Symmetry and Geometry in Physics II

Dear Colleagues,

Nature organizes itself using the language of symmetry. In particular, the symmetry group of special relativity theory is the Lorentz transformation group SO (1,3). A physical system has Lorentz symmetry if the relevant laws of physics are invariant under Lorentz transformations. Lorentz symmetry is one of the cornerstones of modern physics. However, entangled particles involve Lorentz symmetry violation. Understanding entanglement in relativistic settings has been a key question in quantum mechanics. Remarkably, a plausible candidate for the symmetry group of the spacetime of a system of m n-dimensional entangled particles is the Lorentz group SO (m, n) of signature (m, n), for any m, n ∈ ℕ.

Lorentz groups involve relativistically admissible velocities governed by hyperbolic geometry and controlled by Einstein velocity addition. The resulting Einstein addition is a binary operation which is neither commutative nor associative. As such, it is a non-group gyrogroup operation that gives rise to gyrocommutative gyrogroups and gyrovector spaces. The latter, in turn, form the algebraic setting for hyperbolic geometry, just as vector spaces form the algebraic setting for Euclidean geometry.

Papers that study any of the following topics are welcome:

1. Differential or hyperbolic geometry associated with Einstein addition;
2. Einstein and Einstein-related gyrogroups and gyrovector spaces and their hyperbolic geometry;
3. Quantum entanglement that involves Lorentz violation; and
4. Physical applications of any Lorentz group SO (m, n) of signature (m, n), m, n > 1.

Prof. Dr. Abraham A. Ungar
Guest Editor

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