϵϬͿ D ȿ;ϭϴϮϬͿ ϯ ȿ;ϭϲϳϬͿ ȿ;ϭϲϵϬͿ ȿ;ϭϱϮϬͿ Ϯ ȿ;ϭϰϬϱͿ ϭ ȿ;ϭϬϱϬͿ Ϭ Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ Ϯ Ϯ͘ϱ ϯ ϯ͘ϱ ϰ ϰ͘ϱ ϱ : M 2 =0.831J +2.357 M 2 =0.882J +1.533 M 2 =1.073J +0.668 Figure VIII :- RTs for three separate series of Λ baryons along with their straight line equations and string tensions. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 193 ϱ ϰ͘ϱ ɇ;ϮϭϬϬͿ ɇ;ϮϬϯϬͿ ϰ ɇ;ϭϵϭϱͿ ϯ͘ϱ ɇ;ϭϳϳϱͿ ϯ ɇ;ϭϲϳϬͿ Ϯ Ϯ͘ϱ D Ϯ ɇ;ϭϯϵϬͿ ɇ;ϭϯϴϱͿ ϭ͘ϱ ϭ Ϭ͘ϱ Ϭ Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ Ϯ Ϯ͘ϱ ϯ ϯ͘ϱ ϰ : M 2 =0.81J +1.595 M 2 =1.101J +0.309 Figure IX :- RTs for two separate series of Σ baryons along with their straight line equations and string tensions. 194 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 179–194 References [1] M.H.Mac Gregor, Nuovo Cim. A, 103, 983 (1990). [2] M.H.Mac Gregor, Phys. Rev., D9, 1259-1329 (1974). [3] M.H.Mac Gregor, Phys. Rev., D10, 850-883 (1974). [4] N. Hothi, S. Bisht, Hadronic Journal, 38, 465-476(2015). [5] L.Chiatti, Phys. Essays, 27, 1,143-145 (2014). [6] K.A.Olive et al.Particle Data Group, Chin. Phys. CPhys. Essays, 38, 090001 (2014). [7] G. D. Coughlan, J. E. Dodd,The Ideas of Particle Physics, Second Edition, Cambridge University Press, London (1991). [8] A.O.Barut, Atoms with magnetic charges as models of hadrons, in Topic in Modern Physics, edited by Wesley.E..Brittin and Halis Odabasi, Boulder : Colorado Associated Univesity Press, Colorado (1971). [9] A.O.Barut, Phys. Rev., D3, 1747 (1971). [10] N. Hothi, S. Bisht, Indian Jour. of Phys., 85, 12, 1833-1842 (2011). [11] S. Bisht, N. Hothi and G. Bhakuni, Electronic Jour. of Theo. Phys., 24, 299 (2011). [12] D. Akers, hep-ph/0303139 (2003) [13] D. Akers, hep-ph/0303261 (2003). [14] K. Hagiwara et al.Particle Data Group, Phys. Rev., D66, 010001-1 (2002). [15] T.Sawada, hep-ph/0004080 (2000). [16] R. Mignani and E. Recami, Phys. Letters, B62, 41-43 (1976). [17] D. Akers, hep-ph/0309075 (2003). EJTP 14, No. 37 (2018) 195–212 Electronic Journal of Theoretical Physics Neimark-Sacker and Closed Invariant Curve Bifurcations of A Two Dimensional Map Used For Cryptography Yaniss Yahiaoui ∗ and Nourredine Akroune† Laboratoire de Math´ematiques Appliqu´ees, Facult´e des Sciences Exactes, Universit´ede Bejaia, 06000 Bejaia, Algeria Received 4 June 2017, Accepted 20 August 2017, Published 20 April 2018 Abstract: The purpose of this paper is to study dynamics and bifurcations of a family of two- dimensional noninvertible maps used for cryptosystems. Especially, the bifurcation of Neimark- Sacker is analysed algebraically and illustrated by numerical simulations. Furthermore, global bifurcations caused when a closed invariant curve intersects the critical lines are observed by simulation. On the other hand, several chaotic attractors have been observed in the phase plane for some particular values of the parameters. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Discrete dynamical systems; noninvertible maps; critical curves; Neimark-Sacker bifurcation; basins of attraction; global bifurcations; chaotic attractors 2010 MSC: Primary 37Gxx; Secondary 37Exx; 37Cxx; 37G35; 39A28; 39A30 PACS (2010): 02.30.Oz; 05.45.-a; 05.45.Gg; 05.45.Pq; 47.10.Fg 1 Introduction and basic terminologies 2 Let us consider the recurrence relation: Xn+1 = Tξ(Xn)=F (Xn,ξ), where ξ =(a, b) ∈ R 2 denotes a parameter, Xn =(xn,yn) ∈ R and F =(f,g) is a smooth function. Explicitly, Tξ has the form: ⎧ ⎨⎪ xn+1 = f(xn,yn,ξ) Tξ : (1) ⎩⎪ yn+1 = g(xn,yn,ξ) ∈ R2 k For a given positive integer k,theimage of rank k of X is, by definition, X = Tξ (X). ∈ R2 ∈ R2 k Similarly, a preimage of rank k of X is a point X satisfying Tξ (X)=X . ∗ Email: [email protected] † Email: akroune [email protected] 196 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 ∈ N ∈ R2 −k For a given k and a given X ,wedenotebyTξ (X ) the set of the preimages −1 of rank k of X . When the inverse mapping x = Tξ (x) is uniquely determined, we say that Tξ is invertible; otherwise, Tξ is said to be a noninvertible map. If the map Tξ is noninvertible, we have the notion of critical variety (critical point, critical line, critical curve, etc): 1.1 Critical curves The so-called critical curves are specific to noninvertible two-dimensional maps. It was introduced by C. Mira in 1964 (see [5]). If T = Tξ is noninvertible, We call critical curve of T of rank 1, the locus LC (abreviation of critical line) of points having at least two coincident preimages of rank 1. The locus of points which are coincident preimages of points of LC is denoted by LC−1 and called the set of merging preimages.When 2 especially the map T is differentiable, the set LC−1 is simply the set of the points of R in which the jacobian determinant of T vanishes. For k ∈ N, The critical curve of rank k k of T , denoted LCk−1, is defined by LCk−1 := T (LC−1). The critical curve LC can be constituted of one or several branches. These branches separate the plane of phases in opened regions denoted by Zi;eachregionZi corresponds to the set of points which have exactly i preimages of rank 1 by T . We refer to [5] for more precisions. 1.2 Phase plane of type (Z1 − Z3 − Z1) The so-called two-dimensional (Z1 − Z3 − Z1) maps (see [5], [4]) are such that the plane is divided into three unbounded open regions: a region Z3, whose points generate three real rank-one preimages, bordered by two regions Z1, whose points generate only one real rank-one preimage. For such noninvertible maps, the critical curve LC is discontinuous and constituted of two disjoint straight lines L and L dividing the plane of phases into 1 2 three regions: Z1 , Z1 and Z3. 1.3 Bifurcation of Ne¨ımark-Sacker The Neimark-Sacker bifurcation occurs when a stable focus of order k loses its stability as a bifurcation parameter is varied with the consequent birth of a closed invariant curve. The bifurcation can be supercritical or subcritical, resulting in a stable or unstable (within an invariant two-dimensional manifold) closed invariant curve, respectively (see [6, Chap 4]). Let us consider a as the bifurcation parameter and b fixed;sowehavethetwo following cases: — In the subcritical case: A repelling closed invariant curve Γ exists surrounding the stable fixed point, for a size and shrinks merging with the fixed point at a = a0; leaving a repelling focus. — In the supercritical case: At a = a0 the fixed point becomes an unstable focus and for a>a0 an attracting closed invariant curve Γ exists, surrounding the unstable Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 197 fixed point. If the smooth function F has the fixed point x = x∗(a) with simple eigenvalues λ1,2 = ±iθ0(a) r(a)e ,0<θ0(a) <π, then the Ne¨ımark-Sacker bifurcation is obtained by solving the equation r(a) = 1 and then by verifying that the solution a = ac satisfies the non- degeneracy conditions: dr (a ) =0 and eikθ(ac) =1for k =1, 2, 3, 4 da c (see Theorem A of Section 2.2 below). 1.4 Bifurcation of a closed invariant curve This bifurcation is generated by the transformation of an invariant closed curve, born from a focus fixed point of a non-invertible map T = Tξ of the plane via a supercritical Ne¨ımark- Sacker bifurcation, when some parameter is gradually moved away from its bifurcation value. Just after the Ne¨ımark-Sacker bifurcation, an attracting invariant closed curve, say Γ, appears around the unstable focus. While Γ ∩ LC−1 = ∅, the curve Γ, as well as the area of the phase plane enclosed by Γ, say A (Γ), is both forward invariant (under T ) and backward invariant (under T −1,whereT −1 is an inverse of T ). The situation changes when Γ touches the set of merging preimages LC−1, that is when Γ ∩ LC−1 = ∅. As soon as Γ comes into contact with LC−1 in a point A0,thatisΓ∩ LC−1 = {A0}, this bifurcation appears for some value a0 of the bifurcation parameter a. The image A1 of A0 by T is a point of contact between Γ and LC.Fora = a0 + ,where>0is ∩ { 1 2} A sufficiently small, we obtain that Γ LC−1 = A0,A0 and the area (Γ) is no longer forward invariant; in addition, we will observe the creation of convolutions of Γ. Those properties are related to the fact that curves crossing LC−1 are folded along LC.The iterate of rank n of the point A0, denoted An, is a tangential point of contact between Γ n and T (LC−1)=LCn−1. This bifurcation creates oscillations of Γ along the LCn’s and it is responsible of the changes of the form of Γ. When the bifurcation parameter grows more, another phenomenon can be observed; that is the appearance of knots, or loops or self intersections of the unstable set of the saddle belonging to the closed invariant curve. Then this situation is followed by homoclinic situations (intersections between the stable and unstable sets of the saddle) which leads to a chaotic attractor (see e.g., [2], [1]). 2 A study of a nonlinear two dimensional map used for cryp- tography Using nonlinear non-invertible two dimensional maps, it is possible to generate pseudo- random sequences, from which interesting cryptosystems can be born. Such cryptosys- tems rely on the consideration of chaotic signal properties resulting from those nonlinear maps. The model we present here is proposed in the study of chaotic signals in the area of communications as well as in the signal processing. This model is implemented on 198 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 a Digital Signal Processor (DSP), which resists all the attacks we have thought of (see [3]); it consists on the dynamic system generated by the cubic map Tξ :(x, y) → (x ,y), defined by: ⎧ ⎪ ⎨ x = y Tξ : ⎪ , ⎩ y = a(x − x3)+b(y − y3) (where ξ =(a, b) ∈ R2 is a parameter, with a =0). We begin with the following: Proposition 2.1. The map Tξ is non-invertible of the type Z1 − Z3 − Z1. Precisely, we have: √ 2 3 1 2 LC− = (x, y) ∈ R : x = ± and LC = L ∪ L , 1 3 1 2 where L and √ L are the curves√ respectively defined by the equations − 3 2 3 − 3 − 2 3 y = b(x x )+ 9 a and y = b(x x ) 9 a. Proof 2.2. The locus LC−1 of the coincident preimages of rank 1 of Tξ is given by 2 LC−1 = (x, y) ∈ R :det(JTξ(x, y)) = 0 , where JTξ(x, y) denotes the Jacobian matrix of Tξ at the point (x, y). The calculation gives: ⎛ ⎞ ⎜ 01⎟ JTξ(x, y)=⎝ ⎠ a(1 − 3x2) b(1 − 3y2) and then 2 det(JTξ(x, y)) = a(3x − 1). Hence: √ 2 2 2 3 LC− = (x, y) ∈ R :3x − 1=0 = (x, y) ∈ R : x = ± . 1 3 1 2 1 { ∈ So, the locus√ LC−1 is constituted of two branches√ L−1 and L−1,whereL−1 = (x, y) R2 3 } 2 { ∈ R2 − 3 } : x = 3 and L−1 = (x, y) : x = 3 . Consequently, the critical line LC = 1 1 2 2 Tξ(LC−1) is constituted of the two distinct curves L := Tξ(L−1)andL := Tξ(L−1), limiting an open region Z3 for which any point has three preimages of rank 1 and two open regions Z1 for which any point has a unique preimage of rank 1. The calculation gives: √ 2 3 L1 := T (L1 )= (x, y) ∈ R2 : y = b(x − x3)+ a ξ −1 9 √ 2 3 L2 := T (L2 )= (x, y) ∈ R2 : y = b(x − x3) − a . ξ −1 9 This completes the proof of the proposition. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 199 Graphical representation of the critical lines Figure 1 The critical lines of Tξ for a = 2 and b = −1 2.1 Existence of fixed points and local stability of the origin The results about the existence of fixed points of Tξ are given by the following proposition: Proposition 2.3. (i) If a + b ∈ [0, 1), then the map Tξ has a unique fixed point O(0, 0), which is simple. (ii) If a + b =1, then the map Tξ has a unique fixed point O(0, 0), which is triple. ∈ −∞ ∪ ∞ (iii) If a + b ( , 0) (1, + ), then the map Tξ has three simple fixed points: O(0, 0), a+b−1 a+b−1 − a+b−1 − a+b−1 P a+b , a+b and Q a+b , a+b . 3 3 Proof 2.4. Afixedpointx =(x, y)ofTξ satisfies (y, a(x − x )+b(y − y )) = (x, y). Simplifying this, we get that y = x and x satisfies the polynomial equation: x (a + b − 1) − (a + b)x2 =0. The results of the proposition follow. 2.1.1 Stability of the origin Now, we will state the topological classification of the fixed point O(0, 0) according to the values of the parameters a and b. Proposition 2.5. For the fixed point O(0, 0), the following topological classifications hold: (1) O(0, 0) is a saddle if one of these conditions is realized: (a) a<0, b ≤−2 and a>b+1. (b) a<0, b>0 and a + b>1. (c) a>0, b ≤−2 and a + b<1. (d) a>0, −2 (f) a>0, −2 Proof 2.6. The jacobian matrix of Tξ at O(0, 0) is given by: ⎛ ⎞ ⎜01⎟ JTξ(0, 0) = ⎝ ⎠ . ab The characteristic polynomial of JTξ(0, 0) is then given by: P (λ)=λ2 − bλ − a and has the discriminant Δ = b2 +4a. The results follow from the study of the sign of Δ and from the comparison of the modules of the two complexes eigenvalues of JTξ(0, 0) to 1. 2.2 Neimark-Sacker bifurcation We begin by finding the necessary and sufficiently condition for the equilibrium P to be non-hyperbolic. Note that the results obtained below (Proposition 2.7 and Theorem 2.9) become true for the equilibrium Q. 2 Proposition 2.7. Let (a, b) ∈ R such that a + b ∈ (−∞, 0) ∪ (1, +∞). Then the a+b−1 a+b−1 fixed point P a+b , a+b is non-hyperbolic if and only if one of the two following conditions holds: • a2 − b2 − a +2b =0. • ∈ −∞ − ∪ ∞ −2a2+4a a ( , 1) (1, + ) and b = 2a−1 . Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 201 Proof 2.8. The jacobian matrix of Tξ at P is given by: ⎛ ⎞ ⎜ 01⎟ JTξ(P )=⎝ ⎠ −2a−2b+3 −2a−2b+3 a a+b b a+b and has the characteristic polynomial 2a +2b − 3 2a +2b − 3 χ(λ)=λ2 + b λ + a . a + b a + b Now, we distinguish the three following cases according to the sign of the discriminant Δofχ. 1st case: (If Δ ≥ 0). In this case, the point P is non-hyperbolic if and only if one of the roots of χ is equal to 1 or −1; that is χ(−1) = 0 or χ(1) = 0. Since a + b ∈ (−∞, 0) ∪ (1, +∞), it is easy to see that χ(1) =0.So P is non-hyperbolic if and only if χ(−1) = 0; which gives the condition a2 − b2 − a +2b =0. nd 2 case: (If Δ < 0). In this case, χ has two complex conjugate roots λ1 and λ2 (λ2 = λ1). So, P is non-hyperbolic if and only if |λ1| = 1. But since 2 |λ1| =1⇔|λ1| =1⇔ λ1λ1 =1⇔ λ1λ2 =1 2a+2b−3 and λ1λ2 = a a+b , it follows that P is non-hyperbolic if and only if 2a +2b − 3 a =1 (1) a + b which gives −2a2 +4a b = (2) 2a − 1 On the other hand, the condition Δ < 0 is equivalent to: 2a +2b − 3 2 2a +2b − 3 b2 − 4a < 0. a + b a + b 2a+2b−3 1 By substituting in this last equation a+b by a (according to (1)), we get (after sim- plifying): b2 − 4a2 < 0, that is (b +2a)(b − 2a) < 0. −2a2+4a Next, by substituting in this last inequality b by 2a−1 (according to (2)), we get 2a2 +2a −6a2 +6a < 0, 2a − 1 2a − 1 202 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 that is (1 + a)(1 − a) < 0, which holds if and only if a ∈ (−∞, −1) ∪ (1, +∞). In conclusion, P is non-hyperbolic (in this case) if and only if a ∈ (−∞, −1) ∪ (1, +∞) −2a2+4a and b = 2a−1 . This completes the proof of the proposition. a+b−1 a+b−1 From Proposition 2.7, it is established that P ( a+b , a+b ) is non-hyperbolic ∈ −∞ − ∪ ∞ −2a2+4a when a ( , 1) (1, + )andb = 2a−1 . Henceforth, we choose a as a bifurcation parameter to study Neimark-Sacker bifurcation of T in the small neighbourhood of P . We obtain the following: 2 5 −2a0+4a0 Theorem 2.9. For any a0 ∈ (−∞, −1) ∪ (1, +∞) \{ , 2} and b0 = , we have 4 2a0− 1 a0+b0−1 a0+b0−1 a Neimark-Sacker bifurcation of T 0 0 about the equilibrium P0( , )= (a ,b ) a0+b0 a0+b0 ( a0+1 , a0+1 ). In addition, this bifurcation is supercritical if a ∈ (−∞,r ) ∪ (1, 5 ) ∪ 3a0 3a0 0 1 4 5 ∪ ∞ ∈ − − ( 4 , 2) (2, + ) and it is subcritical if a0 (r1, 1) (where r1 3.415895). To prove this theorem, we use the following theorem for Neimark-Sacker bifurcation in two dimensions: Theorem A (Generic Neimark-Sacker bifurcation [6, Chap 4]) For any generic two-dimensional one parameter system X → F (X, a), ±iθ0 having at a =0the fixed point X0 =(0, 0) with complex multipliers λ1,2 = e , there is a neighbourhood of X0 in which a unique closed invariant curve bifurcates from X0 as a passes through zero. The system has to satisfy the following genericity conditions: ±iθ(a) (1) r (0) =0 , where λ1,2(a)=r(a)e , r(0) = 1, θ(0) = θ0. (2) e±ikθ0 =1 for k =1, 2, 3, 4. (3) d(0) =0 , where d(0) is the first Lyapunov coefficient which is given by: −iθ0 iθ0 −2iθ0 e g21 (1 − 2e )e 1 2 1 2 d(0) = & −& g20g11 − |g11| − |g02| , 2 2(1 − eiθ0 ) 2 4 i j where gij (i, j =0, 1, 2) is the coefficient of Z Z in a specific form of F (see [6, Chap 4] for more details). In addition, if d(0) < 0 then we have a supercritical Neimark-Sacker bifurcation and if d(0) > 0, we have a subcritical Neimark-Sacker bifurcation. Proof 2.10 (Proof of Theorem 2.9). Let a be a parameter which varies in a small neighbourhood of a0.Letλ1,2(a) be the multipliers corresponding to the couple (a, b0) Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 203 | | and r(a):= λ1,2(a) .Soλ1,2(a) are the roots of the characteristic polynomial of JTξ(P ), where P ( a+b0−1 , a+b0−1 ). That is λ (a) are the roots of a+b0 a+b0 1,2 − − 2 2a +2b0 3 2a +2b0 3 χ(λ)=λ + b0 λ + a . a + b0 a + b0 If a is sufficiently close to a0, this polynomial has a negative discriminant, so λ1,2(a)are complex conjugates. Then we get − 2 2a +2b0 3 r(a) = λ1(a)λ1(a)=λ1(a)λ2(a)=a . a + b0 Thus 2a +2b − 3 r(a)= a 0 . a + b0 By deriving this with respect to a,weget: 2a+2b0−3 3 + 2 a a+b0 (a+b0) r (a)= . 2 a 2a+2b0−3 a+b0 Particularly, we have: 2a0+2b0−3 3 + 2 a a0+b0 (a0+b0) 0 r (a0)= . 2 a 2a0+2b0−3 0 a0+b0 − 2 Replacing b by 2a0+4a0 ,weget: 0 2a0−1 2 2 − r (a0)= a0 a0 +1 , 3a0 showing that r (a0) =0 (3) Next, solving χ(λ)=0fora = a0, we obtain the explicit formulas: −b b2 0 ± − 0 λ1,2(a0)= i 1 2 . 2a0 4a0 Since the discriminant of χ(λ)(fora = a0) is negative, we have λ1,2(a0) =1and 2 3 4 λ1,2(a0) = 1. It remains to show that λ1,2(a0) =1andthat λ1,2(a0) =1.Wehave √ ± 2πi 1 3 b0 1 3 3 λ1,2(a0) =1⇐⇒ λ1,2(a0)=e = − ± i ⇐⇒ − = − 2 2 2a0 2 −2a2 +4a ⇐⇒ ⇐⇒ 0 0 ⇐⇒ 2 − b0 = a0 = a0 4a0 5a0 =0 2a0 − 1 5 ⇐⇒ a =0ora = . 0 0 4 204 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 5 3 Since the values 0 and 4 are excluded for a0, we have certainly λ1,2(a0) =1. Similarly, we have: 4 b0 λ1,2(a0) =1⇐⇒ λ1,2(a0)=±i ⇐⇒ − =0⇐⇒ b0 =0 2a0 − 2 2a0 +4a0 ⇐⇒ =0⇐⇒ a0 ∈ {0, 2}. 2a0 − 1 4 Since the values 0 and 2 are excluded for a0,wehaveλ1,2(a0) = 1. In conclusion, we have: k λ1,2(a0) =1for k =1, 2, 3, 4(4) The relations (3) and (4) respectively correspond to the conditions 1. and 2. of Theorem A. Now, we are going to verify the condition 3. of Theorem A. To do so, we first transform the equilibrium P ( a0+b0−1 , a0+b0−1 )ofT to the origin. Putting τ := a0+b0−1 , 0 a0+b0 a0+b0 (a0,b0) a0+b0 the calculation gives the new system: ⎧ ⎪ ⎨ x = y . ⎩⎪ 3 3 y = a0 [x + τ − (x + τ) ]+b0 [(y + τ) − (y + τ) ] − τ Since 1 − 3τ 2 = − 1 , the last system is equivalent to a0 ⎧ ⎨⎪ x = y T : ⎪ . ⎩ y = −x − b0 y − 3a τx2 − 3b τy2 − a x3 − b y3 a0 0 0 0 0 Then, following Kuznetsov [6], we must transform T by putting ⎛ ⎞ ⎜x⎟ ⎝ ⎠ =2& ((X + iY )q) , y 2 b0 b0 where q is the eigenvector corresponding to the eigenvalue λ1 = − + i 1 − 2 of the 2a0 4a0 jacobian matrix of T at the origin. The calculation gives: ⎛ ⎞ ⎜ 1 ⎟ q = ⎝ ⎠ . − b − b2 2a + i 1 4a2 So, the transformation we need to use is: ⎧ ⎪ ⎨ x =2X 2 . ⎩⎪ b0 b0 y = − X − 2 1 − 2 Y a0 4a0 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 205 Using this last one, T is transformed to: ⎧ 2 ⎪ b0 b0 ⎪ X = − X − 1 − 2 Y ⎨⎪ 2a0 4a0 T : , ⎪ b2 b ⎪ Y = 1 − 0 X − 0 Y + c XiY j ⎩ 4a2 2a ij 0 0 2≤i+j≤3 where the cij’s are the real numbers given by: 3 3 2 2 12 a0+3b0 6b0 b0 c20 = τ ; c11 = τ ; c02 =6b0τ 1 − 2 ; 2 a0 4a0 2 b0 2a0 1− 2 4a0 4− 4 3 2 2 8 a0 b0 b0 b0 b0 c30 = ; c21 = −3 2 ; c12 = −6 1 − 2 ; 2 a0 a0 4a0 3 b0 2a0 1− 2 4a0 2 b0 c03 = −4b0 1 − 2 . 4a0 Next, by introducing the complex variables Z = X + iY and Z = X + iY , the system T is transformed to: k Z = λ1Z + gkZ Z , 2≤k+≤3 where the gk’s are the complex numbers given by: i 1 − i i i g20 = 4 c20 + 4 c11 4 c02 ; g11 = 2 c20 + 2 c02 ; i − 1 − i i 1 − i − 1 g02 = 4 c20 4 c11 4 c02 ; g30 = 8 c30 + 8 c21 8 c12 8 c03 ; 3 1 i 3 3 − 1 i − 3 g21 = 8 ic30 + 8 c21 + 8 c12 + 8 c03 ; g12 = 8 ic30 8 c21 + 8 c12 8 c03 ; i − 1 − i 1 g03 = 8 c30 8 c21 8 c12 + 8 c03. The expressions of g20, g11, g02 and g21 in terms of a0 and b0 are given by: 2 3 3− 2 g = 3b0 τ + i 12a0+6 b0 12a0b0 τ ; g = 3( a0+b0) τi ; 20 2a0 2 11 2 2 b0 b0 8a0 1− 2 1− 2 4a0 4a0 2 3 3− 2 2− 2 g = − 3b0 τ + i 6a0+3 b0 6a0b0 τ ; g = − 3 b + i 3(2 a0 b0) . 02 2a0 2 21 2 0 2 2 b0 b0 4a0 1− 2 4a0 1− 2 4a0 4a0 Then, using those formulas, we find that the first Lyapunov coefficient d(0) is given by: −iθ0 iθ0 −2iθ0 e g21 (1 − 2e )e 1 2 1 2 d(0) := & −& g20g11 − |g11| − |g02| 2 2(1 − eiθ0 ) 2 4 − 2 λ2g21 (1 2λ1)λ2 1 2 1 2 = & −& g20g11 − |g11| − |g02| 2 2(1 − λ1) 2 4 3 a = 0 49a3 +40a2b − 4a b2 − 4b3 − 45a4 2 − 2 0 0 0 0 0 0 0 4 (4a0 b0)(a0 + b0) − 3 − 2 2 3 4 81a0b0 33a0b0 +6a0b0 +3b0 . 206 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 − 2 By substituting b by 2a0+4a0 in the last expression of d(0), we get: 0 2a0−1 3 2 − − − − − 3 a0(8a0 +24a0 14a0 9) 3 (a0 r1)(a0 r2)(a0 r3) d(0) = − = − a0 , 16 (2a0 − 1)(a0 − 1) 2 (2a0 − 1)(a0 − 1) where r1 = −3.415895 ..., r2 = −0.402449 ... and r3 =0.818345 .... Consequently, we have: d(0) < 0ifa ∈ (−∞,r ) ∪ (1, 5 ) ∪ ( 5 , 2) ∪ (2, +∞) 0 1 4 4 , d(0) > 0ifa0 ∈ (r1, −1) showing (according to Theorem A) that we have a Neimark-Sacker bifurcation of T(a0,b0) about the equilibrium P ( a0+b0−1 , a0+b0−1 ), which is supercritical if a ∈ (−∞,r ) ∪ 0 a0+b0 a0+b0 0 1 5 ∪ 5 ∪ ∞ ∈ − (1, 4 ) ( 4 , 2) (2, + ) and subcritical if a0 (r1, 1). This completes the proof of the theorem. 2.3 Numerical simulations Now, we will give some numerical simulations for the system Tξ to support our theoretical − 2 results. for a =3∈ (−∞, −1) ∪ (1, +∞) \{5 , 2}, we find b = 2a0+4a0 = −1.2. So, If 0 4 0 2a0−1 we fix b = −1.2, then a = 3 is a bifurcation value. Figure 2 shows that the equilibrium is stable if a<3, loses its stability at a = 3 and an attracting invariant closed curve appears from the equilibrium when a>3. This is a supercritical Neimark-Sacker bifurcation (see Figure 2 below). a =2.999 and initial value a = 3 and initial value a =3.001 and initial value (0.001, 0.002). (0.001, 0.002). (0.001, 0.002). Figure 2 Phase portraits of system Tξ − 2 Next, for a = −3 ∈ (−∞, −1)∪(1, +∞)\{5 , 2}, we find b = 2a0+4a0 =30/7. So, if 0 4 0 2a0−1 we fix b =30/7thena = −3 is a bifurcation value. Figure 5 shows that the equilibrium is asymptotically stable if a<−3 and unstable for a ≥−3 (weakly at a = −3), while a unique and unstable closed invariant curve exists for a<−3. This is a subcritical Neimark-Sacker bifurcation (see Figure 3 below). Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 207 a = −3.001 and initial value a = −3 and initial value a = −2.999 and initial value (0.5, 0.5). (0.5, 0.5). (0.5, 0.5). Figure 3 Phase portraits of system Tξ 2.4 The existence and the stability of cycles of the system Tξ 2.4.1 The case of cycles of order 2 2 A cycle of order 2 of Tξ is a point M(x, y)ofR , satisfying Tξ ◦ Tξ(M)=M and Tξ(M) = M. The calculation of Tξ ◦ Tξ gives Tξ ◦ Tξ(x, y)=(x ,y ) such that: ⎧ ⎪ ⎨ x = a(x − x3)+b(y − y3) ⎪ . ⎩ y = a(y − y3)+b a(x − x3)+b(y − y3) − (a(x − x3)+b(y − y3))3 So, a cycle M(x, y)oforder2ofTξ must satisfy the system ⎧ ⎪ ⎨ x = a(x − x3)+b(y − y3) ⎪ , ⎩ y = a(y − y3)+b(x − x3) with the condition x = y. We observe that if we impose to M the additional condition y = −x (with x =0),a simple calculation gives (if a − b ∈ (−∞, 0) ∪ (1, +∞)) the two solutions: 1 1 1 1 (x, y)= 1 − , − 1 − and − 1 − , 1 − . a − b a − b a − b a − b So, we obtain the following Proposition 2.11. If a − b ∈ (−∞, 0) ∪ (1, +∞) then the map Tξ has at least the cycle of order 2: C = {M1,M2}, where 1 1 1 1 M 1 − , − 1 − and M − 1 − , 1 − . 1 a − b a − b 2 a − b a − b The following proposition establishes a necessary and sufficient condition for this cycle C = {M1,M2} of Tξ to be non-hyperbolic. 208 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 Proposition 2.12. Let (a, b) ∈ R2 such that a − b ∈ (−∞, 0) ∪ (1, +∞). Then the preceding cycle C = {M1,M2} of Tξ is non-hyperbolic if and only if one of the following conditions holds: • a2 − b2 − a − 2b =0. • ∈ −∞ − ∪ ∞ 2a2−4a a ( , 1) (1, + ) and b = 2a−1 . Proof 2.13. The jacobian matrix of Tξ at Mi (i =1, 2) is given by: ⎛ ⎞ ⎜ 01⎟ JTξ(Mi)=⎝ ⎠ , − 3 − 3 a 2+ a−b b 2+ a−b which has the characteristic polynomial 3 3 χ(λ)=λ2 + b 2 − λ + a 2 − . a − b a − b Now, we distinguish the three following cases according to the sign of the discriminant Δofχ. 1st case: (If Δ ≥ 0). In this case, the cycle C is non-hyperbolic if and only if one of the roots of χ is equal to 1 or −1; that is χ(−1) = 0 or χ(1) = 0. Since a − b ∈ (−∞, 0) ∪ (1, +∞), it is easy to see that χ(−1) =0.So C is non-hyperbolic if and only if χ(1) = 0; which gives the condition a2 − b2 − a − 2b =0. nd 2 case: (If Δ < 0). In this case, χ has two complex conjugate roots λ1 and λ2 (λ2 = λ1). So, the cycle C is non-hyperbolic if and only if |λ1| = 1. But since 2 |λ1| =1⇔|λ1| =1⇔ λ1λ1 =1⇔ λ1λ2 =1 − 3 and λ1λ2 = a 2 a−b , it follows that C is non-hyperbolic if and only if 3 a 2 − =1 (5) a − b which gives 2a2 − 4a b = (6) 2a − 1 On the other hand, the condition Δ < 0 is equivalent to: 3 2 3 b2 2 − − 4a 2 − < 0. a − b a − b − 3 1 By substituting in this last equation 2 a−b by a (according to (5)), we get (after simplifying): b2 − 4a2 < 0, Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 209 that is (b +2a)(b − 2a) < 0. 2a2−4a Next, by substituting in this last inequality b by 2a−1 (according to (6)), we get 6a2 − 6a −2a2 − 2a < 0, 2a − 1 2a − 1 that is (a +1)(a − 1) > 0, which is realized if and only if a ∈ (−∞, −1) ∪ (1, +∞). In conclusion, for this case, C is non-hyperbolic if and only if a ∈ (−∞, −1) ∪ (1, +∞) 2a2−4a and b = 2a−1 . This completes the proof of the proposition. 2.4.2 The general case Figure 4 below shows the existence of attractive cycles of orders 1 to 14. The black areas correspond to the values of the couple (a, b) for which there is no cycle of order ≤ 14. These black areas may correspond to the existence of chaotic attractors and the white areas correspond to the non-existence of attractor in the phase plane. Figure 4 Stability and existence of attractive cycles of Tξ 2.5 Numerical simulations concerning some other types of bifurcations Now, we present some numerical simulations including Neimark-Sacker bifurcations of cycles of order 2 and bifurcations of closed invariant curves of Tξ. Furthermore, several chaotic attractors are observed for some particular values of a and b. 2− Let a =1.417 ∈ (−∞, −1) ∪ (1, +∞)andb = 2a0 4a0 = −0.9. By fixing b equal to 0 0 2a0−1 b0 and varying the parameter a from the value 1.415 (1) For a =1.415, we have a stable cycle of order 2 of focus type: C = {M1,M2},where M1(−0.7536 ...,0.7536 ...)andM2(0.7536 ...,−0.7536 ...) (see Figure (a)). (2) For a0 := 1.417, we observe the appearance of a Neimark-Sacker bifurcation from the preceding cycle C (see Figure (b)). 1 2 (3) For a0 (7) For a5 := 1.999 ≤ a ≤ 2.18, we observe the appearance of chaotic attractors (see Figures (h), (i) and (j)). (8) For (a, b) ∈ {(−2.14, −0.90), (−3.41, 4.70), (3.16, −1.20)}, other chaotic attractors appear in the phase plane (see Figures (k), (l) and (m)). (a):a =1.415 (b):a =1.417 (c):a =1.45 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 211 (d):a =1.527 (d1):a =1.527 (e):a =1.55 (e1):a =1.55 (e2):a =1.55 (f):a =1.6 (f1):a =1.6 (g):a =1.65 (g1):a =1.65 (h):a =1.999 (i):a =2.09 (j):a =2.18 212 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 195–212 (k):a = −2.14 and b = −0.9 (l):a = −3.412 and b =4.70 (m):a =3.16 and b = −1.2 Figure 5 Phase portraits of system (T ) References [1] A. Agliari, G. Bischi and R. Dieci. Global Bifurcations of Closed Invariant Curves In Tow-Dimensional Maps: A Computer Assisted Study, International Journal of Bifurcation and Chaos, 1 (2005), p. 1285-1328. [2] A. Agliari, G. Italo Bischi and L. Gardini. One Some Methods for the Global Analysis of Closed Invariant Curves in Two-Dimensional Maps, In Business Cycle Dynamics: Models and Tools, T. Puu and I. Sushko, Springer-Verlag, Berlin, 2006. [3] V. Guglielmi, P. Pinel, D. Fournier-Prunaret and A. Taha. Chaos-based cryptosystem on DSP, Chaos, Solutions and Fractals, 42 (2009), p. 2135-2144. [4] C. Mira, G. Bischi and L. Gardini. Basin Fractalization Generated By a Two- dimensional Family of (Z1 − Z3 − Z1)Maps,International Journal of Bifurcations and Chaos, 16, (2006), p. 647-669. [5] C. Mira, L. Gardini, A. Barugola and J.C. Cathala. Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, Singapore, Series on Nonlinear Science,SeriesA,vol.20, (1996). [6] Y.A. Kuznetsov. Elements of applied bifurcation theory, 2nd edition, Springer- Verlag, New York, 1998. EJTP 14, No. 37 (2018) 213–249 Electronic Journal of Theoretical Physics Physics of Currents and Potentials IV. Dirac Space and Dirac Vectors in the Quantum Relativistic Theory V.A. Temnenko∗ Tavrian National University, Vernadsky prospect 4, 95022 Simferopol, Crimea Received 16 November 2017, Accepted 25 December 2017, Published 20 April 2018 Abstract: There has been presented an attempt to transfer the fundamental ideas of physics of continual currents and potentials, described in the previous articles of this series [1], [2], [3], from the classical theory to the quantum relativistic theory. The concept of multidimensional Dirac space, which should contain wave equations of the relativistic quantum theory, has been introduced. Dirac space dimension d is determined by Yang-Mills multiplicity of the sector of physics: d = 8 for the singlet (quantum electrodynamic states); d = 20 for the two-sector singlet-triplet states; d = 52 for the three-sector singlet - triplet- octuplet states. It has been shown that the quantum relativistic state can not be described by the unique wave function (four-component Dirac vector). Singlet states are described by a pair of Dirac vectors, two-sector singlet-triplet states are described by four Dirac vectors, eight Dirac vectors are necessary for description of the three-sector singlet-triplet-octuplet states. It has been shown that the necessity to consider the Riemann curvature of space causes additional difficulties in the process of construction of the quantum relativistic theory. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Dirac space; Dirac Vectors; Relativistic Fourier-Transform; Fourier-Phase In Riemann Space PACS (2010): 03.65.Ta; 03.70.+k; 04.62.+v; 12.90.+b 1 Introduction. From Classical Field Theory to Quantum Relativistic Theory. Dirac Space The classical field theory with continual currents, presented in the preceding articles of this series [1], [2], [3], did not require in its formulation any radical deviation from the ∗ Email: [email protected] 214 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 fundamental ideas formed in theoretical physics of the XXth century. The main broaden- ing of the well-established ideas in these articles consisted of rejection of the mechanical interpretation of 4-currents appearing in the theory: the space-like 4-current is not gen- erated by the motion of the material (”ponderable”, as Albert Einstein would have put it) charge carrier; the space-like current is primary and can not be reduced to any other, more simple entity. It took the physicists of the XIXth century several decades to abandon the attempts to provide mechanical interpretation of the electromagnetic field. J. Maxwell was per- sistently searching for these interpretations; the traces of this search can be found in H. Hertz’s works, but the works of G.A. Lorenz are already free from the attempts of mechanical interpretation of field: field is primary and can not be reduced to anything simpler. Future historians of science may find it difficult to understand why the rejection of me- chanical interpretation of the second half of the electromagnetic dyad, 4-current, was a full century late. This broadening of Lorentz approach to electromagnetic theory seems quite natural and expected. Undoubtedly, it becomes natural and even trivial only with the simultaneous recognition of dyadic nature of a field: ”a current and a potential” as the two inseparable and equal halves of the continual field dyad, the yin and yang of electrodynamics. The field thought of the most outstanding representative of the ”field ideology” of the XXth century physics, Albert Einstein, was distinctively monadic. Einstein perceived the field as a monad described by a single physical-geometric object, and he hoped to see the particles as bunches of field energy; perhaps – as the field singularities, characteristics and motion of which are completely determined by the field and can not be set arbitrarily. For thirty-six years of thinking and working on this series of articles, I have often asked myself a question: ”Would Albert Einstein have accepted the embodiment of Maxwell field program, which is presented in the first article of this series [1]?” Now I believe, that after the period of ”monadic resistance” and stern grumbling, Einstein would have supported such version of electrodynamics. This is really a completely field and totally classical theory, and, besides, it incorporates the Einstein gravitation theory. In a sense, the theory presented in articles [1], [2], [3], is the embodiment of Einstein’s ideal of the classical field theory achieved at the cost of the rejection of monadic descrip- tion of fields and the rejection of visual and naive mechanical interpretation of currents. Even if the field pattern of physical reality, presented in articles [1], [2], [3], is not the final version of Einstein’s ideal, it, anyway, indicates a decisive step towards this ideal. And, after all, does not the idea of the primary nature of space-like 4-current open not less exciting intellectual perspective for theoretical physics than it once opened – ungrounded and arising from nowhere – Louis de Broglies idea of matter waves; the idea, which was decisively and immediately supported by Einstein. Albert Einstein was convinced that the nonlinearity of field equations was necessary for existence of particles as some bunches of field energy. However, this is true only for the monadic theory. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 215 Dyadic field electrodynamics, presented in article [1], is non-linear (even if we omit the requirement of the Riemann nature of geometry) despite the linearity of field equations: the boundary conditions for currents at the boundaries of current zones are non-linear, and the mere presence of previously unknown boundaries of current zones in the theory makes the problem nonlinear (even if the boundary conditions were linear). In the quantum version of physics of currents and potentials, we have to turn decisively off the beaten ”field” track of physics of the XXth century. In the classical version of the theory, presented in the preceding articles of this series, we were just slightly com- plementing this beaten track dating back to Faraday, Maxwell and Lorentz. We tried to use the ”classical” asphalt to connect this track of electrodynamics with the ”branch” of Yang-Mills theory which for decades had been positioning itself as purely quantum pathway that had emerged from quantum wasteland, without any classical basis. Consistent, shrunk into itself, free from any self-contradictions and divergences, the quan- tum formulation of the field theory requires a rather radical break from the existing ap- proach of modern theoretical physics - such a radical break that perhaps it could not have been approved by any of the genius founding fathers of quantum mechanics and quantum field theory. The essence of this break is to change the interpretation of the concept of physical reality itself. For classical relativistic physics, field dyads in the three sectors of physics are the physical reality which is immersed in the four-dimensional space-time continuum, the Riemann geometry of which obeys the Einstein equations. In quantum relativistic physics, we apparently have to assume that semi-components of the field dyads (currents or poten- tials), along with the space-time coordinates, are not unknown physical quantities, but just the arguments of the Dirac wave functions. In other words, during the transition from classical to quantum version of the theory, the fields are no longer an element of physical reality and they turn into the capacitance for reality – the reality of an en- semble of field realizations; the capacitance supplementing the space-time or momentum four-dimensional continuum with its dimensions. Mathematically, the ensemble is described by some set of wave functions - the Dirac spinors. These spinors in quantum electrodynamics depend on eight arguments: four space-time coordinates (or four coordinates in the momentum space) plus four current components (or four potential components). In the complete three-sector problem of physics (”the problem of the Standard Model”), each spinor has 52 arguments: 4 space- time coordinates (or 4 coordinates in the momentum space) plus four components of each of the twelve currents of the Standard Model (or each of the twelve potentials, or arbitrary combination of currents and potentials, constructed so that spinor arguments contain only one semi-component – current or potential – of each of the field sector dyad of the three sectors of physics). Wave equations for the Dirac spinors should be formu- lated as nonlinear equations in partial derivatives in this fifty-two-dimensional space. On the account of the missing of any meaningful term that would adequately describe this 52-dimensional continuum, we shall call it the Dirac space. In singlet (electrodynamic) states, dimension d of the Dirac space is equal to eight; in pure triplet Yang-Mills states 216 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 d = 16, in mixed singlet-triplet (electroweak) states d = 20, in pure octuplet (chromody- namical) Yang-Mills states d = 36; in total three-sector states d = 52. For quantum version of the theory, in contrast to classical version, it is impossible to intro- duce the concept of field tensor – the operation of potential differentiation by coordinates does not have any physical sense: coordinates and potentials are equal arguments of the Dirac spinors in the Dirac space. Accordingly, the quantum versions of the Maxwell and Yang-Mills equations can only be some operator relations. Basic wave equations for the Dirac spinors must be constructed as some radical generalizations of the Dirac equations. The Dirac equation itself, losing the status of the exact equation of physics, in the frame- work of this theory can be some approximate relation which appears after some procedure of an approximate integration of the wave equations by the field arguments, like in the classical electrodynamics of continual currents described in article [1], the Lorenz equa- tion (equation of motion of a point charge carrier) appears as an approximate equation after the approximate integration of the equations of continual theory by the volume occupied by currents, in the approximation of a ”weak external field” (see [1]). We did not manage to construct the basic wave equations of the theory in this work. Moreover, according to the ”rule of Tridentine prudence” [3], we would prefer to avoid premature discussion of the difficult questions of geometry: what space should the equa- tions of gravitation, controlling the metric, be entered in – in the usual four-dimensional continuum , or in 52-dimensional Dirac continuum? How should the energy-momentum tensor be constructed in the theory which does not have a field tensor?, etc. Within the framework of this theory, we have an intention to describe the method, by means of which it is possible to construct the Dirac spinors on the grounds of observa- tional data1 and to describe the connection between spinors. We shall preface this description with presentation of non-relativistic quantum theory in such formulation which allows a natural transfer to the relativistic realm. Discussion of the foundations of quantum mechanics, learnt already at the University freshman class, can make a skilled reader bored and irritated. By putting up with this, the author hopes that this form of presentation of a well-known non-relativistic scheme will make it easier for the reader to further perceive the relativistic relations in the Dirac multi-dimensional space. 2 Non-relativistic Scheme (Born-Heisenberg-Schrodinger Program) 2.1 Born Density. De Broglie and Born Postulates Non-relativistic quantum mechanics is based on the Newtonian concept of physical system as an object consisting of point particles n, coupled by forces of a long-range interaction, but, in contrast to Newtonian mechanics, in quantum mechanics, the n – particle system 1 Here it would be appropriate to remind of the uncompromising Einsteinian maxim: ”Only the theory determines what can be observable”. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 217 itself is not an element of reality allowing a mathematical description. The ensemble of identical to each other n-particle systems is the element of reality, i.e. an infinite number of the systems, from which it is possible to extract individual copies of such systems for making instantaneous coordinate measurements of all the particles of the system. Co- ordinate measurement in the frames of non-relativistic conception of reality can be, in principle, made with unlimitedly high precision. The process of this measurement de- stroys the system completely, and it is impossible to make further measurements with the same copy of the system. In contrast to Newtonian mechanics, where only point coordinates are observable quantities, quantum mechanics is based on the postulate of the existence of one more observable quantity for each particle – momentum vector k, which has the dimension [length]−1. Within the framework of the non-relativistic concept, particle momentums can be measured, in principle, with unlimited precision. The process of this measurement destroys the system completely, and it is impossible to make further measurements with the same copy of the system. The fundamental postulate of measurability of particle momentum and existence of an infinite three-dimensional momentum space with Euclidean metric, apparently, should be formulated explicitly and should be called de Broglie postulate. This postulate trans- forms geometric spatial monad of Newtonian mechanics into space-momentum dyad of quantum mechanics {x|k}. This is an important step on the way of relativistic trans- formation of space-time four-dimensional Minkowski monad xν into the dyad consisting of 4-coordinates xν and 4-momentums kν: {xν kν}. The essential feature of relativistic quantum theory is that we can not interpret semi-components of this dyad – xν and kν – as coordinates and momentums of the point object: point objects do not exist in relativistic physics. In quantum relativistic physics we should just talk about coordinate space xν and momentum space kν and assert that with corresponding measurements, we have an opportunity to discover something in the neighborhood of point xν (or point kν) on a small but finite element of the oriented three-dimensional hyper-surface σ in four-dimensional coordinate space (or momentum space). But let us revert to non-relativistic physics. For our purposes it is sufficient to concentrate on the accurate formulation of mathemat- ical apparatus which is needed to describe the non-relativistic ensemble of one-particle systems (n=1). Let us suppose that at some instant of time t, we made instantaneous coordinate mea- surement N of copies of the system, and let us assume that ΔN of these measurements xt xt have registered the presence of the particle in some small but finite volume ΔV in the x neighborhood of the point with radius-vector x in a pre-selected inertial frame of refer- ence. We postulate that with the unlimited growth of a number of measurements of N xt there is limit R of the relation ΔN /N . Due to arbitrariness of the shape and volume x t xt xt of ΔV this limit can be expressed as ΔV volume integral of some function ρ(x,t)that x x x 218 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 depends on coordinates x at the instant of time t: ΔN xt R t = lim = ρ(x,t)dV. (1) x N →∞ x xt Nt x x ΔV x It is obvious that function ρ(x,t), determined by relation (1), is nonnegative, integrable x with any small volume ΔV into the neighborhood of any point x and normalized per unit x under integration into (1) over the entire infinite three-dimensional coordinate space. The dimension of ρ(x,t)iscm−3. x In an absolutely similar way, making measurements of momentum for Nt copiesofa k 2 one-particle system at the same instant of time t , we can discover that ΔNt of k these measurements register the presence of a particle in a small but final volume ΔV k of momentum space in the neighborhood of some momentum k. The equipment for measuring momentum k is meant to be fixed relative to the same inertial system in which coordinates x are measured and to agree in the orientation of coordinate axes with the equipment that measures coordinates. We postulate that with the unlimited growth of a number of measurements of Nt there is a limit R t for relation ΔNt/Nt.Dueto k k k k arbitrariness of the shape and size of volume ΔV, this limit can be expressed as a volume k ΔV integral of some function ρ(k,t) that depends on momentums k at the instant of k k time t: ⎛ ⎞ ΔNt ⎝ k ⎠ R t = lim = ρ(k,t)dV. (2) k Nt→∞ k k k Nt k ΔV k Function ρ(k,t), determined by relation (2), is nonnegative, integrable with any small k volume ΔV into the neighborhood of any momentum k and normalized per unit under k integration into (2) over the entire infinite three-dimensional momentum space. The di- mension of ρ(k,t)iscm3. k We shall call functions ρ and ρ3, determined by relations (1) and (2), the Born densities: x k ρ is the Born density in coordinate space, ρ is the Born density in momentum space. x k Let us call the assertion of the existence of limits (1) and (2) and, respectively, the exis- tence of the Born densities, as the Born postulate. The use of terms, established in quantum mechanics, requires the obligatory use of the term ”probability” in naming functions ρ and ρ: ρ is the ”probability density of finding x k x the particle in the coordinate space”, and ρ is the ”probability density of finding a particle k in the momentum space”. 2 It is obvious that this expression is principally non-relativistic; it can not be given Lorentz-invariant sense. However, it does not cause any difficulties, since the choice of instant of time for measurements in momentum space does not matter for the ensemble of isolated systems. 3 The indications on the arguments of these functions (x,t)or(k,t) will further be omitted. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 219 We suppose that the term ”probability” in definition of the Born density is redundant. This term refers to the artificial, archaic and unlawful (or subconscious) preservation of Newton’s idea of a separate copy of the system as an element of physical reality in quan- tum mechanics. In quantum mechanics, infinite ensemble of systems is physical reality: the measurements are made on some individual copies of the system, but mathematical description is only possible for ensemble. In fact, individual system does not have ”being- in-time” which is comprehensible for a macro-observer, but ensemble has such being. All physical information on the ensemble is enclosed in a pair of the Born densities ρ,ρ . We shall call this pair ”the Born pair”. x k 2.2 Heisenberg Postulate Awareness of the Born pair allows to calculate different characteristics of the ensemble, such as the ensemble mean value of i-component of radius vector xi : xi = ρ xidV, (3) x x or the ensemble mean value of i-component of momentum ki : ki = ρ kidV. (4) k k Integration by the entire three-dimensional coordinate space (3) or by the entire three- dimensional momentum space (4) is implied in these relations. In the same way we can calculate the mean ensemble value of the square of i-Cartesian 2 component of radius-vector xi , or the mean ensemble value of the square of i-Cartesian 2 momentum component ki : 2 2 xi = ρ xi dV, x x 2 2 ki = ρ ki dV k k as well as the dispersion value of each coordinate σi and each momentum component σi: x k σ = (x − x )2 = x2 − x 2, xi i i i i 2 2 2 σi = (ki − ki ) = k − ki . k i Let us form 3 × 3-Heisenberg matrix Hij of quantities of dispersions: Hij = σi σj. (5) x k It is obvious that the elements of matrix (5) are nonnegative. However, for diagonal elements, a stronger Heisenberg inequality is also valid: 1 H ≥ δ . (6) ij 2 ij 220 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 This inequality expresses a fundamental postulate of non-relativistic quantum physics of ensembles: There are only such Born pairs ρ,ρ that satisfy Heisenberg inequality (6). x k (”The Heisenberg postulate”). We shall call the Born pair which satisfies inequality (6), the Born ensemble. 2.3 Schr¨odinger Theorem The validity of the statement, which we will call Schr¨odinger theorem, follows from the Heisenberg Postulate: For each Born ensemble there is a pair of real functions S(x,t)andS(k,t), (the ”phase x k pair”) such one that the complex functions ψ and ψ, determined by the relations x k i S ψ = ρ e x, (7) x x i S ψ = ρ e k, (8) k k are Fourier images of each other . ψ FT= ψ. (9) x k In relation (9) symbol ”FT=” describes the procedure of non-relativistic three-dimensional Fourier transform, connecting the functions in the coordinate space with the functions in the momentum space: 1 ψ = ψ ei k·x dV, x (2π)3/2 k k (10) 1 ψ = ψ e−i k·x dV. k (2π)3/2 x x Integrals in (10) are taken over the entire momentum space or, respectively, over the entire coordinate space. The two equations (10) are not independent: if the first one is satisfied, the second one is also satisfied. From the determination of complex functions ψ (7) and (8) follows that: ρ = ψ ψ∗, x x x (11) ρ = ψ ψ∗, k k k where asterisk (∗) is a symbol of complex conjugation. After substitution of expressions (7) and (8) into relations (10), and after separation of the real and imaginary parts, these relations form a system of two nonlinear real integral equations relative to a pair of unknown real phase functions S and S. x k Schr¨odinger theorem is the statement of the existence and uniqueness4 of a solution to 4 With the accuracy to within the arbitrary, time-dependent term. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 221 this system of nonlinear integral equations. Let us name complex functions ψ and ψ, determined by relations (7) and (8), the wave x k functionsofSchr¨odinger or, to be more precise, the Fourier-doublet of Schr¨odinger wave functions. Under conjugation (9), existing between the components of Fourier-doublet, only one of the two components of the doublet is enough to describe the Born ensemble: the second one can be calculated by formula (9). The Schr¨odinger theorem proving is unknown to the author; let us leave the burden of its proving to mathematicians. Modern computational mathematics does not contain any clear and simple procedure for constructing a phase pair S, S by the Born ensemble x k ρ,ρ . x k The statement, converse to the Schr¨odinger theorem, (we shall name it Weyl theorem), is well known, and its proving can be found in any textbook on quantum mechanics: for any Fourier -doublet of Schr¨odinger wave functions, the Heisenberg inequalities (6) are satisfied5. 2.4 Ensemble Mass. Planck Postulate The Born densities are continuous and differentiable time functions which allows to study the dependence of any of the ensemble characteristics on time. Velocity of ensemble v is an important characteristic of the one-particle ensemble: • v = x = x ∂t ρ dV, (12) x x where ∂t ρ is the partial time derivative of the Born density ρ. x x Velocity v is the ensemble characteristic. It makes no sense to speak about ”particle velocity” – this notion has no representation in the apparatus of quantum mechanics. Relation (12) allows us to formulate the postulate of existence of the Born ensemble mass: For any of the Born ensemble there is a positive constant m,suchonethat 1 v = k . (13) m The statement of quantity constancy is the hidden definition of the term ”uniform time”. The physical content of equality (13) is the following: if we choose the clock (”uniform time”) so that quantity m is the constant for some single one-particle ensemble, for any 5 Since student years, the author of this article was indignant by the very fact of Weyl derivability of fundamental and empirically irrefutable uncertainty relation (6) from arbitrarily constructed complex- valued apparatus of quantum mechanics. The scheme of construction of non-relativistic quantum me- chanics, presented here, allows us to derive a mathematical fact of the existence of complex wave function from the Heisenberg physical postulate: understanding of physics precedes the construction of mathematical apparatus. We tend to use the same approach in the relativistic area. 222 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 other one-particle ensemble its mass m, which is determined by (13), will be also constant. In fact, relation (13) contains three physical laws: the mass is scalar, the mass is positive, the mass is constant. As well as velocity, mass is the ensemble characteristic. It makes no sense to speak about the mass of a single particle: this concept does not have a representation in the apparatus of quantum mechanics6. In accordance with (13), the quantity, reciprocal to mass, has a dimension of diffusion coefficient: 1 =cm2/sec. m In Newtonian mechanics, as Ernst Mach already shown, mass is the characteristic of a system of two particles (we mean the mass ratio in the two-particle system, i.e., the dimensionless characteristic). To fix the numerical value of the mass of some particle in classical mechanics, it is necessary to choose an arbitrary standard of mass: mass in classical mechanics has independent dimension. In quantum mechanics, mass characterizes a one-particle ensemble (there is no need to create an ensemble of two-particle systems) and it does not have independent dimension (there is no need for existence of mass standard). Comparison of classical and quantum mass is possible in cases of making not very ac- curate and not destructive measurements – for example, in the cloud chamber – which allow to repeat the observations of the same copy of the system over time. During such measurements, the classical particle mass mc can be determined with some limited ac- curacy. The connection of classical quantity mc and ensemble characteristic m is fixed by the following postulate (the correspondence principle or Planck’s postulate): there is a universal constant (Planck’s constant), such one that for any one-particle Born ensemble, the following relation is satisfied mc = m . (14) Planck’s constant appears only in Planck’s postulate7, and in other related statements. It establishes the connection between not very accurate – in essence, not very accurate 6 Undoubtedly, this statement would have caused the protest of many physicists, not just those per- sistent opponents of quantum mechanics, as Albert Einstein, Erwin Schrodinger and Louis de Broglie. Preparation of an ensemble of identical one-particle systems requires confidence in the fact that we do include the same particles into it – in particular, with the same masses. Description of actual procedures for preparation of the ensemble, as well as description of procedures for comparing actual observations that have a limited accuracy, with predictions of quantum mechanics, which apparatus presupposes infinite accuracy of the measurement – it would be better to leave such description to the competent physicists-experimenters. This description requires the explicit formulation of specific hypotheses about the relationship of micro-and macro- world, which implies not only good knowledge of the capabilities of measuring equipment, but also a philosophical mind of the appropriate writer. 7 Perhaps from the point of view of historical correctness, formulas (13) and (14) should be to associated with the name of Louis de Broglie. But we have already used the term ”de Broglie postulate” before. However, in theoretical physics, any ”I” is secondary; surnames can ”stir up the contents with randomness obscuring the pure image of the truth”, as the religious philosopher Pavel Florensky once said. Good works in any field, including theoretical physics, are similar to ancient Greek tragedies, ”they are not written by whoever wishes to, or whenever he wishes to do it”, as the same Florensky said. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 223 – individual macro-measurements and ensemble characteristics which imply, in principle, unlimited precision of measurements and calculations. The use of Planck’s constant in the apparatus of quantum mechanics is not necessary. Formula (13) does not allow to determine the mass of a one-particle ensemble in the rest frame of the ensemble, in which k =0andv = 0. In this system, the mass of the ensemble can be determined by the velocity of diffuse spreading of the ensemble in the co- ordinate space. Perhaps this is the basic physical meaning of the non-relativistic concept ”one-particle ensemble mass”: the characteristic of the ensemble spreading velocity. 2.5 One-particle Ensemble Dynamics. Wave Equation The Born ensemble as a dynamic system must be described by a pair of equations that determine time derivatives of the Born densities by the Born densities known at this point of time: ∂t ρ = F ρ,ρ , x x x k (15) ∂t ρ = F ρ,ρ . k k x k The right sides of equations (15) are non-linear in both of their functional arguments, and each of them is nonlocal in the conjugated argument: function F contains the integral x operator over the momentum space, function F contains the integral operator over the k coordinate space. Let us name relation (15) the Born dynamic equation, and functions F and F –the x k Born generators. The overt form of the two Born generators is unknown for any of the ensembles8. For one-particle ensemble the following can be taken as a postulate: F =0. (16) k The Born density of one-particle ensemble in the momentum space, according to (15) and (16) is determined only by the initial conditions. Formula (16) contains both the momentum conservation law of the ensemble, and the energy conservation law of the ensemble. Ignorance of the general forms of the Born generators F and F makes it impossible to x k give the explicit real formulation of the equations of quantum mechanics in form (15) – the mechanics, interpreted as ”the theory of dynamic systems in the Born ensembles space”. However, God, while creating the quantum world, was so forgiving that we do not need to know the explicit form of the Born generators. 8 It is easy to write down the expressions for Born generators if we allow the use of phase functions S x and S in the notation, – but the explicit form of dependence of the phase functions on the Born densities k is unknown. 224 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 We can formulate the following postulate of autonomy and linearity (”Schr¨odinger pos- tulate”): The equation, which each of Fourier-doublet components of Schr¨odinger wave functions obeys to (”Schr¨odinger equation” or ”wave equation”), is autonomous relative to the second component of the doublet, linear and homogeneous. This postulate, which should be called the postulate of the divine indulgence, has one convenient consequence: it is sufficient to know only one Schr¨odinger equation – for ex- ample, for the momentum component of Fourier-doublet; but, according to (9), we will obtain the equation for the coordinate component through the Fourier transform of the equation for momentum component. The Schr¨odinger postulate has a purely mathematical formulation. It disguises its true physical sense and creates anxiety. God is such a perfect mathematician, that, possibly, He does not exploit math at all, and, anyway, He does not care about our human prob- lems connected with the solution of nonlinear integral-differential dynamic Born equations (15). Why did God need such indulgence to our limited mathematical abilities? It would be great if there was a formal proof of the fact that ”Schr¨odinger postulate” is a mathematical consequence of the Heisenberg postulate – i.e. ”Schr¨odinger Postulate” is actually not a postulate, but a theorem. In this case, God had just no choice: he could not create quantum ensembles with nonlinear Schrodinger equations. The author does not know such proof. If ”Schr¨odinger postulate” is independent of the Heisenberg postulate, God did have a choice, and the choice made by Him, should be formulated not in the language of a fic- titious mathematical object – Fourier-doublet of complex Schr¨odinger wave functions - but in a more meaningful language , the language of Born generators F and F ,sothat x k the statement of the linearity of Schr¨odinger equation would be a trivial consequence of this formulation. Leaving the problem of linearity of Schr¨odinger equation unclear up to the end in a ”deep non-relativistic rear”, we face the question: should we expect the same divine indulgence in the relativistic theory? Should equation for the wave functions be also linear in rela- tivistic physics? For the one-particle Born ensemble considered here, Schr¨odinger equation in momentum space, as it is known, has the following form: k2 i∂t ψ = ψ. (17) k 2m k It is possible to create a certain illusion of the derivability of (17) based on non-consecutive semi-classical ideas, but it would be better to consider wave equation (17) the quantum postulate. The Schr¨odinger equation in coordinate space arises from (17) by means of the Fourier transform: 1 i∂t ψ = − Δψ, (18) x 2m x Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 225 where Δ is the Laplace operator. Of course, we do not have a necessity to solve equation (18): it is enough to solve (17) and perform the Fourier transform for the obtained solution. A definite and, in a sense, insuperable difficulty is connected with the need to satisfy the initial conditions for equations (17) and (18). Keeping to the framework of rational physical interpretation, we should specify the initial conditions in terms of the Born pair ρ,ρ . Transferring these Born conditions to the initial conditions for Schr¨odinger x k wave functions, we should, at least at the initial moment of time, solve nonlinear integral equations (9) which, in accordance with formulas (7) and (8), determine the phase pair S, S . Direct specifying of the initial conditions for Schr¨odinger’s Fourier-doublet of x k the wave functions, apparently, should be considered impossible: we are preparing the ensemble as a certain Born pair, and not as Schr¨odinger’s Fourier-doublet. Therefore, the non-linear and nonlocal Born nature of quantum mechanics, is seen through Schr¨odinger’s linearity at least as the problem of the initial conditions9. 2.6 Born Pair and Schrodinger Fourier-doublet for the Ensemble of Two-particle Systems The ensemble of two-particle systems is prepared from two ensembles of one-particle systems, which masses, m1 and m2, are known from the previous measurements made on these one-particle ensembles. A single measurement of coordinates x (or, respectively, of momentums k) gives the group of two radius vectors (x1, x2), or two momentums (k1, k2). However, we can not associate a particular radius vector (or momentum) with a particular particle, having number 1 or 2, and, respectively, mass m1 or m2:point particles do not have any tags attached, and they are indistinguishable under individual measurements. Therefore, we have to turn each coordinate (or momentum) measurement into two possible descriptions of the measurement results, going over both of the possible options of assigning numbers to particles. A number of N coordinate measurements made xt N over the ensemble at some instant of time t,turninto2xt sets of measurement descriptions. For each of these sets, we determine a number ΔN – the number of measurements, at xt which the availability of particles 1 and 2 is registered in a small but finite volume ΔV x of the six-dimensional configuration space in the vicinity of a point in this space, which is specified by an ordered pair of three-dimensional radius vectors (x1, x2). Then we can calculate the ratio ΔN /N and study the behavior of this ratio under an unlimited xt xt increase in the number of measurements N . xt If the following two conditions are satisfied: (1) for some sets of measurement descriptions, this ratio with the growth of N changes xt 9 It would be appropriate to ask physicist-experimenters the following question: is it possible to prepare the Born ensemble with a specified initial Schr¨odinger wave function – or, is it only the process of preparing the ensemble with two specified initial Born densities that is physically implemented? 226 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 chaotically: there is no limit for this ratio at N → ∞; xt (2) for other sets of measurement descriptions, such limit R exists, and it is the only x t one, – the two particles in two-particle ensemble are named statistically distinguish- 10 able , and limit R t determines the Born density ρ(x1, x2,t) in the six-dimensional x x configuration space: ΔN xt R t = lim = ρ(x1, x2,t)dV1 dV2. (19) x N →∞ x x xt Nt x x ΔV x In formula (19), the integrals are taken over all the coordinates of the first particle and all the coordinates of the second particle within the allocated six-dimensional volume element in the configuration space. Similar relation, with replacement of coordinates x with momentums k under satisfying the conditions of statistical distinctiveness of the ensemble particles, determines the Born density ρ(k1, k2,t) in the six-dimensional momentum space of the two-particle ensemble: k ⎛ ⎞ ΔNt ⎝ k ⎠ R t = lim = ρ(k1, k2,t)dV1 dV2. (20) k Nt→∞ k k k k Nt k ΔV k Two real, positive, normalized per unit Born densities, ρ and ρ, form the Born pair x k ρ,ρ , for which we can calculate two 3 × 3-Heisenberg’s matrices H ij,Hij (5) for each x k 1 2 particle of the two-particle ensemble. Heisenberg inequality (6) is postulated for each particle of the two-particle ensemble. For the Born two-particle ensemble that satisfies the two inequalities (6), as well as for the one-particle situation, we can formulate the Schr¨odinger theorem – the statement that there is a pair of real phase functions S(x1, x2,t)andS(k1, k2,t), such one that the x k Fourier-doublet components of Schr¨odinger wave functions ψ(x1, x2,t)andψ(k1, k2,t), x k determined by relations (7) and (8), are the Fourier images of each other: 1 i (k1·x1+k2·x2) ψ = 3 ψ e dV1 dV2, x (2π) k k k (21) 1 −i (k1·x1+k2·x2) ψ = 3 ψ e dV1 dV2. k (2π) x x x Integrals in (21) are taken over the entire six-dimensional space. Wave functions ψ of the two-particle ensemble, as it is known, satisfy the Schr¨odinger x equation: 1 1 α12 i∂t ψ = − Δψ − Δψ + ψ, (22) x 2m1 1 x 2m2 2 x r12 x 10 These two conditions will knowingly be satisfied if, for example, a two-particle ensemble is made of one-particle ensembles of different masses. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 227 where Δ is the Laplacian by components of vector x1;Δis the Laplacian by components 1 2 of vector x2; r12 = |x2 − x1|; α12 is the coefficient of the coulomb interaction in the two- particle system – a non-zero phenomenological constant characterizing the two-particle ensemble11. In a sense, (22) should be regarded as the definition of this constant. Considering successively three two-particle systems (1,2), (2,3), (3,1), formed by three one-particle ensembles (1), (2), (3), any two of which are statistically distinguishable from each other (for example, proton, electron and muon), we can formally attribute electrical charges e1, e2, e3 to each ensemble, defining these charges as solutions to the system of three algebraic equations: e1 · e2 = α12, e2 · e3 = α23, (23) e3 · e1 = α31. It is obvious that the solution to system (23) exists if the following condition is satisfied: α12 · α23 · α31 > 0. (24) Condition (24) should be taken as a postulate (”Coulomb postulate”). The physical sense of the Coulomb postulate is simple: we live in the world where electrical charges of the same sign repel each other, and charges with opposite sign attract each other. One can imagine the non-relativistic world with a different sign of inequality (24). (In such world, in algebraic equations (23), which determine electric charges, there would appear a minus sign: e1 · e2 = −α12, etc.). If we assume, following Albert Einstein that the only inter- esting question of physics is: ”Did God have a choice?” the inequality (24), postulated and having no justification within the non-relativistic scheme, shows that God did have a choice. It is easy to see that the solution to system (23), under satisfying Coulomb inequality (24), exists and it is unique (accurate within the arbitrary choice of the charge sign for a single charged particle in the Universe). But if the conditions of statistical distinguishability of two particles of the two-particle ensemble are not satisfied, both the method of constructing the Born densities (there is not one, but two different limits R t and two different limits R t) and the method of x k connection of Fourier-doublet of Schr¨odinger wave functions with the Born densities, are complicated. Description of the two-particle ensemble of two statistically indistinguishable particles requires the introduction of a spin as a phenomenological parameter which is explicitly included into the mathematical description of the Born ensemble. Further descending to particulars of non-relativistic quantum system is irrelevant here. 11 Of all kinds of fundamental interactions, non-relativistic quantum mechanics allows to describe only Coulomb electrostatic interaction as a non-relativistic rudiment of relativistic electromagnetic interac- tion. Any other interactions, traditionally considered in quantum mechanics, either have a conventional, model character (”quantum oscillator”) or are unlawful semi-relativistic approximations that destroy non- relativistic logical harmony and conceptual clarity of non-relativistic theory – but improve the agreement of predictions of the theory with high-precision experimental data of atomic and molecular spectroscopy. 228 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 Relativistic quantum theory, the discussion of which is our objective, as well as classical relativistic theory described in previous articles [1], [2], [3], does not contain particles at the ”entry” to the theory. The particles should appear at the ”exit” as an accurate de- scription of some stationary states, or as an approximate description of some transitional non-stationary states. Spin can not remain phenomenological input parameter, but must also appear at the output of the theory. 3 Principles and Problems of Transition to Quantum Relativistic Theory What can we decidedly borrow from non-relativistic quantum system to transfer to rela- tivistic quantum theory? • De Broglie Postulate, which in the relativistic formulation states that four- dimensional momentum space {kν} exists along with four-dimensional coordinate space {xν}. These two spaces form a dyad {xν|kν}. The semi-components of this dyad are complementary to each other by Niels Bohr: the description of the ensemble of fields is possible either in {xν| or in |kν}. • Born postulate of the existence of a relativistic analogue of the Born densities. • Relativistic analogues of the Heisenberg postulate and the Schr¨odinger theo- rem of the existence of Fourier doublets of relativistic wave functions (Dirac spinors). • Perhaps, some analogue of the linearity postulate of relativistic wave equations. What can we transfer to the quantum relativistic theory from the classical three-sector field theory with continuum currents, which has been presented in the previous articles of this series? • The idea of existence of the three independent sectors of physics – singlet, triplet and octuplet. • The idea of dyadic nature of each sector in the form of a dyad current—potential. Dyad semi-components in the quantum version of the theory should be considered as complementary to each other by Niels Bohr: description of the ensemble of fields in each sector is possible either in terms of current or in terms of potential. • The idea of the Riemann geometry of four-dimensional spatial continuum {xν|.Con- struction of the theory of particles, which have a rest mass, as intrinsic states of the theory, is apparently impossible without considering the Riemann curvature of space12. Transferring of the concept of Riemann geometry of coordinate 4-continuum {xν| to quantum theory immediately gives rise to a number of painful questions, the correct 12 ”Does gravitational field play any role in the matter structure and should a continuum inside the atomic nucleus be considered perceptibly non-Euclidean?” – this is the rhetorical question which Albert Einstein asked himself and the readers of ”Nature” during the epoch of the grandiose triumph of general relativity theory [4]. But he never turned to this subject any more. Neither Yukawa’s meson, no isospin, no neutrino made Einstein recognizes the existence of fields other than gravitational or electromagnetic. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 229 answers to which we just have to guess13. • What should be the geometry of the space of momentums |kν} that are the second semi-component of spatial-momentum dyad {xν|kν}, be like? • Should we make similar ”non-Minkowski” demands to the geometry of multidimen- sional spaces of currents and potentials? • Should we consider the metric tensor of the four-dimensional coordinate space {xν| ”attached”” to each unique, irreproducible field realization or is it appropriate to consider it a part of the description of the ensemble of field realizations?14 • If we accept, following Einstein, the minimalist” interpretation of physical geometry: ”Ricci tensor depends linearly on the tensor of field energy-momentum”, how should the right-hand sides of Einstein’s equations be constructed? Should there be present a certain procedure of the field variables (currents or potentials) integration? How can a physical quantity, the integration of which by field variables generates the energy-momentum tensor, be constructed from Dirac spinors? • How should Fourier-transform procedure, necessary for the introduction of Dirac spinors by formulas that have to be a relativistic analogue of formulas (9) and (10) of the non-relativistic quantum scheme, be formulated in the curved Riemannian space? Leaving aside for some time these difficult questions, we can ask ourselves a more simple question – what can we transfer to the continuum relativistic quantum theory from the version of quantum field theory (but without its point fermions) which was developed in the twentieth century? Undoubtedly, there are two ”transportable” ideas: • The idea of the Dirac spinors as the principal instrument of the mathematical de- scription of physical reality. • The Dirac equation – as some prototype, as a heuristic base for the construction of equations of the new theory. For this theory, the existing Dirac equation should be an approximate consequence of the theory equations eventuating after some procedure of the approximate integration by field variables. 4 Dirac Space and Dirac Vectors in Quantum Electrodynamics 4.1 Born Fluxes In quantum electrodynamics, we are dealing with two independent dyads: ”space / mo- mentum” – {xν|kν}, and ”currents / potentials” – {Jν|Aν}. Combining semi-components of these dyads, we get 4 sets of measurement procedures: x|J , x|A , k|J , k|A .In this notation, the Lorentz indices are omitted, and the notation itself has the following 13 Hardly can these answers allow a direct empirical test. 14 From philosophical point of view, we have to put up with the multiplicity of field realizations, but the space (and momentum), four-dimensional capacitance for these realizations exists, of course, only in a single copy. In this sense, it is not very easy to understand what those physicists who speak of ”quantization of gravitation” mean. 230 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 meaning: x|J – is the measurement procedure in the space-time continuum and in the space of currents; x|A – is the measurement procedure in the space-time continuum and in the space of potentials; k|J – is the measurement procedure in the momentum continuum and in the space of currents; k|A – is the measurement procedure in the momentum continuum and in the space of potentials.15 It is convenient to construct the following system of dyadic notation. We will call the space-momentum dyad {x|k} the first dyad and in the bracketed notations of measure- ment procedures we will put the information on the element of the first dyad into the left side, before the dividing line. The current-potential dyad will be called the second dyad, and information on the element of the second dyad will be presented to the right side from the dividing line in the bracket symbol of the measuring procedure. For each dyad, instead of alphabetic characters it is convenient to use dyadic index which takes two possible values: a =1ora =2. Index 1 corresponds to the first semi-component of the dyad (coordinates x in dyad {x|k}, current J is in dyad {J|A}) , index 2 corresponds to the second semi-component of the dyad (momentum k in dyad {x|k}, potential A is in dyad {J|A}). Notation 1|2 can be used with these dyadic indices to describe, for example, measuring procedure x|A . If some physical quantity q is determined by measurement procedure of type a|b (a, b = 1 or 2), this information can be expressed by using the subscript dyadic indices: q.16 ab Let us assume that in space-time continuum {x| there is selected some infinite non-closed oriented three-dimensional hyper-surface σ with time-like unit normal vector nμ at each μ 0 17 point (n nμ =+1,n > 0). This surface should contain space-like infinity . Let us suppose that at sufficiently large number of points of this hyper-surface, there 15 We can hardly present any explicit descriptions of these procedures, even in Einstein’s genre of his favorite ”mental experiments” – eventually, measurements inside the electron must be considered. In quantum version of physics of currents and potentials, something, apparently, has to be left unfinished, just like Leonardo left unfinished the head of Christ in his painting ”The Last Supper” in Milan Dominican convent of Santa Maria delle Grazie, believing, according to Giorgio Vasari, that ”he would not be able to express in it all the heavenly divinity, required by the image of Christ; but Leonardo gave splendor and simplicity to apostles’ heads”. A theoretical physicist, as well as an artist, must strive for ”splendor and simplicity” in something that allows to describe itself, not daring to claim to describe the ”heavenly divinity”. 16 The right-hand subscripts and superscripts are traditionally used as Lorentz indices; over-letter super- scripts are used in [2] and [3] as Yang-Mills indices: for the dyadic indices there is a space left under the letter denoting a physical quantity. 17 In non-relativistic situation, we would describe it in more simple terms, for example, – ”let us assume that at some instant of time t measurements have been made over the entire infinite three-dimensional space”. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 231 have been made measurements in the current space. Let the total number of all mea- surements be equal to N. Let us suppose that ΔN is a number of such measurements 11 11 that something is found in a small but finite 3-volume Δσ of surface σ in the vicinity of some point x and meanwhile, in a small but finite 4-volume ΔΩ of space of currents in the vicinity of some point of this space. We state that there is a ratio limit ΔN/N at N → ∞, and this limit can be expressed 11 11 11 as an integral of a certain quantity ξ : 11 ΔN lim 11 = ξ dσdΩ. (25) N→∞ 11 N 11 11 ΔσΔΩ In formula (25) dσ is a Lorentz-invariant scalar element of 3-volume of hyper surface σ: ν ν ν dσ = n dσ;dσ = dσ dσν, and dΩ is the element of 4-volume in the current space. The integral in (25) is seven- fold. Statement (25) is the relativistic analogue of the Born postulate (1). Quantity ξ 11 can be called 1|1 -Born density. This density is determined on an arbitrarily chosen hyper-surface σ. It should be assumed that the description of physical phenomena does not depend on the choice of σ, and locally does not depend on local normal vector nν, therefore, there is a 4-vector ρ ν,suchonethat 11 ν ξ = ρ nν. (26) 11 11 4-vector ρ ν is afieldin mathematical sense, i.e. depends on point x, but does not 11 dependent on vector nν. But Born density ξ itself is not a field: it depends not only 11 on x, but also on nν. Let us name 4-vector ρ ν ”the Born flux” in representation 1|1 . 11 This vector depends on two 4-vectors xν and Jν, i.e. is the function of a point in the eight-dimensional measurement space (the Dirac space). In accordance with the above, the four variants of measurement procedures a|b (a, b = 1 or 2) generate four Born fluxes ρ ν, each of which is determined in its own version of the ab eight-dimensional Dirac space. Consequently, in relativistic quantum electrodynamics, the Born postulate is a statement of the existence of the four Born fluxes ρ ν.18 Relativistic Born ab ensemble is the quadruplet of fluxes: sixteen functions liable to determination in the four inter-related realizations of the eight-dimensional Dirac space instead of two non- relativistic scalar Born densities in the two three-dimensional spaces – coordinate and momentum... The radical increase in a number of unknown functions and a number of their arguments... 18 In some situations, this statement is too optimistic and requires clarification, for example, in the situation of collision of two electrons we can not distinguish in which of the two electrons the given point is located. In this case there is more than one limit (25). But, undoubtedly, it is reasonable to avoid discussing such clarifications at such an early phase of the theory construction. 232 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 4.2 Dirac Vectors To move forward on the way projected in the non-relativistic scheme, we have to turn a set of real Born fluxes into a set of complex wave functions. To do this, we have to formulate some relativistic analogue of the non-relativistic Schr¨odinger theorem (see p. 2.3). In order to do it, in accordance with formulas (7) and (8), we must learn to take the square root of 4-vector Born flux. This problem could be considered unsolvable if it had not been solved by P.A.M. Dirac already in 192819. Following Dirac, we state that there is a Lorentz vector in the form of a set of four 4 × 4 ν matrices Γαβ (ν is Lorentz index, α and β are the matrix indices α, β = 1, 4) which ν allows to associate the four complex four-component wave functions ψαβ with the four ν real 4- vectors of Born fluxes ραβ so that ∗ ν ν ψα Γαβ ψβ = ρ . (27) ab ab ab In the formula (27) the asterisk (*) is a sign of complex conjugation, α and β are the matrix indices (summation from 1 to 4 is made for repeated matrix indices); a and b are dyadic indices that do not have a vector character (dyadic indices are just the tags of the Dirac space, with repetition of dyadic indices, summation for them is not made). ∗ Four components ψα at fixed dyadic indices a and b are called Dirac spinor in repre- ab | ν sentation a b , and thus the indices α and β in matrices Γαβ can be named not matrix, but spinor ones. However, instead of the established resounding term ”spinor” we will use, albeit awkward, the term ”Dirac vector” and we will call the indices α and β ”Dirac vector indices”. Classical relativistic physics [1], [2], [3] is formulated in terms of Lorentz (and Yang-Mills) vectors while quantum relativistic physics in terms of Dirac vectors. 19 Perhaps, modern history of science underestimates this part of P.Diracs biography. This achievement, in a sense, puts this physicist, and, perhaps, only him alone, on the level with Isaac Newton: Dirac had to invent a new mathematical apparatus (spinors) in order to express the new physics – just like Isaac Newton had to invent the derivative and integral to express the laws of mechanics discovered by him. Neither Maxwell nor Schr¨odinger had to invent the partial differential equations – they had already existed. And even Einstein did not have to invent the Riemann geometry and tensor analysis: they were in the depths of mathematics as a finished product. Werner Heisenberg re-invented matrices, but this invention reflected not only his brilliant abilities, but also insufficient level of his personal mathematical training: matrices had already existed in mathematics. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 233 ν Matrices Γαβ in (27) have the form: Γ0 = δ , αβ ⎛ αβ ⎞ ⎜ 01 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 10 0 0 ⎟ ˆ1 ⎜ ⎟ Γ = ⎜ ⎟ , ⎜ 00 0 −1 ⎟ ⎝ ⎠ 00−10 ⎛ ⎞ − ⎜ 0 i 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i 000⎟ (28) ˆ2 ⎜ ⎟ Γ = ⎜ ⎟ , ⎜ 00 0i ⎟ ⎝ ⎠ 00−i 0 ⎛ ⎞ ⎜ 10 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 −100⎟ ˆ3 ⎜ ⎟ Γ = ⎜ ⎟ . ⎜ 00−10⎟ ⎝ ⎠ 00 01 Usually, in the notation of Dirac algebraic relations (27), instead of matrices (28), Dirac matricesγ ˆν are used, such ones that: Γˆν =ˆγ0γˆν. (29) In some loose way it is possible to express the essence of equalities (27) as follows: at extracting the square root of a real Lorentz vector we get a complex Dirac vector. Relations (27) are the relativistic version of relations (11) and they do not determine Dirac vector ψα on their own. It is obvious that the four complex functions ψα can not be determined by four real equations (27). To determine ψα, besides (27), we need a relativistic version of the Fourier-relation (9), (10). 4.3 Relativistic Fourier-transform The Fourier transform in non-relativistic quantum theory (9) connects the two compo- nents of Schr¨odinger Fourier-doublet, each of which is determined in one semi-component of a non-relativistic dyad {x|k}. While constructing the relativistic Fourier transform (it will be denoted with symbol RFT), we have to take into account that Dirac vectors ψα have two dyadic indices. For mathematical representation of Fourier-procedures with this ”doubling of dyadic indices”, we will introduce the negation symbol of dyadic index 234 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 a as follows: ⎧ ⎪ ⎨ 2, if a =1, a = ⎪ (30) ⎩ 1, if a =2. The transition from ψ to ψ will be called a dyadic conjugation. ab ab Formal notation RFT, which replaces the formal non-relativistic notation (9), must look as follows: RFT ψα = ψα. (31) ab ab Relation (31) means that ψ is a Fourier-transform of dyad-conjugated vector ψ.Ifwe ab ab manage to give a clear mathematical sense to formal symbol RFT, the integral equa- tion (31) together with algebraic Dirac equations (27), will allow to explicitly determine ν Dirac vectors ψα by the observable Born fluxes ρ . It is obvious that in quantum elec- ab ab trodynamics, we get not one Fourier-doublet of Schr¨odinger wave function, but two inde- pendent Fourier-doublets of Dirac vectors: Fourier-doublet, diagonal by dyadic indices, and Fourier-doublet, non-diagonal by dyadic indices. Accordingly, in contrast to non- relativistic theory, all information on the quantum-electro-dynamic states is contained not in one half of the Fourier-doublet, but in two independent halves of the two inde- pendent Fourier-doublets of the Dirac vectors, for example, ψα and ψα. Accordingly, 11 12 mathematics of quantum electrodynamics requires construction of two independent com- plex wave equations, unlike mathematics of non-relativistic quantum mechanics which requires one independent complex wave equation. Constructing the relativistic analogue of non-relativistic Fourier-transform (10), we will open a formal notation RFT (31) as follows: 1 iφ ψα = 7/2 e ψα dσ dΩ, ab (2π) a b ab (32) 1 −iφ ψα = e ψα dσ dΩ. 7/2 a b ab (2π) ab Integrals in (32) are taken over the entire three-dimensional hyper-surface σ in the co- x ordinate continuum (or, correspondingly, σ in the momentum continuum) – this is a k three-fold integral; as well as over the entire four-dimensional current continuum J (or, correspondingly, continuum√ of potentials A) – this is a four-fold integral. The degree of the Fourier multiplier 2π corresponds to seven-fold integration in (32). Symbol φ in (32) denotes relativistic Fourier-phase, which should be constructed so that it was a reasonable relativistic generalization of non-relativistic Fourier phase in (10). This non-relativistic Fourier phase is the inner product x · k of the two semi-components of non-relativistic space-momentum dyad {x|k}. Regardless of the particular type of phase φ, RFT (32) contains a serious vulnerability: integrals in (32) are taken over arbitrarily chosen hyper-surfaces σ and σ. In contrast to x k Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 235 non-relativistic situation, we can not connect these two hyper-surfaces in two different four-dimensional continuums. In non-relativistic quantum mechanics, for example, we integrated over the entire 3-space of momentums at the same instant of time t at which the wave function in the coordinate space was determined. We can not transfer this construction of absolute time to relativistic scheme. Of course, we have to suppose that all the measuring equipment, by means of which we detect something in the space of currents, potentials, coordinates and momentums, is fixed in some locally Galilean frame of reference in some vicinity of each point of the space-time continuum, but the author has to admit that he does not know how to derive a mathematically faultless coupling between σ and σ from this physically faultless statements. We will use the above cited x k ”rule of Leonardo” and will not detail this important part of relativistic picture. 4.4 Relativistic Fourier-phase Non-relativistic formula for the Fourier phase φ = x · k allows to suggests that the rel- ativistic Fourier-phase should be a linear combination of two scalar products of dyadic semi-components of the both quantum-electro-dynamical dyads {x|k} and {J|A}.How- ever, these two scalar products enter into the phase with different weight: μ μ ψ = x kμ + αJ Aμ, (33) where α is some phenomenological constant of quantum electrodynamics, which we will unhesitatingly identify with the constant of fine structure. Here it is appropriate to remind the reader that in the theory appears a fundamental 20 length r0 [1], which we accept as a length unit . The velocity of light is also accepted as a unit. The quantity of electron charge is accepted as a unit. Accordingly, all the quantities appearing in the theory are dimensionless. As it has been noted in [1], the relativistic theory of matter can not be constructed without considering the Riemann curvature of space. It means that one more dimensionless constant, proportional to gravitational constant G, appears in the theory. In another way this can be expressed as follows: the theory must include the Planck length rp and, correspondingly, the dimensionless constant r0/rp. This constant and constant α, appearing in (33), should of course, be connected with each other. However, at present, embryonic stage of the theory construction, this connection, as well as the other important details of the picture, remains non-detailed. If the expression of relativistic Fourier-phase, introduced by formula (33) is correct, the true physical meanings of the constant of fine structure α is that this quantity is a measure of mixing σ-variables and Ω-variables, the measure of their relative weight. In the world with α<<1, the ”observer” and his ”equipment” are often found in an empty space, the space without currents. In such world, the concept of free point particles with the interpretation of ”interaction” as collision of particles (not without difficulties 20 −26 Constant r0 was estimated in [2] r0 ∝ 10 cm. 236 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 and divergences) is useful. In the world with α>>1, the ”observer”21 and his ”equipment” are often found in spaces occupied by currents: the subject of study would rather be the properties and movement of ”voids”, ”bubbles” which are not occupied by currents. Observers like us feel comfortable only in the world with α<<1. But we do not know the principle by ∼ which God fixes a specific value α = 1 137. 4.5 Fourier-phase in the Riemannian Space The two terms in relativistic Fourier-phase (33) have different geometrical importance. ν The second term, proportional to scalar product J Aν, does not change its type in the curved Riemannian space – only for lowering/raising of Lorentz indices there should be used the components of non-Galilean metric tensor gμν, determined locally, at the same point of the space-time Riemann continuum, where the measurements in J-space or A- space were made. However, the first term in (33) makes no sense at all in the Riemann continuum: coor- dinates xμ do not form a vector (vectors are only coordinate differentials). With some psychological effort, we probably have to say the same about components kμ:theydonot form a vector. It is hard to imagine how x and k form a dyad of equal semi-components if space curvature k does not correspond to the Riemann curvature of 4- continuum x. However, it is even more difficult to imagine a curved momentum space: in modern the- oretical physics, apparently, there is no idea that can suggest how to curve momentum continuum. Let us rewrite expression (33) in the form: μ φ = η + αJ Aμ, (34) (PE) μ where η = x kμ. (35) Symbolic notation (35) means that this equality is valid only within the framework of pseudo-Euclidean geometry of four-dimensional continuums x and k. How should we rearrange the expression for phase η in the Riemann continuum? The expression for η which we suggest below (36), is rather a gesture of despair than a display of physical intuition. Let us identify space x and space k. Of course, we can not identify the measurement procedure in these spaces – they are inter-complementary by Niels Bohr. But let us try to interpret xν and kν as coordinates of two points in one four-dimensional continuum. As part of this (which is not easy to imagine, and which is impossible to accept), we can choose some arbitrary point O as the origin (common for x and k) and determine the lengths of the three geodesic lines Cox, Cok, Cxk: 21 In such world there hardly can be ”observers” like us. Cosmologist A.L. Zel’manov once said that we are the witnesses of the processes of a certain type, because other processes take place without witnesses. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 237 • μ ν sox = gμνdx dx – the length of geodesic line Cox connecting point O with Cox point x; • μ ν sok = gμνdx dx – the length of geodesic line Cok connecting point O with Cok point k ; • μ ν sxk = gμνdx dx – the length of geodesic line Cxk connecting point x with point Cxk k. Knowing the lengths of the three geodesic lines, we can construct phase η as follows: 1 η = s2 + s2 − s2 . (36) 2 ox ok xk Expression (36)22 coincides with (35) for the pseudo-Euclidean geometry with using the Minkowski coordinates; expression (36) will be also valid in arbitrary curvilinear coordi- nates introduced in the pseudo-Euclidean continuum, and, finally, we can postulate (36) for the Riemann continuum, if we simultaneously require the metric tensor to obey the Einstein equations (so far disregarding the fact that we have not been able to construct a quantum-electro-dynamic energy-momentum tensor and, therefore, can not even enter the explicit form of the Einstein equations). The idea of identification of spaces x and k, is, undoubtedly, disgusting, and evidently contradicts the initial interpretation of x and k as ”semi-components of dyad {x|k} com- plementary by Bohr”. Perhaps, among the readers of the article there will be found a mathematician who, in contrast to the author, knows the subject of the Fourier-transform in a larger scope than the classical book by Ian Sneddon can provide [5]. The reader, who is able to provide a more convincing mathematical interpretation for phase η in the Riemann continuum. But we, due to the lack of an alternative, will consider formula (36) acceptable for relativistic quantum theory. 4.6 Some Concluding Remarks on Dirac Mathematics of Quantum Electrodynamics • By analogy with the non-relativistic quantum theory, we can assert that the system of equations (27) and (32) that determines Dirac vectors by Born fluxes, is solvable not for any arbitrary specified quadruple of Born fluxes, but only for such quadruples that satisfy some a priori solvability conditions. These conditions are the relativistic generalization of non-relativistic Heisenberg inequalities (6). Taking into account the fact that in the relativistic scheme there are two independent Fourier-doublets of the Dirac vectors, there must be more relativistic solvability conditions than in the non-relativistic scheme. And if we take into account the Riemann complexity of the form of the relativistic Fourier-phase (36), the very possibility of entering the explicit form of relativistic version of the Heisenberg postulate (6) becomes unobvi- 22 The reader will be right if he suspects a school ”cosine theorem” in (36). 238 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 ous23. The very fact of existence of the Fourier-coupling (32) between Dirac vectors in the dyadically-coupled representations (the coupling between ψ and ψ) means, in partic- ab ab ular, that there is no quantum-electro-dynamic state which could be interpreted as a free electromagnetic field, i.e., the ensemble of photons. If wave function ψ is not 22 identically zero, ψ can not be identically zero either: if there are photons, somewhere 11 there are sources of these photons. This is quite natural, but also paradoxical result which is poorly consistent with the usual research methods in theoretical physics. • In classical electrodynamics of continual currents [1], the a priori condition of space- likeliness is imposed on currents Jν: ν J Jν ≤ 0. (37) Hardly can this condition be appropriate in quantum electrodynamics as a hard a priori constraint of the Dirac space geometry. However, condition (37) will actu- ally control the arrangement of quantum-electrodynamic states, if Dirac vector ψα 11 ν contains a multiplier of the form exp(−λJνJ )withλ>0. Under satisfying this condition, components ψα will decrease exponentially in the area of positive values 11 of the pseudo-Euclidean square of the current module. Availability of such exponen- tial multiplier can be provided if we interpret the current equation of the classical version of the theory [1]: Jν +Aν =0, as operator relation: Jˆ ν + Aˆ ν =0, (38) and suggest that ∂ Aˆ ν = in J-representation. (39) ∂Jν The equation for Dirac vector ψα follows from (31) and (39) : 11 ∂ψα 11 ν +J ψα =0 ∂Jν 11 with solution − 1 ν 2 J Jν ψα = Cα(x)e , (40) 11 where Cα(x) is the function that depends on space coordinates. Undoubtedly, relation (40) seems too simple and poor to contain the whole quantum electrodynamics. Besides, operator relation (39) is incompatible with the form of 23 The author uses these remarks to disguise his mathematical weakness, his inability to construct a relativistic analogue of Weyl theorem. Undoubtedly, among the readers of this article there may be found a more successful mathematician able to formulate Weyl theorem for quantum electro-dynamics. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 239 Fourier-phase (33): the lack of imaginary unit i in (39) requires its availability before the second term in (33). This, in its turn, changes the interpretation of equations (32): RTF becomes the Fourier transform only within the first dyad {x|k}. The situation becomes even more complicated if we remember that in the classical version of electrodynamics of continual currents [1], constructed here, there is one more current module restriction: ν ≥−2 J Jν j0, (41) where j0 is some (unknown) fundamental constant. Representation of restriction (41) with use of the corresponding exponential multi- plier in Dirac vector ψα requires a more complicated expression for potential operator Aˆ ν than (39), and, therefore, a more complicated expressions for Fourier phase than (33). By the end of this observation we find ourselves in front of the ruins of the newly constructed quantum relativistic scheme: it is badly compared with classical elec- trodynamics of continuous currents [1]. What can be considered reliable in the constructed scheme? Probably, the concept of the Dirac space and Dirac vectors itself, connected with the Born fluxes by equations (27) and with each other – by relativistic Fourier transform (32) with non-specified relativistic Fourier phase φ. Probably, the idea of partition of Fourier phase into two terms, each of which is con- nected with one dyad, is reliable. But neither of the terms in the notation of specific expression for phase φ (34) is reliable. The first term in form (36) is doubtful because of rather arbitrary method of accounting the Riemann curvature. The second term does not allow to account for current module restrictions that we substantiated in the classical version of the theory [1]. • The Dirac equation in its usual form: μ μ iγαβ ∂μψβ = mψα + eAμ γαβ ψβ, (42) hardly can be directly applied to the constructed here relativistic quantum scheme. It contains particle mass m and particle charge e, i.e. the integral characteristics of some stationary quantum state, while their appearance in local relation (42) seems irrelevant. Besides, this equation does not contain any dyadic indices and does not allow determining which of the two independent Dirac vectors it could be related with. Instead of mass m in the first term on the right side of (42) there should appear a ˆ 24 multiplier connected with the energy-momentum operator T μν. μ ν Its Lorentz indices can be ”extinguished” by the product of Dirac matrices γβγ γγδ. Free Dirac indices of this product can be extinguished by the ”plate” of two Dirac ∗ vectors ψβ ...ψδ. To keep some memory of the ”linearity and autonomy” of the 24 We will afford to neglect the fact that we do not have an expression for this operator yet. 240 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 wave equations, we have to assume that this non-linear ”plate” is formed of the vector which is dyad-conjugated by one of the dyadic indices to vector ψ appearing in the equation. In addition, this multiplier with bilinear ”plate” must imply the integration by the semi-component of the dyad {current|potential}, which is coupled with the semi-component entering the argument of vector ψ. As a result, the Dirac equation can be re-written in the following form: μ ∗ μ ˆ ν iγ ∂μψβ = ψα ψβ γ T μν γγδ ψδ dΩ + .... (43) αβ βγ a 1a 1a 1a 1a In equation (43) the last term on the right side of (42), in which we have to do similar manipulations to remove the symbol of electric charge e from the equation notation, is not written out. The coefficient with ψα on the right side of (43) is the 1a coordinate function x. Therefore, relation (43) is linear by ψα, but integrally the 1a quantum state is described nonlinearly, since the coefficient with ψα is functionally 1a dependent on the second Dirac vector. Instead of charge e in the last term on the right side of (42), we probably have ˆ to write the current operator Jν. Its Lorentz index can be balanced by the Dirac ν matrix γγδ. Free Dirac indices of this matrix can be balanced by the plate of the two Dirac vectors. To ensure the ”linearity” of the Dirac equation, we should provide for a transition to a coupled dyadic index and integration by the corresponding semi-component of the dyad {current|potential}. As a result, we form up the Dirac equation (42) as follows: μ ∗ μ ˆ ν ˆ μ ∗ˆ ν iγ ∂μψβ = ψα ψβγ T μν γγδψδdΩ + Aμγ ψβ ψγJνγγδψδdΩ . (44) αβ βγ a αβ a 1a 1a 1a 1a 1a 1a 1a We have fixed in (44) the first dyadic index equal to 1 taking into consideration the form of the left-side of equation (44), which contains the operator of differentiation by xμ. Under transition to the momentum representation on the left side of (44), μ we should write γαβ kμψβ. 2a Equation (44) is linear relative to the Dirac vector ψ, but in general the problem of 1a quantum electrodynamics is non-linear and even considerably nonlinear (does not allow linearization). By cubic nonlinearities (44), it resembles a ”fundamental field equation” investigated in the later works of Werner Heisenberg [6]. But the equation of W. Heisenberg did not contain any integration operations. Unlike Albert Einstein, who worked on the classical unified field theory, Werner Heisenberg was looking for a quantum version of the field theory, but his thinking, as well as Albert Einstein’s thinking, was clearly monadic: he believed that the field must be described by one wave function (Weyl two-component spinor or Dirac four-component spinor). Our description of quantum electrodynamics requires the use of two independent Dirac vectors, while multi-sector quantum relativistic states, located not only in the singlet (Maxwellian) sector, but also in the triplet and octuplet Yang-Mills sectors Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 241 of physics, require even a larger set of the Dirac vectors. If we fix the free dyadic index a in (44) supposing that a = 2, the operator of ˆ potential Aμ on the right side of (44) will just be reduced to a simple multiplication by potential Aμ. If we assume that dyadic index in (44) can take on both values a =1anda = 2, equations (44) form a complete system of two (vector) Dirac equations for two independent Dirac vectors ψ and ψ. But in this case there arises a question: where 11 12 and how should Maxwell equations show their worth? Maxwellian ”trace” should ˆ certainly manifest itself in the formation of operator Tμν which contains the quantum version of electromagnetic field tensor – but is it sufficient for (44) to be accepted as a complete description of quantum electrodynamics? Of course, all these hard doubts under the construction of quantum electrodynamics equations are generated by the fact that we are trying to construct equations of the theory per se with the lack of any general guiding physical principle. In classical physics, this principle is the principle of least action, and the accompanying aesthetic requirements of simplicity and symmetry claimed to the Lagrangian. Apparently, there is no such principle in the relativistic quantum theory, and the equations of the theory should be just discerned25. • If in the classical expression for electromagnetic field tensor Fμν = ∂μ Aν − ∂ν Aμ, ˆ we make a change of ∂μ → i kμ, but will interpret potential Aμ, as well as momentum ˆ kμ as an operator acting on the Dirac vectors, field operator Fμν can be presented in the form: ˆ ˆ ˆ ˆ ˆ Fμν = i kμ Aν − kν Aμ . (45) ˆ ˆ Operators kμ and Aν act on different groups of the Dirac vector arguments and must commute. If, within the same approach, we treat the Maxwell equations as operator relations: μν ν ˆ ˆ μν ˆ ν ∂μF =4π J → i kμF =4π J , ∂ and accept that in A-representation Jˆ ν = , instead of Maxwell equations, we ∂ Aν 25 The reader must have noticed that we tacitly ignore the Feynman approach to the derivation of relations of quantum physics. This approach is based on the integration by all the field realizations of exponent of the classical action functional (and in the non-relativistic situation – on integrating by all the classical trajectories of the particles). Such course of actions creates the illusion of reality of existence of classical fields as both continuous and differentiable time and place functions (but in the non-relativistic problem – the illusion of existence of smooth trajectories of point particles). Support of such illusion seems unacceptable from the philosophical point of view – despite the technical success of the Feynman approach. Besides, some mathematicians believe that in the pseudo-Euclidean continuum of relativistic theory it is absolutely impossible to give any sense to Feynman integrals by field realizations. 242 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 can enter the equation for the second Dirac vector: ∂ 1 ν − μ ν γαβ ψβ = kμk Aνγαβψβ. (46) ∂Aν 22 4π 22 In notation (46) the operator relation is accepted: ˆ ˆ μ kμ A =0, (47) which replaces the classical condition imposed on the divergence of potential Aν. Operator relation (47) looks rather strange and, of course, makes sense only under the integral sign by σ- and Ω-variables in the ”plates” of ψ-vectors. Equation (46) should be taken as postulated, without any reference to equation (47). We will interpret the classical expression for electrodynamic energy-momentum tensor Tμν [1] as an operator acting on the Dirac vectors: μν → ˆ μν ˆ μν ˆ μν T T = T cur + T f ; ˆ μν ˆ μˆ ν 1 μν ˆ ˆ λ T = J J − g JλJ ; cur 2 (48) 1 1 Tˆ μν = −Fˆ μλFˆ ν + gμνFˆ ληFˆ . f 2 λ 4 λη Relation (48) implies that the electrodynamic energy-momentum operator consists of ˆ μν the energy-momentum operator of currents T cur and the energy-momentum operator ˆ μν of free field T f . Supposing that in (44) a = 2, we transform (44) into the equation for vector ψα, 12 which coefficients on the right side of (44) depend on the integrals of current space containing the operators acting on vector ψα under the integral sign. This vector 11 is a relativistic Fourier-transform of vector ψα. Vector ψα obeys the linear and 22 22 autonomous equation (46). While constructing the right side of the Einstein gravitational equations, we have to use operator (48) instead of the classical energy-momentum tensor, perhaps, in the following form: μν ∗ ˆ μν ∗ ˆ μν T = ψαT curψαdΩ + ψαT f ψαdΩ. (49) 11 11 J 12 12 A The account of the Riemann curvature of the space / time coordinates (and, hence, the momentum space) makes the geometric meaning of momentum operator kν which appears in (45) unclear and, accordingly, devalues the formulated mathematical con- struction. So, what remains after all these futile attempts to construct a system of equations of the relativistic quantum theory?26 26 Albert Einstein wrote to Maurice Solovine that with his persistent desire to understand the basic principles, ”the most part of his time was wasted on futile efforts” ([7], a letter from 30. .1924). The failure, of course, is an inalienable part of the attempts to ”understand the basic principles”. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 243 There remains some contour, outline of the non-embodied intention. And accord- ing to this contour, the quantum electrodynamics equations should consist of three groups: (1) The equation growing from the Dirac equation (42) for one of the two inde- pendent Dirac vectors. Information on the second vector is included into this equation in the form of coefficients containing Ω or Ω integrals of the quadratic J A forms that contain the second Dirac vector. (2) The equation for the second Dirac vector growing from the Maxwell equations. Perhaps, this equation should also be quasi-linear and contain information on the first Dirac vector in coefficients represented by Ω or Ω integrals. We have J A constructed the equation (46) as linear, and this linearity violates the ”Dirac equality” of vectors ψ and ψ, which causes some distrust to this equation. 12 22 (3) Einstein’s classical gravity equations, the right side of which represents Ω or Ω J A integrals of quadratic forms that contain Dirac vectors (49). Operator relations, replacing the current equations of classical electrodynamics with continual currents [1], must be added to these equations. What might be considered the criterion of such program success? • The existence of a variety of stationary solutions that correspond to massive leptons [1]. • The correct result of mass calculation of the muon and the triton. • Predicting of the masses of more massive leptons (if there are more than three states) in the spectrum of stationary states. 5 Dirac Space and Dirac Vectors in Quantum Singlet-triplet (Electroweak) Theory In the quantum singlet-triplet theory, the two dyads of quantum electrodynamics – ”space/momentum” {x|k} and {singlet current | singlet potential}{J|W} –arecom- plemented with the third dyad – {triplet current | triplet potential}{J|W}. Corre- spondingly, all of the observable physical quantities, for example, the Born fluxes, get the third dyadic index a =1ora = 2. For the three dyads we obtain eight variants for possible measurement procedures or, respectively, eight variants of the Dirac space with the arguments of the first or second semi-components of each dyad: 1|1|1 ←→ 2|2|2 , 1|1|2 ←→ 2|2|1 , (50) 1|2|1 ←→ 2|1|2 , 1|2|2 ←→ 2|1|1 . The set of dyadic indices in the right column (50) is a dyadic negation of the set of indices of the left column. Dirac vectors with a set of indices of the left column (50) are the dyad-conjugated Dirac vectors with a set of indices on the right column. Dyad-conjugated 244 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 Dirac vectors of the left and right columns should be connected with the singlet- triplet relativistic Fourier-transform. Of course, as in quantum electrodynamics, the Dirac vectors as constructible quanti- ties, are preceded by the Born densities as observable quantities. For example, let us assume that N is the total number of all measurements within the framework of the 111 set of measurement procedures 1|1|1 , made over some infinite three-dimensional hyper- surface σ (with the time-like unit vector nμ at each point) and that ΔN is the number of 111 measurements which reveal something in a small but finite 3-volume Δσ of surface σ, and simultaneously, in a small but finite 4-zone of singlet 4-current ΔΩ,andsimul- S taneously in a small but finite 12-dimensional volume ΔΩ of the triplet current space. T The Born density ξ is determined by the relation similar to (25): 111 ΔN lim 111 = ξ dσdΩdΩ. (51) N →∞ S T 111 N 111 111 ΔσΔΩΔΩ S T The integral in (51) is a 19-fold. As in quantum electrodynamics, we postulate the fact of existence of the Born flux ρν 111 with the formula similar to (26): ν ξ = ρ nν. (52) 111 111 Similarly, within the framework of the eight possible variants for singlet-triplet states of the measurement procedures, we will construct all eight Lorentz vectors of Born fluxes ρν (a, b, c = 1or2). Each of the eight Born fluxes is the function of its intrinsic set abc of arguments, i.e. defined in its intrinsic version of Dirac space27. Dirac space of the singlet-triplet theory is twenty-dimensional: four dimensions are generated by the first and the second (singlet) dyads each, and 4 × 3 dimensions are generated by the third triplet-dyad. Dirac algebraic equations (27), connecting the Dirac vectors with the Born densities, take the third dyadic index, but preserve their form: ∗ ν ν ψα Γαβ ψβ = ρ . (53) abc abc abc Equations (53) are not enough to determine the set of Dirac vectors ψ and, as above, abc in quantum electrodynamics, we prescribe the presence of Fourier-coupling between the dyad-coupled Dirac vectors: RF T (ST) ψα = ψα. (54) abc abc 27 Neutrino problems, requiring isotropization of singlet current Jν or one of the three Yang-Mills currents Jν , are easily and elegantly solved in the classical field theory with continual currents [1], [2]. In the quantum version of the theory, neutrino situation is connected with the appearance of singularity of one 3 3 ν ν or two Born fluxes on cone J Jν = 0 or, for example, on cone J Jν = 0 in current spaces. The correct construction of the Dirac vectors in such singular situation is not an easy task. Therefore, the quantum description of neutrino seems a more difficult problem than its classical description, presented in [1], [2]. Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 245 Symbolic notation (54), as well as its singlet analogue (31), implies the relativistic Fourier transform, which should be structured for quantum version of the singlet-triplet theory. By analogy with singlet formula (32), we can expand the formal notation (54) as follows: 1 iφ ψα = 19/2 e ψα dσ dΩ dΩ, abc (2π) a b c abc (55) 1 −iφ ψα = e ψα dσ dΩ dΩ. 19/2 a b c abc (2π) abc Relativistic singlet-triplet Fourier-phase will be constructed by analogy with the singlet formulas (33) – (36): 1 ν 1 ν φ = η + α J Wν + J · Wν , (56) pS pT where pS and pT are Weinberg parameters [2]. Relation (56) means that the Fourier-phase φ is formed by a linear combination of scalar products of two semi-components of each field dyad. The relative weights of these prod- ucts are borrowed from the interaction Lagrangian of the classical singlet- triplet theory [2]. The space-momentum part of the Fourier-phase η can be entered either in a usual form (35) – if we tend to ignore the problems of the Riemannian curvature, or in a hypothetical form (36) which allows to account these Riemann problems in some form. As in quantum electrodynamics, the Fourier-phase formula (56) is open to criticism (does not account the constraints on currents and the couplings between currents [2]; contains an arbitrary construction (36) for phase η, etc.). However, now we can not suggest any other formula for the Fourier-phase. Integral equations (54) together with algebraic Dirac equations (53) allow to construct eight complex Dirac vectors28 by eight of the observable real Born fluxes, but this eight of the Dirac vectors forms four Fourier-doublets, and, therefore, for a complete description of the singlet-triplet state it is sufficient to know only four Dirac vectors – one vector of each of the Fourier-doublet. Consequently, the quantum theory of the singlet-triplet states should contain four fun- damental equations controlling these four independent Dirac vectors (plus geometrical Einstein equations that control metric tensor). One of these fundamental equations arises from the Dirac equation (entered with regard for the triplet sector). The possible way to construct this equation has been shown above in the discussion of fundamental equations of quantum electro-dynamics. Two more equa- tions arise from Maxwell and Yang-Mills classical field equations. One more equation, which does not have any source in prior physics, remains ”unsup- ported”. Perhaps, we have to dare to use the Dirac equation twice, for both Fourier vectors. However, we have already gone too far into the undeveloped and unfriendly quantum 28 Under satisfying the solvability conditions which are the singlet-triplet relativistic generalizations of the Heisenberg inequalities. There must be more of these conditions than in quantum electrodynamics. 246 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 territory. It is time to build a fort, fortify our positions and wait for the saving cavalry to come. 6 Dirac Space and Dirac Vectors in Quantum Singlet-triplet- octuplet Theory (The Standard Model) In the quantum version of the singlet-triplet-octuplet theory (STO-theory, or the Stan- dard Model) the three dyads of the singlet-triplet theory are complemented with the fourth dyad ”octuplet current/octuplet potential” {J|W}. Accordingly, all of the ob- servable physical quantities, for example, the Born fluxes, get the fourth dyadic index a =1ora = 2. For the four dyads there appear 16 different possible measurement proce- dures or, respectively, 16 variants of the Dirac space. These 16 variants can be arranged into eight dyad-conjugated pairs: 1|1|1|1 ←→ 2|2|2|2 , 1|1|1|2 ←→ 2|2|2|1 , 1|1|2|1 ←→ 2|2|1|2 , 1|1|2|2 ←→ 2|2|1|1 , (57) 1|2|1|1 ←→ 2|1|2|2 , 1|2|2|1 ←→ 2|1|2|1 , 1|2|2|1 ←→ 2|1|1|2 , 1|2|2|2 ←→ 2|1|1|1 . The set of dyadic indices in the right column is the dyadic negation of the set of indices of the left column. The Dirac vectors with a set of indices of the left and right columns are dyad-conjugated and, therefore, must be connected by relativistic (STO)-Fourier- transform. As well as in quantum electrodynamics or the quantum singlet-triplet theory, the Dirac vectors (as constructible quantities) are preceded by the Born-densities (as observable quantities). Without deciphering the notations which are already obvious to the reader, we can rewrite formula (51) of the ST-theory for the STO-theory. Δ N lim 1111 = ξ dσdΩdΩdΩ. (58) N →∞ S T O 1111 N 1111 1111 ΔσΔΩΔΩΔΩ S T O The integral in (58) is 51-fold: the integration by the 32-dimensional space of octuplet vectors has been added to integral dimension (51). Using the same formula (52), which was used for the ST-theory, we are constructing Born flux ρν from Born density ξ : 1111 1111 ν ξ = ρ nν. (59) 1111 1111 By changing the dyadic indices by the formulas similar to (58) and (59), we construct all 16 Lorentz vectors of Born fluxes ρν (a, b, c, d = 1 or 2) within the framework abcd Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 247 of the sixteen variants of measurement procedures possible in the STO- states. Each of the Born fluxes is a function of its intrinsic set of arguments, i.e. each vector is determined in its intrinsic version of the Dirac space. The Dirac space for STO-state is 52-dimensional: four dimensions are generated by the first and second dyad each (space- momentum and singlet), 12 dimensions are generated by the third dyad (triplet) and 32 dimensions correspond to the fourth dyad (octuplet). Dirac algebraic equations (27) or (53), take the fourth dyadic index, but do not change their form: ∗ ν ν ψα Γαβ ψβ = ρ . (60) abcd abcd abcd In addition to these equations, which are not sufficient to determine the sixteen complex four-component Dirac vectors ψ by the sixteen real four-component Lorentz vectors ρ, we prescribe the existence of the Fourier-coupling between the dyad-conjugated Dirac vectors: RF T (STO) ψα = ψα . (61) abcd abcd Symbolic notation (61), as well as its electrodynamic analogue (31) and its ST-analogue (54), implies the relativistic Fourier-transform, which must be reasonably constructed for quantum version of the STO- theory. By analogy with electrodynamic formula (32), and ST-formula (55), we can open the formal notation (61) as follows: 1 iφ ψα = 51/2 e ψα dσ dΩ dΩ dΩ, abcd (2π) a b c d abcd (62) 1 −iφ ψα = e ψα dσ dΩ dΩ dΩ. 51/2 a b c d abcd (2π) abcd Relativistic Fourier phase for STO-theory will be constructed by analogy with (56): 1 ν 1 ν 1 ν ν φ = η + α J Wν + J · Wν + J · W , (63) pS pT pO ( pS, pT , pO are Weinberg parameters [3]). Formula (63), as well as its analogues (56) and (34), is probably incomplete: it does not account the couplings between currents and restrictions to current modules29. Integral equations (62) together with the Dirac algebraic equations (60) allow to construct sixteen complex Dirac vectors by the sixteen observable real Born fluxes. However, these sixteen Dirac vectors form eight Fourier-doublets and, consequently, for complete descrip- tion of the STO-state it is sufficient to know only eight Dirac vectors – one vector for each Fourier-doublet. Therefore, the quantum version of the theory of STO-states should con- tain eight independent fundamental equations that control this eight independent Dirac 29 Within the framework of the constructed here apparatus of quantum physics of currents and potentials there is just nowhere else to introduce these restrictions and couplings – if they are, in general, important: besides relativistic Fourier-phase there is no other ”free space” for their accounting in the theory. The only alternative is the direct restriction on the geometry of Dirac spaces. 248 Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 vectors (plus geometric Einstein equation controlling the metric tensor). One of these equations grows from the Dirac equation which must be written now with the account of the octuplet sector. Three more equations are based on the three clas- sical field equations (Maxwell equations in the singlet sector and Yang- Mills equations in the triplet and octuplet sector). Half of the required fundamental equations of the STO-theory ”hang” without any classical basis. We have three more current equations of the classical field theory with continuum currents in reserve – for the singlet ,triplet and octuplet sectors ( see [1], [2], [3]). However, in quantum theory, these equations are most likely to become operator relations similar to (38), rather than the missing fundamental equations for the Dirac vectors. Apparently, the Dirac equation should appear in the theory more than once while controlling the behavior of several Dirac vectors. But if we take into account that beyond the STO-theory there is nothing at all (this is the whole known physics), the theory must also contain as one of the solutions and the Big Bang model (the birth of the Universe from nothing) – and, inevitably, in thinking about this theory there may appear a Pascal feeling of ”looking into the abyss”. 7 Conclusion This article presents a perplexing unfinished project of construction the quantum version of physics of currents and potentials30. If historical analogy with the era of construction non-relativistic quantum mechanics is permissible, the project has been presented to the reader in the early ”de Broglie” stage, in the stage of an inspiring but vague idea, long before the ”Schr¨odinger” stage where appears a clear and complete mathematical construction. However, we do have some advantages over de Broglie’s era: we already know Born’s statistical interpretation of the Dirac wave functions, we have Bohr’s idea of complementarity and we even know the dimension of the mathematical spaces in which not yet written wave equations of the theory should work. But we have no idea how massive particles in the form of the intrinsic states of the Dirac vectors can appear within the framework of this concept... The most radical statement which can be formulated on the basis of this article is as follows: In contrast to non-relativistic quantum theory, quantum relativistic states can not be described by one wave function (one Dirac vector); the number of wave functions is determined by sector multiplicity of the condition – two Dirac vectors for the singlet state (and for any one-sector state); four Dirac vectors for the ST-states (and for any two-sector states); eight Dirac vectors for three-sector STO-states. The number of fundamental wave equations of the theory is equal to the number of independent Dirac vectors. Some of the fundamental equations of the theory, similarly to the Dirac equation, have no ”precursors” in classical physics. 30 However, as Leonardo da Vinci once said: ”La prima pittura fu sol di una linea”. (”The first painting consisted just of a single line”). Electronic Journal of Theoretical Physics 14, No. 37 (2018) 213–249 249 Development of the ”Schr¨odinger” stage, where the fundamental wave equation should appear, turned out for the author to be the ”work exceeding our powers and our hopes”31. We do not know whether this project, this concept of 52-dimensional Dirac capacitance for eight Dirac vectors, is the stone, the unshakable foundation upon which there may be erected a solid construction of theoretical physics, which is not shaken by regularizations, renormalizations, anomalies and Higgs fields. It is appropriate to repeat the words of Saint Augustine: ”We will be searching as if we can find, and we will find if our search is endless”. References [1] Temnenko V.A., Physics of currents and potentials. I. Classical electrodynamics with non-point charge. – Electronic Journal of Theoretical Physics, 11, No. 31, 2014. – pp. 221–256. [2] Temnenko V.A., Physics of currents and potentials. II. Classical singlet-triplet electroweak theory with non-point particles. – Electronic Journal of Theoretical Physics, 12, No. 32, 2015. – pp. 179–294. [3] Temnenko V.A., Physics of currents and potentials. III. Octupletical sector of classical field theor with non-point particles. – Electronic Journal of Theoretical Physics, 13, No. 36, 2016. – pp. 69–98. [4] Einstein A., A Brief Outline of the Development of the Theory of Relativity. - Nature, 1921, 106. – pp. 782–784. [5] Sneddon I.N., Fourier Transforms (First Edition). - McGraw-Hill, New York, Toronto, London, 1951. - 542 p. [6] Heisenberg W., Introduction to the unified field theory of elementary particles.- Interscience Publishers; London, New York, Sydney, 1st Ed edition (January 1, 1966). - 192 p. (ch.3) [7] Einstein A., Briefe an Maurice Solovine. – Berlin & Paris 1960. 31 The words of the Lord Chancellor Francis Bacon.