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Spectral decomposition of field operators and causal measurement in quantum field theory

Robert Oeckl

TL;DR

The paper develops a POVM-based framework for measuring bosonic quantum field observables by expressing the spectral content of linear field operators as a limit of 1-parameter POVMs, preserving locality. It defines two quantum-operation schemes for field measurements (discrete and continuous outcomes), proves locality and causal-transparency for the continuous-outcome scheme, and provides a rigorous functional-calculus for observables built from linear fields. The approach reconciles relativistic causality with measurement in QFT by avoiding generic projection-based signaling (Sorkin's objection) and suggests spacetime-generalizable probes via path integrals. Together, these results advance a relativistically consistent measurement theory for quantum fields and invite covariant, spacetime formulations and extensions to nonlinear functions of fields.

Abstract

We construct the spectral decomposition of field operators in bosonic quantum field theory as a limit of a strongly continuous family of positive-operator-valued measure decompositions. The latter arise from integrals over families of bounded positive operators. Crucially, these operators have the same locality properties as the underlying field operators. We use the decompositions to construct families of quantum operations implementing measurements of the field observables. Again, the quantum operations have the same locality properties as the field operators. What is more, we show that these quantum operations do not lead to superluminal signaling and are possible measurements on quantum fields in the sense of Sorkin.

Spectral decomposition of field operators and causal measurement in quantum field theory

TL;DR

The paper develops a POVM-based framework for measuring bosonic quantum field observables by expressing the spectral content of linear field operators as a limit of 1-parameter POVMs, preserving locality. It defines two quantum-operation schemes for field measurements (discrete and continuous outcomes), proves locality and causal-transparency for the continuous-outcome scheme, and provides a rigorous functional-calculus for observables built from linear fields. The approach reconciles relativistic causality with measurement in QFT by avoiding generic projection-based signaling (Sorkin's objection) and suggests spacetime-generalizable probes via path integrals. Together, these results advance a relativistically consistent measurement theory for quantum fields and invite covariant, spacetime formulations and extensions to nonlinear functions of fields.

Abstract

We construct the spectral decomposition of field operators in bosonic quantum field theory as a limit of a strongly continuous family of positive-operator-valued measure decompositions. The latter arise from integrals over families of bounded positive operators. Crucially, these operators have the same locality properties as the underlying field operators. We use the decompositions to construct families of quantum operations implementing measurements of the field observables. Again, the quantum operations have the same locality properties as the field operators. What is more, we show that these quantum operations do not lead to superluminal signaling and are possible measurements on quantum fields in the sense of Sorkin.
Paper Structure (13 sections, 29 theorems, 86 equations, 2 figures)

This paper contains 13 sections, 29 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. POVM and spectral decompositions
  4. Pointwise measure
  5. Functional calculus
  6. Quantum operations
  7. Discrete outcomes
  8. Continuous outcomes
  9. Locality and Causality
  10. Locality
  11. Causal transparency
  12. Discussion and Outlook
  13. Appendix

Key Result

Lemma 3.1

Figures (2)

  • Figure 1: Setup with two measurements $N$ and $M$. $N$ is non-selective, while $M$ is selective. By locality, the outcome of $M$ should not depend on whether or not $N$ is performed.
  • Figure 2: Setup with three measurements $N$, $I$ and $M$. $N$ and $I$ are non-selective, while $M$ is selective. Given locality, causal transparency of $I$ means that the outcome of $M$ should not depend on whether or not $N$ is performed.

Theorems & Definitions (58)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 48 more