Causal State Updates in Real Scalar Quantum Field Theory

TL;DR

This paper establishes a causality-respecting framework for measurements in real scalar quantum field theory, showing that only smeared fields and the identity are compatible with causality, and provides methods to infer complex operator expectations causally.

Contribution

It formulates a necessary and sufficient causality condition for quantum operations in real scalar QFT, identifying the limitations on measurable observables and how to infer complex operators causally.

Findings

Only smeared fields and the identity are causal observables in real scalar QFT.

Complex operators like products of smeared fields lead to acausal state updates.

Expectation values of complex operators can be inferred from causal measurements of smeared fields.

Abstract

In relativistic Quantum Field Theory (QFT) ideal measurements of certain observables are physically impossible without violating causality. This prompts two questions: i) can a given observable be ideally measured in QFT, and ii) if not, in what sense can it be measured? Here we formulate a necessary and sufficient condition that any measurement, and more generally any state update (quantum operation), must satisfy to respect causality in real scalar QFT. We argue that for unitary `kicks' and operations involving 1-parameter families of Kraus operators, e.g. Gaussian measurements, the only causal observables are smeared fields and the identity -- the basic observables in real scalar QFT. We provide examples with more complicated operators such as products of smeared fields, and show that the associated state updates are acausal, and hence impossible. Despite this, one can still recover expectation values of such operators, and we show how to do this using only causal measurements of smeared fields.