Quantum Reference Frames in Arbitrary Charge Sectors: Accessibility of Global Properties from Internal Perspectives

TL;DR

This paper extends quantum reference frame (QRF) formalism to arbitrary fixed total momentum sectors in a translation-invariant setting, moving beyond the zero-momentum constraint. By generalizing both the perspectival and perspective-neutral approaches, the authors derive a $P$-dependent phase in QRF transformations and analyze how the global charge $P$ can be inferred from internal perspectives under three levels of observational access, including potential inter-frame communication. They show that certain observables remain invariant only in restricted conditions and that full inference of $P$ requires relational tomography plus collaboration, while the extra-particle approach offers direct access to $P$ without communication. The work clarifies how different QRF frameworks encode and reveal global properties, discusses covariance of physical laws in the extended setting, and points to further generalizations, including field-theoretic extensions and superselection considerations. Overall, the results illuminate the interplay between local and global descriptions in quantum theory and provide a structured roadmap for exploring QRFs in more complex symmetry settings.

Abstract

A fundamental question in the field of quantum reference frames concerns what global properties of a system can be determined by observers operating entirely from within that system. We investigate this question by extending both the perspectival and perspective-neutral approaches beyond the commonly studied zero total momentum case to arbitrary fixed charge sectors. When the entire system, including the reference frames, moves at a fixed total momentum $P$ relative to an external frame, this global charge becomes encoded in the quantum states and transformations between reference frames. Our extension leads to modified relative states and observables with QRF transformations that induce an additional $P$-dependent phase, treating all charge sectors as equally fundamental. By granting the internal observers successively more access and resources, we identify under which conditions they can infer the total momentum. These results clarify the relationship between major QRF approaches -- perspectival, perspective-neutral, operational, and extra-particle -- showing how their differing conclusions stem from different assumptions about which observables are deemed accessible from within. Our findings cast light on the relation between local and global perspectives and raise fundamental questions about scenarios where no global perspective exists, contributing to a deeper understanding of relationality and the role of perspectives in quantum theory.

Figures (2)

• Figure 1: Extended perspective-neutral construction. The construction follows the usual perspective-neutral approach by starting from the kinematical Hilbert space $\mathcal{H}_{{\mathrm{kin}}}$, projecting on the physical Hilbert space $\mathcal{H}_{{\mathrm{phys}}}$, and reducing into the perspective of $A$ and $C$, respectively, using the reduction maps $\mathcal{R}^{(A)}_\mathbf{S}$ and $\mathcal{R}^{(C)}_\mathbf{S}$. Our modification lies in the first step (highlighted in purple) through a different choice of projector $\Pi_{{\mathrm{phys}}}=\delta(\hat{P}-P)$ for non-zero total momentum $P$.
• Figure 2: Illustration of the game run by an external referee Eve. At the beginning of each run, she prepares the reference frames $A$ and $C$, and an additional system $B$ in a configuration with total momentum $\hat{P}=\hat{p}_A+\hat{p}_B+\hat{p}_C= P$, depicted as the (orange) arrow from the centre of mass. Then, either Alice or Charlie perform a measurement on the remaining systems on the tracks. Over time, they collect data, based on which they try to infer the total momentum $P$. Whether this strategy succeeds depends on the specific observables and resources available to them, characterised by three levels of increased accessibility in the main text.

Theorems & Definitions (8)

• Theorem 1
• Proposition 1
• Example 1
• Example 2
• Theorem 1
• proof
• Proposition 1
• proof