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Symmetry
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Quantum Symmetry
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Quantum Symmetry

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 January 2011) | Viewed by 53304

Special Issue Editor

Dr. Dean Rickles

E-Mail Website
Guest Editor
Unit for History and Philosophy of Science, Faculty of Science, Sydney, NSW 2006, Australia
Interests: philosophy of symmetry; quantum gravity; foundations of physics; spacetime physics; econophysics

Special Issue Information

Dear Colleagues,

There are several differences between classical and quantum theories that have an impact on the possible symmetries:
  • Physical states represented in Hilbert space rather than phase space.
  • Quantum mechanics defines symmetries as mappings between physical states that preserve transition amplitudes. (As Wigner proved, these symmetries can be represented in Hilbert space by unitary and anti-unitary operators.)
  • Quantum mechanics assigns complex numbers to these transition amplitudes.
  • The algebra of observables in quantum mechanics is non-commutative.
  • Quantum particles are indistinguishable.
  • Composite quantum systems are not represented by a Cartesian product structure, but by a linear tensor structure.
Quantum symmetries may also include gauge redundancies and dualities. Gauge redundancies can be understood as multiple representations of the same physical state. Dualities can be understood as isomorphisms holding between pairs of Hilbert spaces together with (canonical) operators. The possibilities for quantum symmetries are tightly constrained by the number of spacetime dimensions and by the dimensionality of the objects of the theory (including whether they are extensionless or structured). Quantum symmetries also refer to quantum groups, which aren't groups as such but algebras that reduce to groups in the limit as a deformation parameter (playing the part of Planck's constant) goes to 1 (returning multiplication to normal).

Contributions are invited on all aspects of quantum symmetries. Those that involve foundational issues or the intersection of theoretical physics and pure mathematics are especially welcomed. Possible themes (not ranked in order preference) include:

  • 2D Conformal Field Theory, Modular Invariance, Statistical Mechanics.
  • Dualities in Quantum Theories.
  • Mirror Symmetry in String Theory.
  • Emergent Quantum Symmetries, Symmetry Breaking, Effective Field Theory, Renormalization Group.
  • Hopf Algebras, Quantum Groups and Low Dimensional Physics.
  • Quantum Geometry (including Non-Commutative Geometry).
  • Spin-Statistics, Anyons, Fractional Quantum Hall Effect.
  • Connections between Quantum Symmetries and Spacetime/Object Dimensionality.
  • Quantum Symmetries in Computation.
  • Relationship between Classical and Quantum Symmetries.

Dr. Dean Rickles
Guest Editor

Keywords

  • quantum symmetry
  • S-duality
  • symmetry breaking
  • anyons
  • braid group
  • quantum groups
  • conformal field theory
  • modular invariance

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Published Papers (7 papers)

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Research

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322 KiB  
Article
Action Duality: A Constructive Principle for Quantum Foundations
by Ken B. Wharton, David J. Miller and Huw Price
Symmetry 2011, 3(3), 524-540; https://doi.org/10.3390/sym3030524 - 27 Jul 2011
Cited by 21 | Viewed by 6803
Abstract
An analysis of the path integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act [...] Read more.
An analysis of the path integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act as a constructive principle, allowing any realistic model of one experiment to generate a corresponding model for its action-dual. Two pairs of action-dual experiments are presented, including one experiment that violates the Bell inequality and yet is action-dual to a single particle. The implications generally support retrodictive and retrocausal interpretations. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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266 KiB  
Article
Is the Notion of Time Really Fundamental?
by Florian Girelli, Stefano Liberati and Lorenzo Sindoni
Symmetry 2011, 3(3), 389-401; https://doi.org/10.3390/sym3030389 - 29 Jun 2011
Cited by 7 | Viewed by 6036
Abstract
From the physics point of view, time is now best described through General Relativity as part of space-time, which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics and quantum mechanical effects. This irreversibility can look puzzling [...] Read more.
From the physics point of view, time is now best described through General Relativity as part of space-time, which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics and quantum mechanical effects. This irreversibility can look puzzling since time-like loops (and hence time machines) can appear in General Relativity (for example in the Gödel universe, a solution of Einstein’s equations). We take this apparent discrepancy as a warning bell, pointing out that time as we understand it might not be fundamental and that whatever theory lying beyond General Relativity may not include time as we know it as a fundamental structure. We propose therefore, following the philosophy of analog models of gravity, that time and gravity might not be fundamental per se, but only emergent features. We illustrate our proposal using a toy-model where we show how the Lorentzian signature and Nordström gravity (a diffeomorphisms invariant scalar gravity theory) can emerge from a timeless non-dynamical space. This article received the fourth prize at the essay competition of the Foundational Questions Institute on the nature of time. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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658 KiB  
Article
Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry
by Philip Goyal and Kevin H. Knuth
Symmetry 2011, 3(2), 171-206; https://doi.org/10.3390/sym3020171 - 27 Apr 2011
Cited by 26 | Viewed by 15080
Abstract
Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with [...] Read more.
Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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391 KiB  
Article
Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?
by Karim P. Y. Thébault
Symmetry 2011, 3(2), 134-154; https://doi.org/10.3390/sym3020134 - 31 Mar 2011
Cited by 3 | Viewed by 6085
Abstract
Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can [...] Read more.
Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity. Full article
(This article belongs to the Special Issue Quantum Symmetry)
418 KiB  
Article
Positive Cosmological Constant and Quantum Theory
by Felix M. Lev
Symmetry 2010, 2(4), 1945-1980; https://doi.org/10.3390/sym2041945 - 19 Nov 2010
Cited by 6 | Viewed by 6031
Abstract
We argue that quantum theory should proceed not from a spacetime background but from a Lie algebra, which is treated as a symmetry algebra. Then the fact that the cosmological constant is positive means not that the spacetime background is curved but that [...] Read more.
We argue that quantum theory should proceed not from a spacetime background but from a Lie algebra, which is treated as a symmetry algebra. Then the fact that the cosmological constant is positive means not that the spacetime background is curved but that the de Sitter (dS) algebra as the symmetry algebra is more relevant than the Poincare or anti de Sitter ones. The physical interpretation of irreducible representations (IRs) of the dS algebra is considerably different from that for the other two algebras. One IR of the dS algebra splits into independent IRs for a particle and its antiparticle only when Poincare approximation works with a high accuracy. Only in this case additive quantum numbers such as electric, baryon and lepton charges are conserved, while at early stages of the Universe they could not be conserved. Another property of IRs of the dS algebra is that only fermions can be elementary and there can be no neutral elementary particles. The cosmological repulsion is a simple kinematical consequence of dS symmetry on quantum level when quasiclassical approximation is valid. Therefore the cosmological constant problem does not exist and there is no need to involve dark energy or other fields for explaining this phenomenon (in agreement with a similar conclusion by Bianchi and Rovelli). Full article
(This article belongs to the Special Issue Quantum Symmetry)
286 KiB  
Article
Introduction to a Quantum Theory over a Galois Field
by Felix M. Lev
Symmetry 2010, 2(4), 1810-1845; https://doi.org/10.3390/sym2041810 - 1 Nov 2010
Cited by 8 | Viewed by 6004
Abstract
We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only [...] Read more.
We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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Review

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251 KiB  
Review
Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications
by Young S. Kim and Marilyn E. Noz
Symmetry 2011, 3(1), 16-36; https://doi.org/10.3390/sym3010016 - 14 Feb 2011
Cited by 8 | Viewed by 6068
Abstract
Among the symmetries in physics, the rotation symmetry is most familiar to us. It is known that the spherical harmonics serve useful purposes when the world is rotated. Squeeze transformations are also becoming more prominent in physics, particularly in optical sciences and in [...] Read more.
Among the symmetries in physics, the rotation symmetry is most familiar to us. It is known that the spherical harmonics serve useful purposes when the world is rotated. Squeeze transformations are also becoming more prominent in physics, particularly in optical sciences and in high-energy physics. As can be seen from Dirac’s light-cone coordinate system, Lorentz boosts are squeeze transformations. Thus the squeeze transformation is one of the fundamental transformations in Einstein’s Lorentz-covariant world. It is possible to define a complete set of orthonormal functions defined for one Lorentz frame. It is shown that the same set can be used for other Lorentz frames. Transformation properties are discussed. Physical applications are discussed in both optics and high-energy physics. It is shown that the Lorentz harmonics provide the mathematical basis for squeezed states of light. It is shown also that the same set of harmonics can be used for understanding Lorentz-boosted hadrons in high-energy physics. It is thus possible to transmit physics from one branch of physics to the other branch using the mathematical basis common to them. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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