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Editorial

2 pages, 171 KiB

Open AccessEditorial

Editorial to the Special Issue “80 Years of Professor Wigner’s Seminal Work: On Unitary Representations of the Inhomogeneous Lorentz Group”

by Julio Marny Hoff da Silva

Universe 2021, 7(8), 310; https://doi.org/10.3390/universe7080310 - 21 Aug 2021

Viewed by 1398

Abstract

The present Special Issue is dedicated to celebrate 80 years of the Professor Eugene Paul Wigner paper “On Unitary Representations of the Inhomogeneous Lorentz Group”, published in 1939 [...] Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

Research

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11 pages, 10077 KiB

Open AccessArticle

Lepton-Antineutrino Entanglement and Chiral Oscillations

by Victor A. S. V. Bittencourt, Alex E. Bernardini and Massimo Blasone

Universe 2021, 7(8), 293; https://doi.org/10.3390/universe7080293 - 9 Aug 2021

Cited by 4 | Viewed by 1994

Abstract

Dirac bispinors belong to an irreducible representation of the complete Lorentz group, which includes parity as a symmetry yielding two intrinsic discrete degrees of freedom: chirality and spin. For massive particles, chirality is not dynamically conserved, which leads to chiral oscillations. In this [...] Read more.

Dirac bispinors belong to an irreducible representation of the complete Lorentz group, which includes parity as a symmetry yielding two intrinsic discrete degrees of freedom: chirality and spin. For massive particles, chirality is not dynamically conserved, which leads to chiral oscillations. In this contribution, we describe the effects of this intrinsic structure of Dirac bispinors on the quantum entanglement encoded in a lepton-antineutrino pair. We consider that the pair is generated through weak interactions, which are intrinsically chiral, such that in the initial state the lepton and the antineutrino have definite chirality but their spins are entangled. We show that chiral oscillations induce spin entanglement oscillations and redistribute the spin entanglement to chirality-spin correlations. Such a phenomenon is prominent if the momentum of the lepton is comparable with or smaller than its mass. We further show that a Bell-like spin observable exhibits the same behavior of the spin entanglement. Such correlations do not require the knowledge of the full density matrix. Our results show novel effects of the intrinsic bispinor structure and can be used as a basis for designing experiments to probe chiral oscillations via spin correlation measurements. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

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10 pages, 245 KiB

Open AccessCommunication

On Gauge Invariance of the Bosonic Measure in Chiral Gauge Theories

by Gabriel de Lima e Silva, Thalis José Girardi and Sebastião Alves Dias

Universe 2021, 7(8), 283; https://doi.org/10.3390/universe7080283 - 4 Aug 2021

Cited by 2 | Viewed by 1394

Abstract

Gauge invariance of the measure associated with the gauge field is usually taken for granted, in a general gauge theory. We furnish a proof of this invariance, within Fujikawa’s approach. To stress the importance of this fact, we briefly review gauge anomaly cancellation [...] Read more.

Gauge invariance of the measure associated with the gauge field is usually taken for granted, in a general gauge theory. We furnish a proof of this invariance, within Fujikawa’s approach. To stress the importance of this fact, we briefly review gauge anomaly cancellation as a consequence of gauge invariance of the bosonic measure and compare this cancellation to usual results from algebraic renormalization, showing that there are no potential inconsistencies. Then, using a path integral argument, we show that a possible Jacobian for the gauge transformation has to be the identity operator, in the physical Hilbert space. We extend the argument to the complete Hilbert space by a direct calculation. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

14 pages, 273 KiB

Open AccessArticle

Relativistic Combination of Non-Collinear 3-Velocities Using Quaternions

by Thomas Berry and Matt Visser

Universe 2020, 6(12), 237; https://doi.org/10.3390/universe6120237 - 11 Dec 2020

Cited by 4 | Viewed by 1757

Abstract

Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of [...] Read more.

Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by

v = \frac{2 w}{1 + w^{2}}

, so that

w = \frac{v}{1 + \sqrt{1 - v^{2}}} = \frac{v}{2} + O (v^{3})

, and promote them to quaternions using

w = w \hat{n}

, where

\hat{n}

is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by

w_{1 \oplus 2} \equiv w_{1} \oplus w_{2} \equiv {(1 - w_{1} w_{2})}^{- 1} (w_{1} + w_{2})

, and then show that this is also equivalent to

w_{1 \oplus 2} = (w_{1} + w_{2}) {(1 - w_{2} w_{1})}^{- 1}

. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of

w_{1}

with

w_{2}

. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle

Ω

and the direction of the combined velocities:

e^{Ω} = e^{Ω \hat{Ω}} = {(1 - w_{1} w_{2})}^{- 1} (1 - w_{2} w_{1})

, and

{\hat{w}}_{1 \oplus 2} = e^{Ω / 2} \frac{w_{1} + w_{2}}{| w_{1} + w_{2} |}

. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the non-commutative algebra of quaternions. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

27 pages, 379 KiB

Open AccessArticle

Making a Quantum Universe: Symmetry and Gravity

by Houri Ziaeepour

Universe 2020, 6(11), 194; https://doi.org/10.3390/universe6110194 - 23 Oct 2020

Cited by 8 | Viewed by 2497

Abstract

So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the preliminary results [...] Read more.

So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the preliminary results for a model of quantum universe, in which gravity is fundamentally and by construction quantic. The model is based on three well motivated assumptions with compelling observational and theoretical evidence: quantum mechanics is valid at all scales; quantum systems are described by their symmetries; universe has infinite independent degrees of freedom. The last assumption means that the Hilbert space of the Universe has

S U (N \to \infty) ≅ area preserving Diff . (S_{2})

symmetry, which is parameterized by two angular variables. We show that, in the absence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum fluctuations break the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference—observer—and another as clock, two more continuous parameters arise, which can be interpreted as distance and time. We identify the classical spacetime with parameter space of the Hilbert space of the Universe. Therefore, its quantization is meaningless. In this view, the Einstein equation presents the projection of quantum dynamics in the Hilbert space into its parameter space. Finite dimensional symmetries of elementary particles emerge as a consequence of symmetry breaking when the Universe is divided to subsystems/particles, without having any implication for the infinite dimensional symmetry and its associated interaction-percived as gravity. This explains why gravity is a universal force. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

Review

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19 pages, 353 KiB

Open AccessReview

Strongly Continuous Representations in the Hilbert Space: A Far-Reaching Concept

by Julio Marny Hoff da Silva and Gabriel Marcondes Caires da Rocha

Universe 2021, 7(8), 285; https://doi.org/10.3390/universe7080285 - 4 Aug 2021

Cited by 1 | Viewed by 1560

Abstract

We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations [...] Read more.

We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations of Wigner’s work. We conclude the paper by reviewing the connection between continuity, the possibility of defining certain local groups, and their relation to projective representations. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

31 pages, 463 KiB

Open AccessReview

Fundamental Symmetries and Spacetime Geometries in Gauge Theories of Gravity—Prospects for Unified Field Theories

by Francisco Cabral, Francisco S. N. Lobo and Diego Rubiera-Garcia

Universe 2020, 6(12), 238; https://doi.org/10.3390/universe6120238 - 11 Dec 2020

Cited by 29 | Viewed by 3036

Abstract

Gravity can be formulated as a gauge theory by combining symmetry principles and geometrical methods in a consistent mathematical framework. The gauge approach to gravity leads directly to non-Euclidean, post-Riemannian spacetime geometries, providing the adequate formalism for metric-affine theories of gravity with curvature, [...] Read more.

Gravity can be formulated as a gauge theory by combining symmetry principles and geometrical methods in a consistent mathematical framework. The gauge approach to gravity leads directly to non-Euclidean, post-Riemannian spacetime geometries, providing the adequate formalism for metric-affine theories of gravity with curvature, torsion and non-metricity. In this paper, we analyze the structure of gauge theories of gravity and consider the relation between fundamental geometrical objects and symmetry principles as well as different spacetime paradigms. Special attention is given to Poincaré gauge theories of gravity, their field equations and Noether conserved currents, which are the sources of gravity. We then discuss several topics of the gauge approach to gravitational phenomena, namely, quadratic Poincaré gauge models, the Einstein-Cartan-Sciama-Kibble theory, the teleparallel equivalent of general relativity, quadratic metric-affine Lagrangians, non-Lorentzian connections, and the breaking of Lorentz invariance in the presence of non-metricity. We also highlight the probing of post-Riemannian geometries with test matter. Finally, we briefly discuss some perspectives regarding the role of both geometrical methods and symmetry principles towards unified field theories and a new spacetime paradigm, motivated from the gauge approach to gravity. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

34 pages, 518 KiB

Open AccessReview

Dirac, Majorana, Weyl in 4D

by Loriano Bonora, Roberto Soldati and Stav Zalel

Universe 2020, 6(8), 111; https://doi.org/10.3390/universe6080111 - 4 Aug 2020

Cited by 8 | Viewed by 2824

Abstract

This is a review of some elementary properties of Dirac, Weyl and Majorana spinors in 4D. We focus in particular on the differences between massless Dirac and Majorana fermions, on one side, and Weyl fermions, on the other. We review in detail the [...] Read more.

This is a review of some elementary properties of Dirac, Weyl and Majorana spinors in 4D. We focus in particular on the differences between massless Dirac and Majorana fermions, on one side, and Weyl fermions, on the other. We review in detail the definition of their effective actions, when coupled to (vector and axial) gauge fields, and revisit the corresponding anomalies using the Feynman diagram method with different regularisations. Among various well known results we stress in particular the regularisation independence in perturbative approaches, while not all the regularisations fit the non-perturbative ones. As for anomalies, we highlight in particular one perhaps not so well known feature: the rigid relation between chiral and trace anomalies. Full article

(This article belongs to the Special Issue 80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group")

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