DOCSLIB.ORG

Quantum Criticality and Dark Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

CONTENTS 1 Criticality and dark matter M. Pitk¨anen, June 29, 2020 Email: [email protected]. http://tgdtheory.com/public_html/. Recent postal address: Rinnekatu 2-4 A 8, 03620, Karkkila, Finland. Contents 1 Introduction 5 2 Criticality In TGD Framework 6 2.1 Mathematical Approach To Criticality . .6 2.1.1 Basic Building Bricks Of TGD Vision . .6 2.1.2 Hierarchy Of Criticalities And Hierarchy Breakings Of Conformal Invariance7 2.1.3 Emergence Of The Covering Spaces Associated With The Hierarchy Of Planck Constants . .8 2.1.4 Other Connections . .8 2.1.5 Hierarchy of Planck constants, space-time surfaces as covering spaces, and adelic physics . .9 2.2 Phenomenological Approach To Criticality . 10 2.3 Do The Magnetic Flux Quanta Associated With Criticality Carry Monopole Flux? 11 3 What's New In TGD Inspired View About Phase Transitions? 12 3.1 About Thermal And Quantum Phase Transitions . 13 3.2 Some Examples Of Quantum Phase Transitions In TGD Framework . 13 3.3 ZEO Inspired View About Phase Transitions . 15 3.3.1 Question related to TGD inspired description of phase transitions . 15 3.3.2 Symmetries and phase transitions . 16 3.3.3 Quantum phase transitions and 4-D spin glass energy landscape . 18 3.3.4 What ZEO can give to the description of criticality? . 19 3.4 Maxwell's lever rule and expansion of water in freezing: two poorly understood phenomena . 20 3.4.1 Maxwell's lever rule as an indication for the presence of magnetic flux tubes 20 3.4.2 Strangeness in the freezing of water . 23 3.5 TGD based view about ferromagnetism . 24 CONTENTS 2 3.5.1 The ideas related to the work of Li et al . 25 3.5.2 Some reasons to get interested . 25 3.5.3 TGD based view . 26 3.5.4 The ideas related to the work of Li et al . 29 3.5.5 Some reasons to get interested . 29 3.5.6 TGD based view . 30 4 Summary about applications of hierarchy of Planck constants to quantum crit- icality 33 4.1 Particle Physics . 33 4.2 Condensed Matter Physics . 34 4.3 Living Matter . 34 4.4 Fringe Physics . 35 4.5 Proposed Mechanisms For Generating Large heff Phase . 37 5 Applications to Condensed Matter 39 5.1 Mysterious Action At Distance Between Liquid Containers . 39 5.2 The Behavior Of Superfluids In Gravitational Field . 40 5.2.1 Fountain effect . 40 5.2.2 TGD inspired model for the fountain effect . 41 5.2.3 Superfluids dissipate! . 43 5.2.4 An improvement of the flux tube picture . 44 5.2.5 What about Sun? . 45 5.2.6 Evidence for macroscopic quantum coherence of fluid flow at criticality . 45 5.3 The mysterious dichloromethane droplet, which refuses to sink in water and begins to spin . 46 5.4 The mysterious dichloromethane droplet, which refuses to sink in water and begins to spin . 46 5.5 Does The Physics Of SmB6 Make The Fundamental Dynamics Of TGD Directly Visible? . 48 5.5.1 Observations . 48 5.5.2 Could TGD help to understand the strange behavior of SmB6?....... 48 5.6 Quantization of thermal conductance and quantum thermodynamics . 52 5.7 Are monopoles found? . 54 5.8 Badly behaving photons and space-time as 4-surface . 55 5.9 Deviation from the prediction of standard quantum theory for radiative energy trans- fer in faraway region . 58 5.10 Time crystals, macroscopic quantum coherence, and adelic physics . 58 5.11 Non-local production of photon pairs as support for heff =h = n hypothesis . 60 5.12 Exciton-polariton Bose-Einstein condensate at room temperature and heff hierarchy 60 5.13 New findings related to high Tc super-conductivity . 62 5.13.1 High Tc superconductivity at room temperature and pressure . 62 5.13.2 Transition to high Tc superconductivity involves positive feedback . 65 5.13.3 BCS super conductivity at almost room temperature . 66 5.14 Quantum scarring from TGD point of view . 66 5.14.1 General TGD based considerations . 68 5.14.2 A concrete TGD inspired model for MBQS . 71 5.15 Three surprising condensed matter findings . 72 5.15.1 The strange behavior of light . 72 5.15.2 New surprises related to super-conductors . 74 5.15.3 Conductors of electricity, which are poor conductors of heat . 74 6 Biological Applications 74 6.1 Worrying About The Consistency With The TGD Inspired Quantum Biology . 74 6.1.1 Some background . 75 6.1.2 Does hgr = heff hypothesis predict that the energy range of dark photons is that of biophotons? . 75 CONTENTS 3 6.1.3 Should one modify the hgr = heff hypothesis? . 76 6.1.4 Did animal mitochondrial evolution have a long period of stagnation? . 77 6.1.5 An island at which body size shrinks . 78 6.2 Why metabolism and what happens in bio-catalysis? . 79 6.2.1 Why metabolic energy is needed? . 79 6.2.2 Conditions on bio-catalysis . 80 6.2.3 Phase transition reducing the value of heff =h = n as a basic step in bio- catalysis . 80 6.2.4 How molecules in cells “find” one another and organize into structures? . 81 6.2.5 Are mysteriously disappearing electrons dark in TGD sense? . 83 6.2.6 About possible implications . 84 6.2.7 Misbehaving Ruthenium atoms . 84 6.3 Does valence bond theory relate to the hierarchy of Planck constants? . 85 6.3.1 Transition from atomic physics to molecular physics and chemistry . 86 6.3.2 Valence bond theory very briefly . 87 6.3.3 Deducing an estimate for the value of n = heff =h from bond lengths . 88 6.3.4 About biological interpretation . 90 6.3.5 About the biological role of low valence ions . 94 6.3.6 Quantum criticality . 96 6.3.7 Quantum criticality and TGD inspired quantum biology . 97 6.3.8 A new mechanism of quantum criticality . 97 6.3.9 A new mechanism of quantum bio-control . 98 6.4 Are Bacteria Able To Induce Super-Fluidity? . 99 6.5 Bacteria behave like spin system: Why? . 100 6.5.1 The first step: binding of dark protons to formamido-pyrimidine . 103 6.5.2 Second step: Could shock waves due to extraterrestrial impacts have pro- duced RNA bases? . 105 6.6 Could the replication of mirror DNA teach something about chiral selection? . 106 6.6.1 Dark matter and chirality selection . 107 6.6.2 Model for the replication of DNA . 108 6.6.3 Differences between standard model and TGD based description . 111 6.7 Is dark DNA dark also in TGD sense? . 111 7 The analogs of CKM mixing and neutrino oscillations for particle and its dark variants 112 7.1 21-cm anomaly as a motivation for the model of the interaction between different levels of heff hierarchy . 112 7.2 Mixing and oscillations of dark photons . 114 7.2.1 Mixing and oscillations of ordinary and dark particles . 114 7.2.2 Mass squared matrix for photons . 118 7.2.3 Could the mixing with dark photons provide and additional contribution to particle masses? . 118 7.2.4 Description of ordinary-dark scattering diagrams . 119 8 TGD Inspired View About Blackholes And Hawking Radiation 119 8.1 Is Information Lost Or Not In Blackhole Collapse? . 119 8.2 What Are The Problems? . 120 8.3 TGD View About Black Holes And Hawking Radiation . 121 8.3.1 The basic ideas of TGD relevant for blackhole concept . 121 2 8.3.2 Could electric-magnetic duality allow to understand 1=heff dependence of the effective string tension? . 124 8.3.3 The analogs of blackholes in TGD . 125 8.4 More About BMS Supertranslations . 127 CONTENTS 4 9 How to demonstrate quantum superposition of classical gravitational fields? 130 9.1 Is gravitation classical or quantal? . 131 9.2 Zeno effect and weak measurements . 132 9.2.1 Can one test quantum character of gravitation experimentally? . 132 9.2.2 The notion of weak measurement . 133 9.2.3 Weak measurement induced by measurement of classical gravitational fields 134 9.2.4 What about the situation in TGD? . 135 9.3 What Does Topological Order Mean? . 137 9.4 Topological Order And Category Theory . 138 9.5 Category Theoretical Description Of Topological Order In TGD . 139 9.5.1 Topological order in TGD . 139 9.5.2 Stacking, time reversal, and gapless states in TGD framework? . 140 9.5.3 Category theory.
Recommended publications
  • Majorana Fermions in Mesoscopic Topological Superconductors: from Quantum Transport to Topological Quantum Computation

    Majorana Fermions in Mesoscopic Topological Superconductors: from Quantum Transport to Topological Quantum Computation

    Majorana Fermions in Mesoscopic Topological Superconductors: From Quantum Transport to Topological Quantum Computation Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Stephan Plugge D¨usseldorf, April 2018 Aus dem Institut f¨ur Theoretische Physik, Lehrstuhl IV der Heinrich-Heine-Universit¨at D¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨at D¨usseldorf Berichterstatter: 1. Prof. Dr. Reinhold Egger 2. PD Dr. Hermann Kampermann 3. Prof. Felix von Oppen, PhD Tag der m¨undlichen Pr¨ufung: 15. Juni 2018 List of included publications S. Plugge, A. Zazunov, P. Sodano, and R. Egger, Majorana entanglement bridge,Phys. Rev. B 91, 214507 (2015). [Selected as Editors’ Suggestion] L.A. Landau, S. Plugge, E. Sela, A. Altland, S.M. Albrecht, and R. Egger, Towards real- istic implementations of a Majorana surface code, Phys. Rev. Lett. 116, 050501 (2016). S. Plugge, A. Zazunov, E. Eriksson, A.M. Tsvelik, and R. Egger, Kondo physics from quasiparticle poisoning in Majorana devices,Phys.Rev.B93, 104524 (2016). S. Plugge, L.A. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger, Roadmap to Majorana surface codes,Phys.Rev.B94, 174514 (2016). [Selected as Editors’ Suggestion] S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg, Majorana box qubits, New J. Phys. 19 012001 (2017). [Fast-Track Communication; selected for Highlights of 2017 collection] T. Karzig, C. Knapp, R. Lutchyn, P. Bonderson, M. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. Marcus, and M. H. Freedman, Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes,Phys.Rev.B95, 235305 (2017).
  • LIST of PUBLICATIONS Jürg Fröhlich 1. on the Infrared Problem in A

    LIST of PUBLICATIONS Jürg Fröhlich 1. on the Infrared Problem in A

    LIST OF PUBLICATIONS J¨urgFr¨ohlich 1. On the Infrared Problem in a Model of Scalar Electrons and Massless Scalar Bosons, Ph.D. Thesis, ETH 1972, published in Annales de l'Inst. Henri Poincar´e, 19, 1-103 (1974) 2. Existence of Dressed One-Electron States in a Class of Persistent Models, ETH 1972, published in Fortschritte der Physik, 22, 159-198 (1974) 3. with J.-P. Eckmann : Unitary Equivalence of Local Algebras in the Quasi-Free Represen- tation, Annales de l'Inst. Henri Poincar´e 20, 201-209 (1974) 4. Schwinger Functions and Their Generating Functionals, I, Helv. Phys. Acta, 47, 265-306 (1974) 5. Schwinger Functions and Their Generating Functionals, II, Adv. of Math. 23, 119-180 (1977) 6. Verification of Axioms for Euclidean and Relativistic Fields and Haag's Theorem in a Class of P (φ)2 Models, Ann. Inst. H. Poincar´e 21, 271-317 (1974) 7. with K. Osterwalder : Is there a Euclidean field theory for Fermions? Helv. Phys. Acta 47, 781 (1974) 8. The Reconstruction of Quantum Fields from Euclidean Green's Functions at Arbitrary Temperatures, Helv. Phys. Acta 48, 355-369 (1975); (more details are contained in an unpublished paper, Princeton, Dec. 1974) 9. The Pure Phases, the Irreducible Quantum Fields and Dynamical Symmetry Breaking in Symanzik-Nelson Positive Quantum Field Theories, Annals of Physics 97, 1-54 (1975) 10. Quantized "Sine-Gordon" Equation with a Non-Vanishing Mass Term in Two Space-Time Dimensions, Phys. Rev. Letters 34, 833-836 (1975) 11. Classical and Quantum Statistical Mechanics in One and Two Dimensions: Two-Compo- nent Yukawa and Coulomb Systems, Commun.
  • Attempts to Go Beyond Bose and Fermi Statistics

    Attempts to Go Beyond Bose and Fermi Statistics

    Beyond Bose and Fermi Statistics O.W. Greenberg University of Maryland Spin-Statistics 2008 October 21, 2008 Trieste, Italy Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W.
  • An Introduction to Anyons

    An Introduction to Anyons

    An Introduction to Anyons Avinash Deshmukh University of British Columbia In this paper, we introduce anyons, particles that pick up a non-trivial phase under exchange. We start by considering the topological conditions that allow anyons to be created and construct the braid group, which defines anyonic exchange statistics. We proceed to discuss abelian and non- abelian exchange representations and review cyons, fibonacci anyons, and Ising anyons as illustrative examples. Finally, we briefly examine solved condensed matter models in which anyons are produced and review the exchange statistics of these systems. I. INTRODUCTION rigorous version of this argument. It is well known that in three-dimensions, there are two One important consequence of θ being any value is that it is iθ iθ types of particles; bosons are symmetric under exchange and not generally true that e “ e´ . We must then distinguish fermions are anti-symmetric under exchange. However, in between \counterclockwise" exchanges and \clockwise" ex- two-dimensions, it is possible to have particles that are nei- changes, which correspond to the accumulated phases θ and ther anti-symmetric nor symmetric under exchange. These ´θ respectively. particles are known as anyons. It has been shown that the exotic statistics of anyons can be used to encode information for topological quantum computation [1], and it is believed B. The Braid Group that they can be used in the construction of superconductors [2]. Earlier this year, abelian anyons were directly observed Consider a system of n particles that follow θ-statistics in a two-dimensional electronic gas collider [3] and in an elec- 2 with the positions r1; :::; rn P R in the center of mass tronic Fabry-Perot intereferometer [4], showing that these frame, described by the wave function pr1; :::; rnq.
  • Boson-Lattice Construction for Anyon Models

    Boson-Lattice Construction for Anyon Models

    Boson-Lattice Construction for Anyon Models Bel´enParedes Ludwig Maximilian University, Munich This work is an attempt to unveil the skeleton constituents. The set of anyon types emerging from a of anyon models. I present a construction to sys- certain topological order satisfies a collection of fusion tematically generate anyon models. The construc- and braiding rules [10{14], which determine the way in tion uses a set of elementary pieces or fundamental which anyons fuse with each other to give rise to other anyon models, which constitute the building blocks anyon types, and the form in which they braid around to construct other, more complex, anyon models. each other. A set of anyon types together with their fu- A principle of assembly is established that dictates sion and braiding rules define an anyon model, which per- how to articulate the building blocks, setting out the fectly mirrors its corresponding underlying physical state. global blueprint for the whole structure. Remark- What are the possible anyon models that can exist? ably, the construction generates essentially all tabu- From a purely mathematical perspective, it is known that lated anyon models. Moreover, novel anyon models the language underlying anyon models is modular ten- (non-tabulated, to my knowledge) arise. To embody sor category [15{22]. Within this beautiful (and com- the construction I develop a very physical, visual and plex) mathematical formalism the answer to this ques- intuitive lexicon. An anyon model corresponds to a tion can be simply phrased: any possible anyon model system of bosons in a lattice. By varying the num- corresponds to a unitary braided modular tensor cate- ber of bosons and the number of lattice sites, towers gory.
  • Exchange and Exclusion in the Non-Abelian Anyon Gas

    Exchange and Exclusion in the Non-Abelian Anyon Gas

    EXCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS DOUGLAS LUNDHOLM AND VIKTOR QVARFORDT Abstract. We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on N strands. Allowing for arbitrary rank (dependent on N) and non-abelian representations, and letting N ! 1, this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincar´eand Hardy inequalities) and a local exclusion principle (also implying a Lieb{Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial. Contents 1. Introduction2 1.1. Anyons | abelian vs. non-abelian3 1.2. Some mathematically rigorous results for abelian anyons5 1.3. Main new results8 2. Algebraic anyon models 11 2.1. Fusion 11 2.2. Braiding 14 2.3. Pentagon and hexagon equations 15 2.4. Vacuum, bosons and fermions 17 2.5. Abelian anyons 18 2.6. Fibonacci anyons 18 2.7. Ising anyons 20 3. Exchange operators and phases 21 3.1. Braid group representations 22 3.2. Exchange operators 25 3.3. Abelian anyons 26 3.4. Fibonacci anyons 27 3.5. Ising anyons 29 arXiv:2009.12709v1 [math-ph] 26 Sep 2020 3.6. Clifford anyons 32 3.7.
  • Annual Report 2006

    Annual Report 2006

    ANNUAL REPORT 2006 Dutch Research School of Theoretical Physics (DRSTP) Landelijke Onderzoekschool voor Theoretische Natuurkunde (LOTN) Visiting address: Minnaert building Leuvenlaan 4 3584 CE Utrecht Postal address: P.O. Box 80.195 3508 TD Utrecht the Netherlands tel.: +31 (0)30 2535916 fax: +31 (0)30 2535937 e-mail: [email protected] website: http://www1.phys.uu.nl/drstp/ Preface This is the Annual Report 2006 of the Dutch Research School of Theoretical Physics (DRSTP). It provides an overview of the educational and research activities during 2006, intended for a broad spectrum of interested parties. The report also presents two research highlights written by staff members of the Research School. In addition, it offers a wealth of factual information, such as a list of the participating staff, of the PhD students, a comprehensive list of publications, as well as other relevant statistics. The annual report is not the only information that we make available throughout the year. We also publish a monthly newsletter and a yearly guide of our educational activities. Up-to-date information on the DRSTP is also readily available on internet at: http://www1.phys.uu.nl/drstp/. Finally, we should like to thank all of those who contributed to the Research School during this past year. prof. dr. B. de Wit prof. dr. K. Schoutens Scientific director Chair governing board August 2007 Contents 1 The DRSTP in 2006 7 2 Scientific highlights 11 3 PhD programme 19 3.1 Educational programme . 19 3.1.1 DRSTP postgraduate courses (AIO/OIO schools) . 19 3.1.2 Guest lecturers .
  • TWO-DIMENSIONAL QUANTUM GAS S

    TWO-DIMENSIONAL QUANTUM GAS S

    f .'..~«rf£^.« ;»*«4 I INSTITUTO DE FÍSICA TEÓRICA 1TT/P-17/9O TWO-DIMENSIONAL QUANTUM GAS a* R. Aldr ovandl* Instituto de Física Teóricas State Untversity oi 55o Paulo - UNESP Rúa Parriplcna 145 - 01405 -São.Pauio S? - Brazil s Abstract o ícter,!;cal itnprnstfcile perlicles '.n a 2-diíiieosior4! ccnfíg-jralisn serra fcrotá stetisti», Ihí&rír.a;i3'o beUv=eri bosons end . sy.iMügiric croups fte r.ein properties of ao ij^J jis cl su-;-. ^ are presèTiiâl Itw/ CD iaterpolôte U*e prepertiçs et t5so.is erd If i fr.s but cióssicai pyticles as a special case Restriction to 2 c .; *',ens prscVjfcs l4.T»tcü points but ori-jic-sías fl pecv! - . symmetry,responsible in particular for ths'icsrj , fi - boson ft.il iernt:on specific h£c:!S. o 0 * With perttel «uppyl cí CMr « Idex 11-31 370Uii'-.r 5^ .1fi!ôf«00 55ll36i4«? U£SP <• E?JA?£SP. tíí.MET 1.Introduction Quantum mechanics of a given system is controlled Dy the group of classes of closed loops (the fundamental group) or Us configuration space ill and, of course, statistical mechanics will follow suit. The configuration space of a system of N identical particles will have a non-trivial fundamental group [2], that is, will be multiply connected Let M be the configuration space of each particle and consider the N set M • |x • (xt, x2,.... xN)), the cartesian product of manifold M by itself N tifiics. MN will be the configuration space of a gas of N distinguishable particles contained in fi n is usually a 3- dimensional Euclidean box whose volume V is taken to infinity at the thermodynamica! limit.
  • Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation

    Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation

    DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2017 Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation VIKTOR QVARFORDT KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation VIKTOR QVARFORDT Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2017 Supervisor at KTH: Douglas Lundholm Examiner at KTH: Kristian Bjerklöv TRITA-MAT-E 2017:17 ISRN-KTH/MAT/E--17/17--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored.
  • Thermodynamics of Non-Abelian Exclusion Statistics

    Thermodynamics of Non-Abelian Exclusion Statistics

    THERMODYNAMICS OF NON-ABELIAN EXCLUSION STATISTICS Wung-Hong Huang Department of Physics National Cheng Kung University Tainan,70101,Taiwan The thermodynamic potential of ideal gases described by the simplest non-abelian statistics is investigated. I show that the potential is the linear function of the element of the abelian- part statistics matrix. Thus, the factorizable property in the Haldane (abelian) fractional exclusion shown by the author [W. H. Huang, Phys. Rev. Lett. 81, 2392 (1998)] is now extended to the non-abelian case. The complete expansion of the thermodynamic potential is also given. Keywords: fractional exclusion statistics; quantum Hall effect. Classification Number: 05.30.-d, 71.10.+x E-mail: [email protected] arXiv:hep-th/0008153v1 19 Aug 2000 1 I. INTRODUCTION It is know that the quasiparticle appearing in the strong correlation system in the low dimension may obeying the Haldane fractional exclusion statistics [1]. The famous examples are those in the fractional quantized Hall effect and spin 1/2 antiferromagnetic chain [2-4]. The thermodynamics in these system had been investigated by several authors [5-8]. Since the braid group in two space dimensions may be represented by the non-commute matrix, the non-abelian braid statistics could be shown in a real world [9-16]. These include the quasiparticle in the non-abelian fractional quantum Hall state as well as those in the conformal field theory [9-15]. The spinon in the Heisenberg chains could also show the non-abelian exchange statistics [16]. In a recent paper [15], Guruswamy and Schoutens had derived the occupation number distribution functions for the particle obeying the non-abelian statistics.
  • The Quantum Geometry of Spin and Statistics

    The Quantum Geometry of Spin and Statistics

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server The Quantum Geometry of Spin and Statistics Robert Oeckl∗ Centre de Physique Th´eorique, CNRS Luminy, 13288 Marseille, France and Department of Applied Mathematics and Mathematical Physics, University of Cambridge, Cambridge CB3 0WA, UK DAMTP-2000-81 27 January 2001 Abstract Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin- statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose-Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the un- derlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach. MSC: 81R50, 81S05, 81S40, 81T18 keywords: quantum groups, statistics, spin, quantum field theory, anyons, quons ∗email: [email protected] 1 1 Introduction While spin in quantum physics arises from the geometry of space-time, statistics is connected to the geometry of configuration space. Half-integer spin and Bose-Fermi statistics arise in 3 or higher dimensions, while in 2 dimensions more general fractional spin and anyonic statistics are possible (see [?] and references therein). In fact, both, spin and statistics are related to symmetries. In the case of spin this is plainly understood in terms of the symmetries of space-time.
  • Quantum Computing and Quantum Topology

    Quantum Computing and Quantum Topology

    Topological Quantum Computation Zhenghan Wang Microsoft Station Q & UC Sana Barbara Texas, March 26, 2015 Classical Physics Turing Model Quantum field computing is the same as quantum computing. Quantum Mechanics Quantum Computing Quantum Field Theory ??? True for TQFTs String Theory ?????? (Freedman, Kitaev, Larsen, W.) Quantum Computation • There is a serious prospect for quantum physics to change the face of information science. • Theoretically, the story is quite compelling: – Shor’s factoring algorithm (1994) – Fault tolerance ~1996-1997 independently • P. Shor • A. M. Steane • A. Kitaev • But for the last twenty years the most interesting progress has been to build a quantum computer. Why Quantum More Powerful? • Superposition • Entanglement A (classical) bit is given by a Quantum states need not be physical system that can exist products. For example: in one of two distinct states: 0 or 1 A qubit is given by a physical system that can exist in a linear combination of two distinct quantum states: or This is the property that enables quantum state teleportation and Einstein’s “spooky action at a distance.” |휓 > ∈ 퐶푃1 ● Classical information source is modeled by a random variable X The bit---a random variable X {0,1} with equal probability. Physically, it is a switch I (p)= - n p log p , X i=1 i 2 i 1 ● A state of a quantum system is an information source The qubit---a quantum system whose states given by non-zero vectors in ℂퟐ up to non-zero scalars. Physically, it is a 2-level quantum system. Paradox: A qubit contains both more and less than 1 bit of information.