A Spacetime Symmetry Approach to Relativistic Quantum Multi-Particle Entanglement

Abstract

A Lorentz transformation group SO(m,n) of signature (m,n), m,n∈N, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature (m,n). Accordingly, the Lorentz group SO(1,3) is the common Lorentz transformation group from which special relativity theory stems. It is widely acknowledged that special relativity and quantum theories are at odds. In particular, it is known that entangled particles involve Lorentz symmetry violation. We, therefore, review studies that led to the discovery that the Lorentz group SO(m,n) forms the symmetry group by which a multi-particle system of m entangled n-dimensional particles can be understood in an extended sense of relativistic settings. Consequently, we enrich special relativity by incorporating the Lorentz transformation groups of signature (m,3) for all m≥2. The resulting enriched special relativity provides the common symmetry group SO(1,3) of the (1+3)-dimensional spacetime of individual particles, along with the symmetry group SO(m,3) of the (m+3)-dimensional spacetime of multi-particle systems of m entangled 3-dimensional particles, for all m≥2. A unified parametrization of the Lorentz groups SO(m,n) for all m,n∈N, shakes down the underlying matrix algebra into elegant and transparent results. The special case is when (m,n)=(1,3) is supported experimentally by special relativity. It is hoped that this review article will stimulate the search for experimental support when (m,n)=(m,3) for all m≥2.

Isomorphism of Binary Operations in Differential Geometry

“…Following definition (34) of the operation⊕, this operation is isomorphic to the operation ⊕ with the isomorphism ϕ.…”

“…In this section we consider a simple way to get gyrogroups via bijections. We use the following definitions [34]. Definition 6.…”

“…There appeared a theory that may be called gyrogeometry, based on nice algebraic properties of the aforementioned operations and the concepts of gyrogroups, gyrovector spaces, gyrotrigonometry, and gyrogeometric objects [21][22][23][24][25][26][27][28][29][30]. This theory has been extended to the space of matrices with indefinite scalar products, which is closely related to the phenomenon of entanglement in quantum physics [32][33][34], and to harmonic analysis [5][6][7].…”

Entanglement-enhanced test proposal for local Lorentz-symmetry violation via spinor atoms

“…Our test scheme may open up a feasible way to achieve high-precision estimation of the LSV parameter κ. Given the broad interest in testing LSV and quantum-enhanced metrology, our study not only may pave the way to test some theories, such as string theory [6], field theory [7][8][9]12], Einsteinaether theories [46] and quantum gravity [47], but also provides an alternative application of multiparticle entangled states [48]. Meanwhile, the energy shift resulted from LSV is similar to the quadratic Zeeman shifts in atomic clocks [49][50][51] and atomic magnetometry [52][53][54], thus our test proposal also can be used in practical quantum sensors.…”

Differential Geometry and Binary Operations

“…Gyrogroup actions are studied in [8]. An application of Einstein bi-gyrogroups to quantum multi-particle entanglement is presented in [9]. Several recent studies of gyrogroups and gyrovector spaces are presented in [10][11][12].…”