Version 1
: Received: 19 September 2024 / Approved: 20 September 2024 / Online: 20 September 2024 (09:48:01 CEST)
How to cite:
Land, M. Covariant Representation of Spin and Entanglement --- A Review and Reformulation. Preprints2024, 2024091608. https://doi.org/10.20944/preprints202409.1608.v1
Land, M. Covariant Representation of Spin and Entanglement --- A Review and Reformulation. Preprints 2024, 2024091608. https://doi.org/10.20944/preprints202409.1608.v1
Land, M. Covariant Representation of Spin and Entanglement --- A Review and Reformulation. Preprints2024, 2024091608. https://doi.org/10.20944/preprints202409.1608.v1
APA Style
Land, M. (2024). Covariant Representation of Spin and Entanglement --- A Review and Reformulation. Preprints. https://doi.org/10.20944/preprints202409.1608.v1
Chicago/Turabian Style
Land, M. 2024 "Covariant Representation of Spin and Entanglement --- A Review and Reformulation" Preprints. https://doi.org/10.20944/preprints202409.1608.v1
Abstract
A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et.\ al. This approach modifies Wigner's method by introducing an arbitrary timelike unit vector $n^\mu$ on which one induces a representation of $SL(2,C)$, rather than induce on the spacetime momentum $p^\mu$. Generalizing this approach, we construct relativistic spin states on an extended phase space $\{(x^\mu,p^\mu),(\zeta^\mu,\pi^\mu)\}$, inducing a representation on the momentum $\pi^\mu$, thus providing a novel dynamical interpretation of the timelike unit vector $n^\mu = \pi^\mu / M$. Studying the unitary representations of the Poincar\'e group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.
Keywords
quantum entanglement; relativistic quantum theory; problem of time
Subject
Physical Sciences, Particle and Field Physics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.