Perspective-neutral approach to quantum frame covariance for general symmetry groups

TL;DR

The work generalizes the perspective-neutral quantum reference frame framework to arbitrary unimodular Lie groups, modeling frame orientations with coherent states and a covariant POVM. It develops a gauge-invariant, perspective-neutral Hilbert space via group averaging, constructs relational Dirac observables with a clear conditional-probability interpretation, and extends Page-Wootters and symmetry-reduction formalisms to general groups. Two distinct QRF transformation types—gauge-induced quantum coordinate changes and symmetry-induced active changes—are shown to be unitary and to organize relational observables into orbits, yielding novel effects such as isotropy-imposed coarse-graining and frame-orientation–dependent internal subsystems in non-symmetric cases. The paper also provides explicit examples (U(1) and SU(2) with j=1) to illustrate frame-dependent subsystems and contrasts its perspective-neutral approach with purely perspective-dependent methods, outlining avenues toward gauge-field/edge-mode applications and quantum gravity extensions.

Abstract

In the absence of external relata, internal quantum reference frames (QRFs) appear widely in the literature on quantum gravity, gauge theories and quantum foundations. Here, we extend the perspective-neutral approach to QRF covariance to general unimodular Lie groups. This is a framework that links internal QRF perspectives via a manifestly gauge-invariant Hilbert space in the form of "quantum coordinate transformations", and we clarify how it is a quantum extension of special covariance. We model the QRF orientations as coherent states which give rise to a covariant POVM, furnishing a consistent probability interpretation and encompassing non-ideal QRFs whose orientations are not perfectly distinguishable. We generalize the construction of relational observables, establish a variety of their algebraic properties and equip them with a transparent conditional probability interpretation. We import the distinction between gauge transformations and physical symmetries from gauge theories and identify the latter as QRF reorientations. The "quantum coordinate maps" into an internal QRF perspective are constructed via a conditioning on the QRF's orientation, generalizing the Page-Wootters formalism and a symmetry reduction procedure. We find two types of QRF transformations: gauge induced "quantum coordinate transformations" as passive unitary changes of description and symmetry induced active changes of relational observables from one QRF to another. We reveal new effects: (i) QRFs with non-trivial orientation isotropy groups can only resolve isotropy-group-invariant properties of other subsystems; (ii) in the absence of symmetries, the internal perspective Hilbert space "rotates" through the kinematical subsystem Hilbert space as the QRF changes orientation. Finally, we invoke the symmetries to generalize the quantum relativity of subsystems before comparing with other approaches. [Abridged]

Figures (5)

• Figure 1: The set of orthonormal axes in ${\hbox{$\rm R$}}^3$ is acted on regularly by $G \cong \mathrm{SO}(3)$. In the top, we see a reference $R$ and a system $S$ which are acted on by the left regular representation of $G$. This is a gauge transformation, and the relative orientation between $R$ and $S$ is preserved. On the bottom, we see that $R$ is acted on by the right regular action. This commutes with the gauge action $U_{RS}(g')$ and hence is a physical symmetry. The relative orientation between $R$ and $S$ is changed by this action.
• Figure 2: This figure illustrates $G$-orbits of different relational observables relative to $R$ under $\mathcal{V}_R$ in the space $\mathcal{A}_{\rm phys}$. The orbit $\{F_{\mathbf{1}_S,R}(g)\}_{g \in G}$ is a single point. The orbit $\{F_{f'_S,R}(g)\}_{g \in G}$ is the right line. We see that two observables $F_{f'_S,R}(h)$ and $F_{f'_S,R}(h')$ where $h,h'$ belong to the same right coset are equal. The orbit $\{F_{f_S,R}(g)\}_{g \in G}$ is represented by the left line. One can transform between $F_{f_S,R}(g_1)$ and $F_{f_S,R}(g_1')$ on this orbit with the operator $\mathcal{V}_R^{{\rm phys}}({g_1'}^{-1} g_1)$ as represented by the dotted arrow. Different orbits generally being of different dimension, the relational observable families define a stratification of $\mathcal{A}_{\rm phys}$.
• Figure 3: This figure illustrate the relations between different physical and kinematical subsystem Hilbert spaces, as well as their embedding in the larger Hilbert space $\mathcal{H}_{\rm kin} \cong \mathcal{H}_{R_1} \otimes \mathcal{H}_S \otimes \mathcal{H}_{R_2}$ and $\mathcal{H}_{\rm phys}$. In the figure we depict the case $\mathcal{H}_{\rm phys} \subset \mathcal{H}_{\rm kin}$ for simplicity. The physical subsystem space $\mathcal{H}^{\rm phys}_{R_1 S,g_2}$ of subsystems $R_1S$ conditional on $R_2$ in orientation $g_2$ is a subspace of $\mathcal{H}_{R_1} \otimes \mathcal{H}_{S}$. Similarly for $\mathcal{H}^{\rm phys}_{R_2 S,g_1}$ and $\mathcal{H}_{S} \otimes \mathcal{H}_{R_2}$. One can map directly between the physical system spaces of $R_1S$ relative to $R_2$ and $R_2S$ relative to $R_1$ via the change of reference system operator $V_{R_1 \to R_2}(g_2,g_1)$. As $R_2$ changes orientation $g_2$, the subspace $\mathcal{H}_{R_1S,g_2}^{\rm phys}$ "rotates" through the kinematical $\mathcal{H}_{R_1}\otimes\mathcal{H}_S$ if symmetries are absent (similarly with $R_1$ and $R_2$ interchanged).
• Figure 4: In analogy with special relativity (recall Sec. \ref{['ssec_SR']}), one can see the physical Hilbert space $\mathcal{H}_{\rm phys}$ as the global manifold. The reduced Hilbert space $\mathcal{H}_{R_1S,g_2}$ is the "quantum coordinate" description of the physical degrees of freedom from the perspective of $R_2$ in configuration $g_2$, which is analogous to a coordinate patch on the global manifold. The reduced Hilbert space $\mathcal{H}_{R_1S,g_2}^{\rm phys}$ is obtained from $\mathcal{H}_{\rm phys}$ via the map $\mathcal{R}_{\mathbf{S},R_2}(g_2)$ which corresponds to a coordinate map in our analogy. Hence, the map $V_{R_2\to R_1}(g_2,g_1)$ is analogous to a change of coordinates.
• Figure 5: This figure illustrate various orbits of observable in $\mathcal{A}_{\rm phys}$. In blue on the left is the orbit of relational observables $\{F_{f_S,R_1}(g)\}_{g \in G}$ describing the system $S$ relative to the frame $R_1$. As can be seen, any two elements $F_{f_S,R_1}(g_1)$ and $F_{f_S,R_1}(g_1')$ in the same orbit are related by the (configuration-independent) symmetry transformation $\mathcal{V}^{\rm phys}_{R_1}({g_1'} ^{-1} g_1)$ in Eq. \ref{['obsreorient']}. Similarly, on the right in blue, one has the orbit of relational observables $\{F_{f_S,R_2}(g)\}_{g \in G}$ describing the system $S$ relative to the frame $R_2$. One can map between points $F_{f_S,R_1}(g_1)$ and $F_{f_S,R_2}(g_2)$ on different orbits with the operator $\mathcal{V}_{R_1 \to R_2}^{g_1 \to g_2}$, which is a relation-conditional symmetry.

Theorems & Definitions (47)

• Example 1
• Lemma 1
• Example 2
• Example 3: The Weyl-Heisenberg group, part 1 of 2
• Example 4: The Weyl-Heisenberg group, part 2 of 2
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof