Symmetry Principles in Quantum Systems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (30 December 2020) | Viewed by 5844

Special Issue Editor

Special Issue Information

Dear Colleagues,

In 1931, Hermann Weyl published the famous monograph “The Theory of Groups and Quantum Mechanics”. From that moment, symmetry has been one of the most important principles for describing quantum phenomena. This Special Issue invites you to contribute your original research work and review articles on “Symmetry in Quantum Systems” that either advances theoretical and mathematical methods or extends the bounds of existing methodologies and other challenges in quantum theory. Symmetry is usually associated with theory, but we also expect articles discussing the experimental aspects of symmetry.

We hope that this Special Issue will provide an overall picture of the discussed problems and up-to-date findings to both researchers and students and that the readers would ultimately benefit from reading the contributions.

Scope: Potential topics dealing with but not limited to the following subheadings are deemed suitable for publication:

  • Group theoretical methods;
  • Many body quantum systems;
  • Low-dimensional systems;
  • Symmetries of differential equations proposed as mathematical models;
  • Quantum selection rules;
  • Symmetry in quantum information processing;
  • Supersymmetry in quantum systems;
  • Symmetry in soft matter physics;
  • Symmetry in condensed matter physics;
  • Approximate symmetry and symmetry breaking;
  • The gauge principle.

Prof. Jan Sładkowski
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • symmetry principles
  • representations of groups
  • many body systems
  • quantum theory
  • supersymmetry
  • symmetry breaking
  • molecular symmetry

Published Papers (3 papers)

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Research

23 pages, 405 KiB  
Article
Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit
Symmetry 2021, 13(1), 22; https://doi.org/10.3390/sym13010022 - 24 Dec 2020
Cited by 4 | Viewed by 1392
Abstract
We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an [...] Read more.
We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space. Full article
(This article belongs to the Special Issue Symmetry Principles in Quantum Systems)
14 pages, 312 KiB  
Article
Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem
Symmetry 2020, 12(1), 39; https://doi.org/10.3390/sym12010039 - 23 Dec 2019
Cited by 1 | Viewed by 2014
Abstract
The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete [...] Read more.
The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete set of Fock states is obtained, together with the corresponding explicit wavefunctions and their inner product integrals free from any divergence problem and with Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the underlying symmetry group, motivated from the perspective of the Minkowski spacetime as a representation for the Lorentz group, we obtain the natural non-unitary Fock space picture commonly considered, although not formulated and presented in the careful details given here. From a direct derivation of the appropriate basis state wavefunctions of the finite-dimensional irreducible representations of the Lorentz symmetry, the relation between the latter and the Fock state wavefunctions is also explicitly shown. Moreover, the full picture, including the states with a non-positive norm, may give a consistent physics picture as a version of Lorentz covariant quantum mechanics. The probability interpretation for the usual von Neumann measurements is not a problem, as all wavefunctions restricted to a definite value for the `time’ variable are just like those of the usual time independent quantum mechanics. A further understanding from a perspective of the dynamics from the symplectic geometry of the phase space is shortly discussed. Full article
(This article belongs to the Special Issue Symmetry Principles in Quantum Systems)
16 pages, 327 KiB  
Article
Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics
Symmetry 2019, 11(12), 1429; https://doi.org/10.3390/sym11121429 - 20 Nov 2019
Cited by 3 | Viewed by 1976
Abstract
Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries [...] Read more.
Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos T . In the case of the Basel topos and R 4 , the local invariance with respect to the corresponding atlases implies exotic smoothness on R 4 . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic R 4 underlying an exotic S 4 (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local T -invariance breaks tovariance even in the weaker sense. Full article
(This article belongs to the Special Issue Symmetry Principles in Quantum Systems)
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