Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity

TL;DR

This paper develops a conformally invariant quantum theory using 8-dimensional complex domains, employing a gauge-invariant nonlinear sigma-model approach, and explores implications like maximal acceleration and Born's Reciprocity.

Contribution

It introduces a novel conformal quantum framework based on nonlinear sigma models in Cartan domains, incorporating spinning particles and extending Born's Reciprocity Principle.

Findings

Explicit calculation of wave functions and kernels in conformal domains

Extension of Schwinger's Master Theorem for these models

Proposal of a maximal acceleration in conformal relativity

Abstract

We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master Theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.