Relativistic Covariance and Nonlinear Quantum Mechanics: Tomonaga-Schwinger Analysis

TL;DR

This work assesses whether deterministic nonlinear quantum field theories can be relativistically covariant under the Tomonaga–Schwinger (TS) framework. It derives a generalized, state-dependent integrability condition that includes Fréchet-derivative contributions from nonlinear generators and analyzes how various nonlinear forms—local Weinberg-type, nonlocal mean-field, and retarded nonlinearity—impact foliation independence and microcausality. The results show that generic state-dependent evolutions violate the TS integrability condition and can generate instantaneous entanglement between spacelike regions, signaling a breakdown of relativistic locality. These findings challenge the compatibility of deterministic nonlinear quantum dynamics with relativistic covariance and have implications for interpretations invoking wavefunction collapse and nonlocal evolution.

Abstract

We use the Tomonaga-Schwinger (TS) formulation of quantum field theory to determine when state-dependent additions to the local Hamiltonian density (i.e., modifications to linear Schrodinger evolution) violate relativistic covariance. We derive new operator integrability conditions required for foliation independence, including the Frechet derivative terms that arise from state-dependence. Nonlinear modifications of quantum mechanics affect operator relations at spacelike separation, leading to violation of the integrability conditions.