Entanglement on Spin Networks Loop Quantum
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The Pennsylvania State University The Graduate School The Eberly College of Science CORRELATIONS IN QUANTUM GRAVITY AND COSMOLOGY A Dissertation in Physics by Bekir Baytas © 2018 Bekir Baytas Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2018 The dissertation of Bekir Baytas was reviewed and approved∗ by the following: Sarah Shandera Assistant Professor of Physics Dissertation Advisor, Chair of Committee Eugenio Bianchi Assistant Professor of Physics Martin Bojowald Professor of Physics Donghui Jeong Assistant Professor of Astronomy and Astrophysics Nitin Samarth Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. ii Abstract We study what kind of implications and inferences one can deduce by studying correlations which are realized in various physical systems. In particular, this thesis focuses on specific correlations in systems that are considered in quantum gravity (loop quantum gravity) and cosmology. In loop quantum gravity, a spin-network basis state, nodes of the graph describe un-entangled quantum regions of space, quantum polyhedra. We introduce Bell- network states and study correlations of quantum polyhedra in a dipole, a pentagram and a generic graph. We find that vector geometries, structures with neighboring polyhedra having adjacent faces glued back-to-back, arise from Bell-network states. The results present show clearly the role that entanglement plays in the gluing of neighboring quantum regions of space. We introduce a discrete quantum spin system in which canonical effective methods for background independent theories of quantum gravity can be tested with promising results. In particular, features of interacting dynamics are analyzed with an emphasis on homogeneous configurations and the dynamical building- up and stability of long-range correlations. -
1 HESTENES' TETRAD and SPIN CONNECTIONS Frank
HESTENES’ TETRAD AND SPIN CONNECTIONS Frank Reifler and Randall Morris Lockheed Martin Corporation MS2 (137-205) 199 Borton Landing Road Moorestown, New Jersey 08057 ABSTRACT Defining a spin connection is necessary for formulating Dirac’s bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes’ tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of × K ρ SL(2,R) U(1) gauge potentials Fα and a complex scalar field . This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary K linear connection acting on the tensor fields (Fα ,ρ) . We also derive spin connections for which Dirac’s bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection. 1 I. INTRODUCTION Defining a spin connection to replace the partial derivatives in Dirac’s bispinor equation in a Minkowski space-time, is necessary for the formulation of Dirac’s bispinor equation in a curved space-time. All the spin connections acting on bispinors found in the literature first introduce a local orthonormal tetrad field on the space-time manifold and then require that the Dirac Lagrangian be invariant under local change of tetrad [1] – [8]. -
Twistor Theory at Fifty: from Rspa.Royalsocietypublishing.Org Contour Integrals to Twistor Strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J
Downloaded from http://rspa.royalsocietypublishing.org/ on November 10, 2017 Twistor theory at fifty: from rspa.royalsocietypublishing.org contour integrals to twistor strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J. Mason4 Cite this article: Atiyah M, Dunajski M, Mason LJ. 2017 Twistor theory at fifty: from 1School of Mathematics, University of Edinburgh, King’s Buildings, contour integrals to twistor strings. Proc. R. Edinburgh EH9 3JZ, UK Soc. A 473: 20170530. 2Trinity College Cambridge, University of Cambridge, Cambridge http://dx.doi.org/10.1098/rspa.2017.0530 CB21TQ,UK 3Department of Applied Mathematics and Theoretical Physics, Received: 1 August 2017 University of Cambridge, Cambridge CB3 0WA, UK Accepted: 8 September 2017 4The Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford OX2 6GG, UK Subject Areas: MD, 0000-0002-6477-8319 mathematical physics, high-energy physics, geometry We review aspects of twistor theory, its aims and achievements spanning the last five decades. In Keywords: the twistor approach, space–time is secondary twistor theory, instantons, self-duality, with events being derived objects that correspond to integrable systems, twistor strings compact holomorphic curves in a complex threefold— the twistor space. After giving an elementary construction of this space, we demonstrate how Author for correspondence: solutions to linear and nonlinear equations of Maciej Dunajski mathematical physics—anti-self-duality equations e-mail: [email protected] on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang– Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations. -
The Pauli Exclusion Principle and SU(2) Vs. SO(3) in Loop Quantum Gravity
The Pauli Exclusion Principle and SU(2) vs. SO(3) in Loop Quantum Gravity John Swain Department of Physics, Northeastern University, Boston, MA 02115, USA email: [email protected] (Submitted for the Gravity Research Foundation Essay Competition, March 27, 2003) ABSTRACT Recent attempts to resolve the ambiguity in the loop quantum gravity description of the quantization of area has led to the idea that j =1edgesof spin-networks dominate in their contribution to black hole areas as opposed to j =1=2 which would naively be expected. This suggests that the true gauge group involved might be SO(3) rather than SU(2) with attendant difficulties. We argue that the assumption that a version of the Pauli principle is present in loop quantum gravity allows one to maintain SU(2) as the gauge group while still naturally achieving the desired suppression of spin-1/2 punctures. Areas come from j = 1 punctures rather than j =1=2 punctures for much the same reason that photons lead to macroscopic classically observable fields while electrons do not. 1 I. INTRODUCTION The recent successes of the approach to canonical quantum gravity using the Ashtekar variables have been numerous and significant. Among them are the proofs that area and volume operators have discrete spectra, and a derivation of black hole entropy up to an overall undetermined constant [1]. An excellent recent review leading directly to this paper is by Baez [2], and its influence on this introduction will be clear. The basic idea is that a basis for the solution of the quantum constraint equations is given by spin-network states, which are graphs whose edges carry representations j of SU(2). -
Arxiv:1907.02341V1 [Gr-Qc]
Different types of torsion and their effect on the dynamics of fields Subhasish Chakrabarty1, ∗ and Amitabha Lahiri1, † 1S. N. Bose National Centre for Basic Sciences Block - JD, Sector - III, Salt Lake, Kolkata - 700106 One of the formalisms that introduces torsion conveniently in gravity is the vierbein-Einstein- Palatini (VEP) formalism. The independent variables are the vierbein (tetrads) and the components of the spin connection. The latter can be eliminated in favor of the tetrads using the equations of motion in the absence of fermions; otherwise there is an effect of torsion on the dynamics of fields. We find that the conformal transformation of off-shell spin connection is not uniquely determined unless additional assumptions are made. One possibility gives rise to Nieh-Yan theory, another one to conformally invariant torsion; a one-parameter family of conformal transformations interpolates between the two. We also find that for dynamically generated torsion the spin connection does not have well defined conformal properties. In particular, it affects fermions and the non-minimally coupled conformal scalar field. Keywords: Torsion, Conformal transformation, Palatini formulation, Conformal scalar, Fermion arXiv:1907.02341v1 [gr-qc] 4 Jul 2019 ∗ [email protected] † [email protected] 2 I. INTRODUCTION Conventionally, General Relativity (GR) is formulated purely from a metric point of view, in which the connection coefficients are given by the Christoffel symbols and torsion is set to zero a priori. Nevertheless, it is always interesting to consider a more general theory with non-zero torsion. The first attempt to formulate a theory of gravity that included torsion was made by Cartan [1]. -
Spin Networks and Sl(2, C)-Character Varieties
SPIN NETWORKS AND SL(2; C)-CHARACTER VARIETIES SEAN LAWTON AND ELISHA PETERSON Abstract. Denote the free group on 2 letters by F2 and the SL(2; C)-representation variety of F2 by R = Hom(F2; SL(2; C)). The group SL(2; C) acts on R by conjugation, and the ring of in- variants C[R]SL(2;C) is precisely the coordinate ring of the SL(2; C)- character variety of a three-holed sphere. We construct an iso- morphism between the coordinate ring C[SL(2; C)] and the ring of matrix coe±cients, providing an additive basis of C[R]SL(2;C). Our main results use a spin network description of this basis to give a strong symmetry within the basis, a graphical means of computing the product of two basis elements, and an algorithm for comput- ing the basis elements. This provides a concrete description of the regular functions on the SL(2; C)-character variety of F2 and a new proof of a classical result of Fricke, Klein, and Vogt. 1. Introduction The purpose of this work is to demonstrate the utility of a graph- ical calculus in the algebraic study of SL(2; C)-representations of the fundamental group of a surface of Euler characteristic -1. Let F2 be a rank 2 free group, the fundamental group of both the three-holed sphere and the one-holed torus. The set of representations R = Hom(F2; SL(2; C)) inherits the structure of an algebraic set from SL(2; C). The subset of representations that are completely reducible, denoted by Rss, have closed orbits under conjugation. -
3. Introducing Riemannian Geometry
3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4). -
A Speculative Ontological Interpretation of Nonlocal Context-Dependency in Electron Spin
A SPECULATIVE ONTOLOGICAL INTERPRETATION OF NONLOCAL CONTEXT-DEPENDENCY IN ELECTRON SPIN Martin L. Shough1 1st October 2002 Contents Abstract ................................................................................................................... .2 1. Spin Correlations as a Restricted Symmetry Group ............................................... 5 2. Ontological Basis of a Generalised Superspin Symmetry .....................................17 3. Epistemology & Ontology of Spin Measurement ................................................. 24 4. The Superspin Network. Symmetry-breaking and the Emergence of Local String Modes .......................................................................................... 39 5. Superspin Interpretation of the Field ................................................................... 57 6. Reflections and connections ................................................................................ 65 7. Cosmological implications .................................................................................. 74 References ........................................................................................................... 89 1 [email protected] Abstract Intrinsic spin is understood phenomenologically, as a set of symmetry principles. Inter-rotations of fermion and boson spins are similarly described by supersymmetry principles. But in terms of the standard quantum phenomenology an intuitive (ontological) understanding of spin is not to be expected, even though (or rather, -
Weyl's Spin Connection
THE SPIN CONNECTION IN WEYL SPACE c William O. Straub, PhD Pasadena, California “The use of general connections means asking for trouble.” —Abraham Pais In addition to his seminal 1929 exposition on quantum mechanical gauge invariance1, Hermann Weyl demonstrated how the concept of a spinor (essentially a flat-space two-component quantity with non-tensor- like transformation properties) could be carried over to the curved space of general relativity. Prior to Weyl’s paper, spinors were recognized primarily as mathematical objects that transformed in the space of SU (2), but in 1928 Dirac showed that spinors were fundamental to the quantum mechanical description of spin—1/2 particles (electrons). However, the spacetime stage that Dirac’s spinors operated in was still Lorentzian. Because spinors are neither scalars nor vectors, at that time it was unclear how spinors behaved in curved spaces. Weyl’s paper provided a means for this description using tetrads (vierbeins) as the necessary link between Lorentzian space and curved Riemannian space. Weyl’selucidation of spinor behavior in curved space and his development of the so-called spin connection a ab ! band the associated spin vector ! = !ab was noteworthy, but his primary purpose was to demonstrate the profound connection between quantum mechanical gauge invariance and the electromagnetic field. Weyl’s 1929 paper served to complete his earlier (1918) theory2 in which Weyl attempted to derive electrodynamics from the geometrical structure of a generalized Riemannian manifold via a scale-invariant transformation of the metric tensor. This attempt failed, but the manifold he discovered (known as Weyl space), is still a subject of interest in theoretical physics. -
Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results
universe Article Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results Jakub Mielczarek 1,2 1 CPT, Aix-Marseille Université, Université de Toulon, CNRS, F-13288 Marseille, France; [email protected] 2 Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Cracow, Poland Received: 16 April 2019; Accepted: 24 July 2019; Published: 26 July 2019 Abstract: Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is to make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin network. Furthermore, all spin labels of the spin network are assumed to be equal j = 1/2, which is crucial for the introduction of the intertwiner qubits. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use of: IBM’s ibmqx4 5-qubit quantum computer, simulator of quantum computer provided by the same company and QX quantum computer simulator. Finally, values of the vertex probability are determined employing both the QX and the IBM simulators with 20-qubit quantum register and compared with analytical predictions. Keywords: Spin networks; vertex amplitudes; quantum computing 1. Introduction The basic objective of theories of quantum gravity is to calculate transition amplitudes between configurations of the gravitational field. The most straightforward approach to the problem is provided by the Feynman’s path integral Z i (SG+Sf) hY f jYii = D[g]D[f]e } , (1) where SG and Sf are the gravitational and matter actions respectively. -
Spin Connection Resonance in Gravitational General Relativity∗
Vol. 38 (2007) ACTA PHYSICA POLONICA B No 6 SPIN CONNECTION RESONANCE IN GRAVITATIONAL GENERAL RELATIVITY∗ Myron W. Evans Alpha Institute for Advanced Study (AIAS) (Received August 16, 2006) The equations of gravitational general relativity are developed with Cartan geometry using the second Cartan structure equation and the sec- ond Bianchi identity. These two equations combined result in a second order differential equation with resonant solutions. At resonance the force due to gravity is greatly amplified. When expressed in vector notation, one of the equations obtained from the Cartan geometry reduces to the Newton inverse square law. It is shown that the latter is always valid in the off resonance condition, but at resonance, the force due to gravity is greatly amplified even in the Newtonian limit. This is a direct consequence of Cartan geometry. The latter reduces to Riemann geometry when the Cartan torsion vanishes and when the spin connection becomes equivalent to the Christoffel connection. PACS numbers: 95.30.Sf, 03.50.–z, 04.50.+h 1. Introduction It is well known that the gravitational general relativity is based on Rie- mann geometry with a Christoffel connection. This type of geometry is a special case of Cartan geometry [1] when the torsion form is zero. There- fore gravitation general relativity can be expressed in terms of this limit of Cartan geometry. In this paper this procedure is shown to produce a sec- ond order differential equation with resonant solutions [2–20]. Off resonance the mathematical form of the Newton inverse square law is obtained from a well defined approximation to the complete theory, but the relation between the gravitational potential field and the gravitational force field is shown to contain the spin connection in general. -
Supergravities
Chapter 10 Supergravities. When we constructed the spectrum of the closed spinning string, we found in the 3 1 massless sector states with spin 2 , in addition to states with spin 2 and other states i ~j NS ⊗ NS : b 1 j0; piL ⊗ b 1 j0; piR with i; j = 1; 8 yields gij(x) with spin 2 − 2 − 2 (10.1) i 3 NS ⊗ R : b− 1 j0; piL ⊗ j0; p; αiR with i; α = 1; 8 yields χi,α(x) with spin 2 2 The massless states with spin 2 correspond to a gauge field theory: Einstein's theory 3 of General Relativity, but in 9 + 1 dimensions. The massless states with spin 2 are part of another gauge theory, an extension of Einstein gravity, called supergravity. That theory was first discovered in 1976 in d = 3 + 1 dimensions, but supergravity theories also exist in higher dimensions, through (up to and including) eleven dimensions. In this chapter we discuss supergravity theories. They are important in string theory because they are the low-energy limit of string theories. We do not assume that the reader has any familiarity with supergravity. We begin with the N = 1 theory in 3 + 1 dimensions, m also called simple supergravity, which has one gravitational vielbein field eµ , and one m α fermionic partner of eµ , called the gravitino field µ with spinor index α = 1; ··· ; 4 A ¯ _ (or two 2-component gravitinos µ and µ,A_ with A; A = 1; 2) and µ = 0; 1; 2; 3. This theory has one real local supersymmetry with a real (Majorana) 4-component spinorial α A ∗ _ parameter (or two 2-component parameters and ¯A_ = (A) with A; A = 1; 2).