Contractions of the relativistic quantum LCT group and the emergence of spacetime symmetries

TL;DR

This paper explores how spacetime symmetries like de Sitter and Poincaré groups emerge from the contraction of the relativistic quantum LCT group, revealing a fundamental symplectic structure underlying quantum phase space.

Contribution

It explicitly demonstrates the contraction process of the LCT group’s Lie algebra to recover familiar spacetime symmetries from a more fundamental quantum phase space symmetry.

Findings

Contraction of the LCT Lie algebra yields the de Sitter algebra $rak{so}(1,4)$.

In the flat limit, the LCT algebra contracts to the Poincaré algebra $rak{iso}(1,3)$.

The work links quantum phase space symmetries to classical spacetime symmetries via group contraction.

Abstract

Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature $(N_+,N_-)$, the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group $Sp(2N_+,2N_-)$, which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature $(1,4)$, clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the In\"on\"u-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length $\ell$ and a maximum length $L$, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra $\mathfrak{so}(1,4)$ and, in the flat-curvature limit, to the Poincar\'e algebra $\mathfrak{iso}(1,3)$ of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.