Einstein causality of quantum measurements in the Tomonaga-Schwinger picture

TL;DR

The paper develops a relativistic measurement framework based on the Tomonaga–Schwinger picture, generalising Lüders' rule to globally hyperbolic spacetimes with foliations by acausal Cauchy surfaces. It shows selective measurements obey a state-independent anyonic commutation relation over spacelike precompact regions, which for POVMs reduces to bosonic commutativity, ensuring no-signalling in the instantaneous limit. It also analyzes Sorkin's impossible measurements, arguing that measurement irreversibility prevents contradictions, and discusses extensions beyond the interaction picture where Hilbert spaces on different slices may not be unitarily equivalent. The results reinforce the consistency of a unitary, foliation-independent Tomonaga–Schwinger dynamics in relativistic quantum theory and point toward broader applicability in quantum field theory beyond the interaction picture.

Abstract

We investigate a generalisation to Lüders' rule à la Aharonov-Albert in those globally hyperbolic spacetimes which allow unitarily equivalent Hilbert spaces to be defined along Cauchy hypersurfaces, thus relying on the existence of an interaction picture à la Tomonaga-Schwinger. We show that under this rule and under the additional assumptions of the integrability and unitarity of the Tomonaga-Schwinger dynamics and the foliation-independence of rays on acausal Cauchy hypersurfaces, selective quantum measurements satisfy a state-independent anyonic commutation relation over spacelike-separated precompact regions. We highlight that this propagates to positive operator-valued measures, where the commutation is necessarily bosonic. In the instantaneous-measurement idealisation, this implies quantum no-signalling for non-selective measurements. We then examine Sorkin's impossible measurements and show that immediate contradictions can be averted as long as collapse-inducing measurements are irreversible. These results reaffirm the consistency of the Tomonaga-Schwinger picture of relativistic quantum theory, for which unitarity, integrability and foliation-independence of the states exclude superluminal signalling despite the ``instantaneity" of a side-cone measurement collapse rule. We finish by discussing the possibility of extending such results beyond the interaction picture.

Figures (4)

• Figure 1: Foliation of a globally hyperbolic spacetime into two families of Cauchy hypersurfaces $\mathcal{S}_1$ and $\mathcal{S}_2$, both of which contain $\Sigma_i$ and $\Sigma_f$ in the strict causal past and strict causal future, respectively, of spacelike-separated precompact regions $\mathcal{U}$ and $\mathcal{V}$. $\Sigma_1 \in \mathcal{S}_1$ (blue dashed line) lies in the strict causal future of $\mathcal{U}$ and the strict causal past of $\mathcal{V}$; $\Sigma_2 \in \mathcal{S}_2$ (red dotted line) lies in the strict causal past of $\mathcal{U}$ and the strict causal future of $\mathcal{V}$. In the foliation $S_1$, $\ket{\psi[\Sigma_f]}_{12} = c_{12} U_4\mathsf{M}_{2,n}(\mathcal{V})U_3 U_2 \mathsf{M}_{1,m}(\mathcal{U}) U_1 \ket{\psi[\Sigma_i]}$; in the foliation $S_2$, $\ket{\psi[\Sigma_f]}_{21} = c_{21}\tilde{U}_4\mathsf{M}_{1,m}(\mathcal{U})\tilde{U}_3 \tilde{U}_2 \mathsf{M}_{2,n}(\mathcal{V}) \tilde{U}_1 \ket{\psi[\Sigma_i]}$. These are one and the same state as the foliation is arbitrary, i.e. they belong to the same ray.
• Figure 2: Setup of Sorkin's impossible measurements. We consider three foliations $\mathcal{S}_1$, $\mathcal{S}_2$ and $\mathcal{S}_3$ of the globally hyperbolic spacetime $(\mathcal{M},g)$, all of which contain the acausal Cauchy hypersurfaces $\Sigma_\alpha$ and $\Sigma_\beta$ (black). $\mathcal{S}_1$ and $\mathcal{S}_2$ both contain $\Sigma_f$ and $\Sigma_i$ (green). Foliation $\mathcal{S}_1$ contains $\Sigma_1^{(1)}$, $\Sigma_2^{(1)}$ and $\Sigma_3^{(1)}$ (red dashed lines). Foliation $\mathcal{S}_2$ contains $\Sigma_1^{(2)}$, $\Sigma_2^{(2)}$ and $\Sigma_3^{(2)}$ (blue dotted lines). Foliation $\mathcal{S}_3$ contains $\Sigma_1^{(3)}$, $\Sigma_2^{(3)}$, $\Sigma_3^{(3)}$, $\Sigma_4^{(3)}$, $\Sigma_5^{(3)}$, $\Sigma_6^{(3)}$ (purple curved lines). In $\mathcal{S}_1$, the state first undergoes the measurement $\{\mathsf{M}_{2,n}(\mathcal{V})\}$ before the measurement $\{\mathsf{M}_{1,m}(\mathcal{U})\}$, while in $\mathcal{S}_2$ the state first undergoes the measurement $\{\mathsf{M}_{1,m}(\mathcal{U})\}$ before the measurement $\{\mathsf{M}_{2,n}(\mathcal{V})\}$. In $\mathcal{S}_3$, the state first undergoes the measurement $\{\mathsf{M}_{1,m}(\mathcal{U})\}$, then $\{\mathsf{M}_{3,o}(\mathcal{W})\}$, then $\{\mathsf{M}_{2,n}(\mathcal{V})\}$. Note that $\Sigma_f \cap (J^-(B^+(\mathcal{W})) \smallsetminus B^+(\mathcal{W})) \neq \varnothing$, i.e. the collapse induced by the measurement of $\{\mathsf{M}_{3,o}(\mathcal{W})\}$ has not happened on $\Sigma_f$ in either $\mathcal{S}_1$ or $\mathcal{S}_2$.
• Figure 3: Foliation of a globally hyperbolic spacetime into two families of Cauchy hypersurfaces $\mathcal{S}_1$ and $\mathcal{S}_2$, both of which contain $\Sigma_i$ and $\Sigma_f$ in the strict causal past and strict causal future, respectively, of spacelike-separated precompact regions $\mathcal{U}$ and $\mathcal{V}$. In blue dashed lines ($\mathcal{S}_1$), $B^-(\mathcal{U}) \subset \Sigma_1^{12}$ lies in the causal past of $\mathcal{V}$, $B^+(\mathcal{U}) \cup B^-(\mathcal{V}) \subset \Sigma_2^{12}$ and $B^+(\mathcal{V}) \subset \Sigma_3^{12}$ lies in the causal future of $\mathcal{U}$; in red dotted lines ($\mathcal{S}_2$), $B^-(\mathcal{V}) \subset \Sigma_1^{21}$ lies in the causal past of $\mathcal{U}$, $B^+(\mathcal{V}) \cup B^-(\mathcal{U}) \subset \Sigma_2^{21}$ and $B^+(\mathcal{U}) \subset \Sigma_3^{21}$ lies in the causal future of $\mathcal{V}$. The state on $\Sigma_f$ is independent of the choice of foliation, i.e. the final states in both foliations belong to the same ray.
• Figure 4: Construction for intertwining an acausal Cauchy hypersurface between two precompact spacelike-separated regions. Here, we have that $\mathcal{U} = \mathcal{U}_1 \cup \mathcal{U}_2$ (blue filled) where $\mathcal{U}_2$ is a point, and $\mathcal{V} \mathrel{\hbox{$\perp$}\mkern2mu{\perp}} \mathcal{U}$ where $\mathcal{V}$ (red filled) is a causal diamond. Although a causal diamond has an empty domain of dependence, the neighbourhood $K_\mathcal{V}$ (dashed red) has a nontrivial past domain of dependence; likewise, although a point has an empty domain of dependence, the neighbourhood of $\mathcal{U}_2$ has a nontrivial future domain of dependence. Cauchy hypersurfaces $s_1$ (green) and $s_2$ (orange) intersect $D^+(K_{\mathcal{U}_1}) \cup D^+(K_{\mathcal{U}_2})$ and $D^-(K_\mathcal{V})$ at $\sigma_1$ (turquoise) and $\sigma_2$ (brown), respectively. The compact acausal surface $H = \sigma_1 \cup \sigma_2$ can be used to create a Cauchy spacelike hypersurface $\Sigma_1$ (purple) intertwining $\mathcal{U}$ and $\mathcal{V}$.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Proposition 1
• proof
• Definition 3: State update rule in the interaction picture
• Theorem 1
• proof
• Theorem 2: Brooke_2002
• Corollary 1
• Theorem 3