Limits on Relativistic Quantum Measurement

TL;DR

This paper explores the constraints causality imposes on quantum measurements in relativistic quantum field theory, comparing operator, algebraic, and path integral approaches, and proposing a measurement model that respects causality.

Contribution

It introduces a measurement theory in algebraic quantum field theory incorporating the apparatus as a quantum field, ensuring causally consistent measurements in a relativistic setting.

Findings

Causality conditions are essential for measurable operators in relativistic quantum theory.

The apparatus modeled as a quantum field leads to causally well-behaved measurement formalism.

Path integral approach faces similar causality issues as operator-based methods, requiring post-measurement conditions.

Abstract

Requiring causality on measurements in quantum field theory seems to impose strong conditions on a self-adjoint operator to be really measurable. This may seem limiting and artificial in the operator language of algebraic quantum field theory (AQFT), but is essential for a truly relativistic theory. Recent publications attempt to deal with this issue by including the apparatus into the formalism, connecting AQFT with measurement theory, but other options have been suggested. In this essay, I discuss the causality conditions on self-adjoint operators both in the language of AQFT and in the language of quantum information theory. I then present measurement theory in AQFT, modelling the apparatus as a quantum field with coupling to the measured system restricted to a region of spacetime. I highlight how this approach leads to a causally well behaved theory. Finally, I attempt to formulate the causality conditions on measurements in the Feynman path integral approach, using the concept of decoherent histories. I claim that the path integral approach has problems with causality similar to the operator based approaches and that even here causality is an a posteriori condition.