Spacetime symmetries and the qubit Bloch ball: a physical derivation of finite dimensional quantum theory and the number of spatial dimensions

TL;DR

This paper derives finite-dimensional quantum theory and the three-dimensionality of space from spacetime symmetries and physical principles, highlighting a fundamental link between spacetime structure and quantum mechanics.

Contribution

It provides a physical derivation of quantum theory and the number of spatial dimensions based on spacetime symmetries and probabilistic frameworks, with no prior assumption of three spatial dimensions.

Findings

Number of spatial dimensions must be three for consistency with Poincaré invariance.

Reconstruction of the qubit Bloch ball within general probabilistic theories.

Establishment of a fundamental connection between spacetime symmetries and quantum theory.

Abstract

Quantum theory and relativity are the pillar theories on which our understanding of physics is based. Poincar\'e invariance is a fundamental physical principle stating that the experimental results must be the same in all inertial reference frames in Minkowski spacetime. It is a basic condition imposed on quantum theory in order to construct quantum field theories, hence, it plays a fundamental role in the standard model of particle physics too. As is well known, Minkowski spacetime follows from clear physical principles, like the relativity principle and the invariance of the speed of light. Here, we reproduce such a derivation, but leave the number of spatial dimensions $n$ as a free variable. Then, assuming that spacetime is Minkowski in $1+n$ dimensions and within the framework of general probabilistic theories, we reconstruct the qubit Bloch ball and finite dimensional quantum theory, and obtain that the number of spatial dimensions must be $n=3$, from Poincar\'e invariance and other physical postulates. Our results suggest a fundamental physical connection between spacetime and quantum theory.