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30 pages, 5862 KiB

Open AccessArticle

Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant

by Young S. Kim and Marilyn E. Noz

Symmetry 2020, 12(8), 1270; https://doi.org/10.3390/sym12081270 - 1 Aug 2020

Cited by 3 | Viewed by 3413

The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the

O (3, 2)

de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein’s Lorentz-covariant world. Full article

(This article belongs to the Section Physics)

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23 pages, 349 KiB

Open AccessArticle

The Standard Model of Particle Physics with Diracian Neutrino Sector

by Theodorus Maria Nieuwenhuizen

Symmetry 2019, 11(8), 994; https://doi.org/10.3390/sym11080994 - 3 Aug 2019

Cited by 4 | Viewed by 2502

Abstract

The minimally extended standard model of particle physics contains three right handed or sterile neutrinos, coupled to the active ones by a Dirac mass matrix and mutually by a Majorana mass matrix. In the pseudo-Dirac case, the Majorana terms are small and maximal mixing of active and sterile states occurs, which is generally excluded for solar neutrinos. In a “Diracian” limit, the physical masses become pairwise degenerate and the neutrinos attain a Dirac signature. Members of a pair do not oscillate mutually so that their mixing can be undone, and the standard neutrino model follows as a limit. While two Majorana phases become physical Dirac phases and three extra mass parameters occur, a better description of data is offered. Oscillation problems are worked out in vacuum and in matter. With lepton number –1 assigned to the sterile neutrinos, the model still violates lepton number conservation and allows very feeble neutrinoless double beta decay. It supports a sterile neutrino interpretation of Earth-traversing ultra high energy events detected by ANITA. Full article

26 pages, 1588 KiB

Open AccessArticle

Entangled Harmonic Oscillators and Space-Time Entanglement

by Sibel Başkal, Young S. Kim and Marilyn E. Noz

Symmetry 2016, 8(7), 55; https://doi.org/10.3390/sym8070055 - 28 Jun 2016

Cited by 11 | Viewed by 6925

Abstract

The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the

O (3, 2)

de Sitter group containing two

O (3, 1)

Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the

S p (4)

group, which serves as the basic language for two-mode squeezed states. Since the

S p (4)

symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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35 pages, 364 KiB

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Old Game, New Rules: Rethinking the Form of Physics

by Christian Baumgarten

Symmetry 2016, 8(5), 30; https://doi.org/10.3390/sym8050030 - 6 May 2016

Cited by 3 | Viewed by 4991

Abstract

We investigate the modeling capabilities of sets of coupled classical harmonic oscillators (CHO) in the form of a modeling game. The application of the simple but restrictive rules of the game lead to conditions for an isomorphism between Lie-algebras and real Clifford algebras. We show that the correlations between two coupled classical oscillators find their natural description in the Dirac algebra and allow to model aspects of special relativity, inertial motion, electromagnetism and quantum phenomena including spin in one go. The algebraic properties of Hamiltonian motion of low-dimensional systems can generally be related to certain types of interactions and hence to the dimensionality of emergent space-times. We describe the intrinsic connection between phase space volumes of a 2-dimensional oscillator and the Dirac algebra. In this version of a phase space interpretation of quantum mechanics the (components of the) spinor wavefunction in momentum space are abstract canonical coordinates, and the integrals over the squared wave function represents second moments in phase space. The wave function in ordinary space-time can be obtained via Fourier transformation. Within this modeling game, 3+1-dimensional space-time is interpreted as a structural property of electromagnetic interaction. A generalization selects a series of Clifford algebras of specific dimensions with similar properties, specifically also 10- and 26-dimensional real Clifford algebras. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

18 pages, 136 KiB

Open AccessArticle

Dirac Matrices and Feynman’s Rest of the Universe

by Young S. Kim and Marilyn E. Noz

Symmetry 2012, 4(4), 626-643; https://doi.org/10.3390/sym4040626 - 30 Oct 2012

Cited by 6 | Viewed by 6698

Abstract

There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set consists of ten generators of the Sp(4) group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of Sp(4) to that of SL(4, r) if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman’s rest of the universe makes this Sp(4)-to-SL(4, r) transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups SL(4, r) and Sp(4) are locally isomorphic to the Lorentz groups O(3, 3) and O(3, 2) respectively. This allows us to interpret Feynman’s rest of the universe in terms of space-time symmetry. Full article

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