No-Signalling Fixes the Hilbert-Space Inner Product

TL;DR

This paper demonstrates that the standard Hilbert-space inner product in quantum mechanics is uniquely determined by the no-signalling principle, as any modification leads to superluminal signalling, violating causality.

Contribution

It proves that the standard inner product cannot be altered without violating the no-signalling principle, establishing its uniqueness in quantum theory.

Findings

Any nontrivial inner product modification causes superluminal signalling.

The standard inner product is uniquely enforced by no-signalling.

Modifying the inner product contradicts relativistic causality.

Abstract

We investigate whether the inner product structure of quantum mechanics can be modified without violating fundamental physical principles. We consider a generalized inner product defined by a positive operator and assume local unitary dynamics, existence of entangled states and the no-signalling principle. We show that any nontrivial choice of inner product different from standard one inevitably leads to superluminal signalling, in contradiction with relativistic causality. Therefore, the standard Hilbert-space inner product is uniquely enforced by no-signalling.