Quantum reference frame transformations as symmetries and the paradox of the third particle

TL;DR

The paper rehabilitates quantum reference frames by treating QRF transformations as symmetry operations of carefully constructed finite Abelian $ ext{G}$-systems, thereby unifying structural and quantum-information perspectives. It introduces invariant and relational observables, the physical (relational) Hilbert space $ ext{H}_{ m phys}$, and the relational trace to coherently embed and trace relational data across particle groups. The relational framework resolves the paradox of the third particle by focusing on relational observables and a relational partial trace that is independent of arbitrary embeddings, and it connects with constraint quantization concepts like Dirac observables. The results clarify how QRFs induce frame-dependent phenomena (e.g., superselection-like behavior for nonzero total momentum) while preserving a perspective-neutral description, with potential implications for quantum gravity, edge modes, and temporal relational dynamics.

Abstract

In a quantum world, reference frames are ultimately quantum systems too -- but what does it mean to "jump into the perspective of a quantum particle"? In this work, we show that quantum reference frame (QRF) transformations appear naturally as symmetries of simple physical systems. This allows us to rederive and generalize known QRF transformations within an alternative, operationally transparent framework, and to shed new light on their structure and interpretation. We give an explicit description of the observables that are measurable by agents constrained by such quantum symmetries, and apply our results to a puzzle known as the `paradox of the third particle'. We argue that it can be reduced to the question of how to relationally embed fewer into more particles, and give a thorough physical and algebraic analysis of this question. This leads us to a generalization of the partial trace (`relational trace') which arguably resolves the paradox, and it uncovers important structures of constraint quantization within a simple quantum information setting, such as relational observables which are key in this resolution. While we restrict our attention to finite Abelian groups for transparency and mathematical rigor, the intuitive physical appeal of our results makes us expect that they remain valid in more general situations.

Figures (6)

• Figure 1: The simplest example of this article's setup: a discretization of wave functions in one spatial dimension under translation symmetry. The configuration space is the cyclic group $\mathbb{Z}_n$, and the one-particle Hilbert space is $\mathcal{H}=\ell^2(\mathbb{Z}_n)\simeq\mathbb{C}^n$. We have $N$ distinguishable particles in a joint quantum state $|\psi\rangle\in\mathcal{H}^{\otimes N}$, and we study QRF transformations that switch between the "perspectives of the particles".
• Figure 2: Some of the structures we uncover in Section \ref{['SecTechnical']}. We axiomatically derive and analyze the quantum symmetry group $\mathcal{U}_{\rm sym}$, and characterize a class of "alignable states" that can be transformed into a form that is "relative to one of the particles". As described in Refs. GiacominiVanrietveldeHamette, "jumping from the first to the third particle", for example (sketched on top), can transform separable into entangled states, owing to the fact that, as we will show, $\mathcal{U}_{\rm sym}$ is larger than the classical group of translations. We identify two subalgebras of operators that are invariant under all quantum symmetries, $\mathcal{A}_{\rm phys}\subset\mathcal{A}_{\rm inv}$, and corresponding projections that extract the "invariant part" of a state.
• Figure 3: What both approaches have in common: a system $S$ with a symmetry group $\mathcal{G}_{\rm sym}$ acting on its states $\rho\in\mathcal{S}$. States are implicitly defined via some (physical or fictional) external relatum, but internally (that is, for observers without access to the relatum) $\rho$ and $G\rho$ are indistinguishable, for all $G\in\mathcal{G}_{\rm sym}$.
• Figure 4: A communication scenario within the quantum information approach as in Ref. Bartlett. The focus is on sending and recovering actual physical (quantum) states that are defined (as in Assumption 1) with respect to some external relatum, i.e. that may contain unspeakable information. This task becomes interesting if Alice's and Bob's reference frames are initially unaligned.
• Figure 5: A simple communication scenario which we choose for illustrating the operational essence of the structural approach as in Refs. GiacominiVanrietveldeHamette. The focus is on agents agreeing on a (redundancy-free) description of quantum states in the absence of an external relatum.

Theorems & Definitions (49)

• Example 1
• Example 2
• Example 3
• Definition 4: $\mathcal{G}$-system
• Lemma 5
• proof
• Example 6
• Definition 7: Invariant observable
• Example 8
• Lemma 9