The classical limit of a physical theory and the dimensionality of space

TL;DR

This paper explores the relationship between the state space of elementary systems and the physical space in general probabilistic theories, showing that quantum mechanics uniquely embeds in three-dimensional space under pairwise interactions.

Contribution

It demonstrates that only quantum mechanics with pairwise interactions can embed both the elementary system and classical fields in three-dimensional space, highlighting the uniqueness of quantum theory.

Findings

Quantum mechanics uniquely embeds in 3D space with pairwise interactions.

Higher-dimensional embeddings are impossible with only pairwise interactions.

Multi-particle interactions may allow higher-dimensional embeddings.

Abstract

In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the "elementary systems" and the physical space in which "laboratory devices" are embedded. Each of those spaces has its own dimension- the minimal number of real parameters (coordinates) needed to specify the state of system or a point within the physical space. Within an operational framework to a physical theory, the two dimensions coincide in a natural way under the following "closeness" requirement: the dynamics of a single elementary system can be generated by the invariant interaction between the system and the "macroscopic transformation device" that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanics fulfils this requirement since an arbitrary unitary transformation of an elementary system (spin-1/2 or qubit) can be generated by the pairwise invariant interaction between the spin and the constituents of a large coherent state ("classical magnetic field"). Both the spin state space and the "classical field" are then embedded in the Euclidean three-dimensional space. Can we have a general probabilistic theory, other than quantum theory, in which the elementary system ("generalized spin") and the "classical fields" generating its dynamics are embedded in a higher-dimensional physical space? We show that as long as the interaction is pairwise, this is impossible, and quantum mechanics and the three-dimensional space remain the only solution. However, having multi-particle interactions and a generalized notion of "classical field" may open up such a possibility.