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Causal measurement in quantum field theory: spacetime

Robert Oeckl

TL;DR

This work develops a spacetime-compositional framework for regularized measurements of spacetime observables in bosonic QFT, ensuring relativistic causality and causal transparency through spacetime probes within the local positive formalism. It introduces a regularized spectral calculus via Gaussian POVMs and a Schwinger-Keldysh-based operational quantization to describe measurements that extend in time and space, while proving locality, compositionality, and the causal structure of measurement outcomes. A central result is that measurements back-react on themselves and induce correlations in the causal future, with explicit formulas for multi-observable probes and their propagator structure, including Hadamard, retarded, and advanced components. The framework clarifies how to measure spacetime observables, addresses the limitations of removing regularization, and outlines extensions to nonlinear observables and practical connections to energy-momentum tensor measurements and detector models, offering a path toward a relativistically consistent measurement theory in QFT.

Abstract

We provide a framework and explicit construction for the regularized measurement of a large class of spacetime-localized observables in bosonic quantum field theory. The measurements fully satisfy relativistic causality and causal transparency, i.e., avoid unphysical superluminal signaling. We show explicitly how the measurement of time-extended observables back-reacts on itself and induces correlations between other measurements in its causal future. Our framework is fully compositional in spacetime and extends previous results on the measurement of instantaneous observables.

Causal measurement in quantum field theory: spacetime

TL;DR

This work develops a spacetime-compositional framework for regularized measurements of spacetime observables in bosonic QFT, ensuring relativistic causality and causal transparency through spacetime probes within the local positive formalism. It introduces a regularized spectral calculus via Gaussian POVMs and a Schwinger-Keldysh-based operational quantization to describe measurements that extend in time and space, while proving locality, compositionality, and the causal structure of measurement outcomes. A central result is that measurements back-react on themselves and induce correlations in the causal future, with explicit formulas for multi-observable probes and their propagator structure, including Hadamard, retarded, and advanced components. The framework clarifies how to measure spacetime observables, addresses the limitations of removing regularization, and outlines extensions to nonlinear observables and practical connections to energy-momentum tensor measurements and detector models, offering a path toward a relativistically consistent measurement theory in QFT.

Abstract

We provide a framework and explicit construction for the regularized measurement of a large class of spacetime-localized observables in bosonic quantum field theory. The measurements fully satisfy relativistic causality and causal transparency, i.e., avoid unphysical superluminal signaling. We show explicitly how the measurement of time-extended observables back-reacts on itself and induces correlations between other measurements in its causal future. Our framework is fully compositional in spacetime and extends previous results on the measurement of instantaneous observables.

Paper Structure

This paper contains 23 sections, 16 theorems, 122 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Locality and causality of measurement
  3. Causality in non-relativistic quantum mechanics
  4. Relativistic causality in QFT
  5. Causal transparency in QFT
  6. Sorkin's result
  7. Observables, quantization, measurement
  8. Non-relativistic quantum mechanics
  9. QFT
  10. Spacetime
  11. Amplitudes, observables and correlation functions
  12. Slice observables
  13. Generalized spectral decomposition
  14. From operations to probes
  15. Locality and composition
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $S_1,S,S_2$ be subsets of the equal-time hypersurfaces at distinct times $t_1,t, t_2$ such that $S_2$ does not intersect the causal future of $S_1$. Let $N$ be a non-selective quantum operation localizable at $S_1$, $M$ a selective quantum operation localizable at $S_2$ and $A:L_t\to\mathbb{R}$ Here, the symbol $\diamond$ means that operations are ordered according to their temporal order and

Figures (6)

  • Figure 1: Setup with two measurements at different times that are localized in space. The non-selective measurement $N$ is localized in the spatial subset $S_1$ at $t_1$ and the selective measurement $M$ is localized in the spatial subset $S_2$ at $t_2$. $S_1$ does not intersect the causal past of $S_2$. The initial state is $\sigma$ at $t_1$ and the system is discarded after time $t_2$.
  • Figure 2: An additional non-selective measurement $I$ at the intermediate time $t$ is inserted into the setting of Figure \ref{['fig:stlocality']}.
  • Figure 3: If an operator at $t_1$ is localizable in the spatial subset $S_1$, then it is localizable at $t_2$ in $S_2$.
  • Figure 4: Spacetime extended non-selective measurement $N$ in region $R_1$ and selective measurement $M$ in region $R_2$, between initial and final spacelike hypersurfaces $\Sigma_1$ and $\Sigma_2$. A spacelike hypersurface $\Sigma$ separates the two. The initial state at $\Sigma_1$ is $\sigma$ and the system is discarded at $\Sigma_2$.
  • Figure 5: Same setup as in Figure \ref{['fig:relcausality']}, but with an additional non-selective intermediate measurement $I$ in the spacetime region $R$ to test causal transparency.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 3.1: Oe:spectral
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Theorem 4.3
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Proposition 5.4
  • Proposition 5.5
  • ...and 18 more