Syntactic Structure, Quantum Weights

TL;DR

This paper explains the universal emergence of local actions and Euclidean weights in classical, statistical, and quantum theories through minimal syntactic constraints on histories, linking descriptive complexity to physical action forms.

Contribution

It introduces a structural explanation based on minimal, prefix-free code constraints that determine the form of physical actions and weights in various theories.

Findings

Universal form of action fixed by descriptional structure

Exponential redundancy leads to Euclidean weights

Euclidean measure reconstructs Lorentzian amplitudes under reflection positivity

Abstract

Why do local actions and exponential Euclidean weights arise so universally in classical, statistical, and quantum theories? We offer a structural explanation from minimal constraints on finite descriptions of admissible histories. Assume that histories admit finite, self-delimiting (prefix-free) generative codes that can be decoded sequentially in a single forward pass. These purely syntactic requirements define a minimal descriptive cost, interpretable as a smoothed minimal program length, that is additive over local segments. First, any continuous local additive cost whose stationary sector coincides with the empirically identified classical variational sector is forced into a unique Euler--Lagrange equivalence class. Hence the universal form of an action is fixed by descriptional structure alone, while the specific microscopic Lagrangian and couplings remain system-dependent semantic input. Second, independently of microscopic stochasticity, finite prefix-free languages exhibit exponential redundancy: many distinct programs encode the same coarse history, and this redundancy induces a universal exponential multiplicity weight on histories. Requiring this weight to be real and bounded below selects a real Euclidean representative for stable local bosonic systems, yielding the standard Euclidean path-integral form. When Osterwalder--Schrader reflection positivity holds, the Euclidean measure reconstructs a unitary Lorentzian amplitude.