Physics and Symmetries of Commutative and Noncommutative Quantum Field Theory

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

Deadline for manuscript submissions: 15 April 2024 | Viewed by 8000

Special Issue Editor

Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia
Interests: theoretical and mathematical physics

Special Issue Information

Dear Colleagues,

Quantum field theory (QFT) is a theoretical framework that combines classical field theory and quantum mechanical principles to describe the nature around us. It has been extremely successful in constructing physical models in particle physics (especially the QED and Standard model) and condensed matter physics, and has important implications in quantum aspects of gravity theories.


QFT is a result of constructing a theory that describes the many-particle sys-tems in accordance with the quantum mechanical principles together with the principles of special relativity.  The Poincaré symmetry is thus incorporated in the theory from the very beginning.  It is believed that for particles with very high energies the gravity effects become significant, so that these particles no longer "see" the spacetime as smooth and continuous, but instead they see it as quantized or fuzzy. The QFT constructed on such deformed manifolds requires a new framework. Such framework could be provided via the noncommutative geometry, where the search for the diffeomorphisms leaving spacetime invariant leads to deformation of Poincaré symmetry.

 Even though QFT has had much success in explaining phenomenological phenomena, its mathematical rigor (in some aspects) is still a topic of interest and work for many mathematical physicists. Noncommutative geometry and noncommutative QFTs have promising results in explaining the geometry of particle interactions, providing a framework even for quantum aspects of gravity and even lifting the theory to a more rigorous formulation.

This Special Issue will publish contributions on various aspects of commutative and noncommutative QFTs, studying the interplay between them, symmetries and their generalizations, and their applications in the quantum aspects of spacetime. The articles are expected to present some new or recent results in the field along with a general review of the problem and a history of the problem in matter.

Dr. Tajron Jurić
Guest Editor

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Keywords

  • Quantum field theory and the principle of symmetry
  • Noncommutative geometry, spectral triple, and spectral action 
  • Symmetry in mathematical physics
  • Space-time symmetries
  • Generalized symmetries, Hopf algebras, and quantum groups  
  • Noncommutative QFTs and gravity 
  • QFT in curved spaces 
  • Quantization and symmetry
  • Renormalization (Wilson, Epstein–Glaser, Algebraic, etc.) 
  • Axiomatic-, Algebraic-, Constructive-QFT

Published Papers (6 papers)

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Research

13 pages, 308 KiB  
Article
Three Alternative Model-Building Strategies Using Quasi-Hermitian Time-Dependent Observables
by Miloslav Znojil
Symmetry 2023, 15(8), 1596; https://doi.org/10.3390/sym15081596 - 17 Aug 2023
Viewed by 559
Abstract
In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a [...] Read more.
In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the time-dependent states ψ(t) (using Schrödinger-type equations) as well as of the time-dependent observables Λj(t), j=1,2,,K (using Heisenberg-type equations). In the unitary-evolution dynamical regime of our interest, both of the respective generators of the evolution (viz., in our notation, the Schrödingerian generator G(t) and the Heisenbergian generator Σ(t)) have, in general, complex spectra. Only the spectrum of their superposition remains real. Thus, only the observable superposition H(t)=G(t)+Σ(t) (representing the instantaneous energies) should be called Hamiltonian. In applications, nevertheless, the mathematically consistent models can be based not only on the initial knowledge of the energy operator H(t) (forming a “dynamical” model-building strategy) but also, alternatively, on the knowledge of the Coriolis force Σ(t) (forming a “kinematical” model-building strategy), or on the initial knowledge of the Schrödingerian generator G(t) (forming, for some reason, one of the most popular strategies in the literature). In our present paper, every such choice (marked as “one”, “two” or “three”, respectively) is shown to lead to a construction recipe with a specific range of applicability. Full article
13 pages, 1007 KiB  
Article
Comparison of the Lifshitz Theory Using the Nonconventional Fit of Response Functions with Precise Measurements of the Casimir Force
by Galina L. Klimchitskaya and Vladimir M. Mostepanenko
Symmetry 2023, 15(5), 1011; https://doi.org/10.3390/sym15051011 - 01 May 2023
Viewed by 966
Abstract
It is known that the fundamental Lifshitz theory, which is based on the first principles of thermal quantum field theory, experiences difficulties when compared with precise measurements of the Casimir force. We analyzed the nonconventional fit of the response functions of many materials [...] Read more.
It is known that the fundamental Lifshitz theory, which is based on the first principles of thermal quantum field theory, experiences difficulties when compared with precise measurements of the Casimir force. We analyzed the nonconventional fit of the response functions of many materials along the imaginary frequency axis to the empirical model of “modified” oscillators, which was recently proposed in the literature. According to our results, this model is unacceptable because at high frequencies it leads to the asymptotic behavior of the response functions, which is in contradiction with that following from the fundamental physical principles. We calculated the Casimir interaction in the configurations of several precise experiments using the Lifshitz theory and the response functions to the quantized electromagnetic field expressed in terms of modified oscillators and demonstrated that the obtained results are excluded by the measurement data. This invalidated a claim made in the literature that the Casimir–van der Waals forces calculated using these response functions are in remarkable agreement with the experimental values. Possible reasons for a disagreement between experiment and theory are discussed, and the way to improve the situation is indicated. Full article
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19 pages, 333 KiB  
Article
The Origin of the Difference between Space and Time
by Hrvoje Nikolić
Symmetry 2023, 15(4), 957; https://doi.org/10.3390/sym15040957 - 21 Apr 2023
Viewed by 1688
Abstract
All differences between the role of space and time in nature are explained by proposing principles in which none of the spacetime coordinates has an a priori special role. Spacetime is treated as a non-dynamical manifold, with a fixed global RD topology. [...] Read more.
All differences between the role of space and time in nature are explained by proposing principles in which none of the spacetime coordinates has an a priori special role. Spacetime is treated as a non-dynamical manifold, with a fixed global RD topology. The dynamical theory of gravity determines only the metric tensor on a fixed manifold. All dynamics is treated as a Cauchy problem, so it follows that one coordinate takes a special role. It is proposed that any boundary condition that is finite everywhere leads to a solution which is also finite everywhere. This explains the (1,D1) signature of the metric, the boundedness of energy from below, the absence of tachyons, and other related properties of nature. The time arrow is explained by proposing that the boundary condition should be ordered. The quantization is considered as a boundary condition for field operators. Only the physical degrees of freedom are quantized. Full article
20 pages, 401 KiB  
Article
Noncommutative Correction to the Entropy of Charged BTZ Black Hole
by Tajron Jurić and Filip Požar
Symmetry 2023, 15(2), 417; https://doi.org/10.3390/sym15020417 - 04 Feb 2023
Cited by 1 | Viewed by 998
Abstract
Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outlined the formalism of Hopf algebras and its connection to the algebra of infinitesimal diffeomorphisms. Using a Drinfeld twist, we deformed spacetime symmetries, algebra of vector fields and differential [...] Read more.
Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outlined the formalism of Hopf algebras and its connection to the algebra of infinitesimal diffeomorphisms. Using a Drinfeld twist, we deformed spacetime symmetries, algebra of vector fields and differential forms, leading to a formulation of noncommutative Einstein equations. We studied a concrete example of charged BTZ spacetime and deformations steaming from the so-called angular twist. The entropy of the noncommutative charged BTZ black hole was obtained using the brick-wall method. We used a charged scalar field as a probe and obtained its spectrum and density of states via WKB approximation. We provide the method used to calculate corrections to the Bekenstein–Hawking entropy in higher orders in WKB, but we present the final result in the lowest WKB order. The result is that, even in the lowest order in WKB, the entropy, in general, contains higher powers in , and it has logarithmic corrections and polynomials of logarithms of the black hole area. Full article
14 pages, 323 KiB  
Article
Revisiting the Okubo–Marshak Argument
by Christian Gaß, José M. Gracia-Bondía and Jens Mund
Symmetry 2021, 13(9), 1645; https://doi.org/10.3390/sym13091645 - 07 Sep 2021
Cited by 4 | Viewed by 1492
Abstract
Modular localization and the theory of string-localized fields have revolutionized several key aspects of quantum field theory. They reinforce the contention that local symmetry emerges directly from quantum theory, but global gauge invariance remains in general an unwarranted assumption to be examined case [...] Read more.
Modular localization and the theory of string-localized fields have revolutionized several key aspects of quantum field theory. They reinforce the contention that local symmetry emerges directly from quantum theory, but global gauge invariance remains in general an unwarranted assumption to be examined case by case. Armed with those modern tools, we reconsider here the classical Okubo–Marshak argument on the non-existence of a “strong CP problem” in quantum chromodynamics. Full article
16 pages, 311 KiB  
Article
Quantum Supertwistors
by Rita Fioresi and María Antonia Lledó
Symmetry 2021, 13(7), 1241; https://doi.org/10.3390/sym13071241 - 10 Jul 2021
Cited by 1 | Viewed by 1320
Abstract
In this paper, we give an explicit expression for a star product on the super-Minkowski space written in the supertwistor formalism. The big cell of the super-Grassmannian Gr(2|0,4|1) is identified with the chiral, super-Minkowski [...] Read more.
In this paper, we give an explicit expression for a star product on the super-Minkowski space written in the supertwistor formalism. The big cell of the super-Grassmannian Gr(2|0,4|1) is identified with the chiral, super-Minkowski space. The super-Grassmannian is a homogeneous space under the action of the complexification SL(4|1) of SU(2,2|1), the superconformal group in dimension 4, signature (1,3), and supersymmetry N=1. The quantization is done by substituting the groups and homogeneous spaces by their quantum deformed counterparts. The calculations are done in Manin’s formalism. When we restrict to the big cell, we can explicitly compute an expression for the super-star product in the Minkowski superspace associated to this deformation and the choice of a certain basis of monomials. Full article
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