The Trinity of Relational Quantum Dynamics

TL;DR

The Trinity of Relational Quantum Dynamics shows that clock-neutral Dirac observables, Page-Wootters relational dynamics, and relational Heisenberg dynamics from quantum symmetry reduction are different faces of the same underlying relational framework. By employing covariant clock POVMs and G-twirling, the authors construct gauge-invariant relational observables and prove their equivalence to PW and reduced descriptions, resolving long-standing criticisms (notably Kuchař) without relying on ideal clocks or ancillas. They introduce clock-frame changes that consistently transform states and observables through a clock-neutral space, revealing clock-dependent temporal nonlocality and clarifying entanglement’s role as a kinematical feature. The work extends to non-ideal, non-degenerate clocks and provides a robust, gauge-invariant method for changing temporal reference frames, strengthening the foundation for quantum reference frames and relational quantum gravity concepts.

Abstract

The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum dynamics: 1) relational observables in the clock-neutral picture of Dirac quantization, 2) Page and Wootters' (PW) Schrödinger picture formalism, and 3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. We develop a quantization procedure for relational Dirac observables using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of gauge-invariantly extending gauge-fixed quantities. We establish algebraic properties of these relational observables. We extend a recent clock-neutral approach to changing temporal reference frames, transforming relational observables and states, and demonstrate a clock dependent temporal nonlocality effect. We show that Kuchař's criticism, alleging that the conditional probabilities of the PW formalism violate the constraint, is incorrect. They are a quantum analog of a gauge-fixed description of a gauge-invariant quantity and equivalent to the manifestly gauge-invariant evaluation of relational observables in the physical inner product. The trinity furthermore resolves a previously reported normalization ambiguity and clarifies the role of entanglement in the PW formalism. The trinity finally permits us to resolve Kuchař's criticism that the PW formalism yields wrong propagators by showing how conditional probabilities of relational observables give the correct transition probabilities. Unlike previous proposals, our resolution does not invoke approximations, ideal clocks or ancilla systems, is manifestly gauge-invariant, and easily extends to an arbitrary number of conditionings.

Figures (6)

• Figure 1: The trinity of relational quantum dynamics posits that the dynamics described by relational Dirac observables in the clock-neutral picture of Dirac quantization, the relational Schrödinger picture of the Page-Wootters formalism, and the relational Heisenberg picture obtained upon a quantum symmetry reduction of the clock-neutral theory are three manifestations of the same relational quantum theory.
• Figure 2: Depicted is the unconstrained phase space $\mathcal{P}_{\rm kin}$ (rectangular prism), the constraint surface $\mathcal{C}$ (green surface), gauge orbits/dynamical trajectories in $\mathcal{C}$ generated by $C_H$ (black curves on $\mathcal{C}$), the gauge-fixing surface $T= \tau$ (red plane), and the reduced phase space $\mathcal{P}^{\rm red}_{S}$ (thick black line, see Sec. \ref{['Heisenberg']}). The relational Dirac observable $F_{f,T}(\tau)$ is a gauge-invariant function on $\mathcal{C}$ corresponding to the question "what is the value of the function $f$ when the clock $T$ reads $\tau$?" Hence, it corresponds to the value of the function $f$ on the intersection of the gauge-fixing surface $T=\tau$ with $\mathcal{C}$. Letting the parameter $\tau$ run unfolds the relational dynamics and thus corresponds to 'scanning' $\mathcal{C}$ with the family of gauge-fixing surfaces $T=\tau$.
• Figure 3: The trinity of relational quantum dynamics. This figure depicts the reduction maps from the physical Hilbert space $\mathcal{H}_{\rm phys}$ to the physical system Hilbert space $\mathcal{H}_S^{\rm phys}$ and their inverses. These maps are used to transform states and observables between the clock-neutral picture given by Dirac quantization, the relational Schrödinger picture derived from the Page-Wootters formalism, and the relational Heisenberg picture. It is these maps that are used to prove the equivalence between these three relational quantum dynamics comprising the trinity.
• Figure 4: The temporal frame change (TFC) maps in the relational Schrödinger picture (Page-Wootters formalism), $\Lambda^{A \to B}_{\mathbf S}$, and the relational Heisenberg picture, $\Lambda^{A \to B}_{\mathbf H}$, as well as TFC maps acting in-between them, as given in Eq. \ref{['TFC']}. To transform the state of clock $B$ and system $S$ and with respect to clock $A$, to the state of $A$ and $S$ with respect to $B$, we must first pass to the physical Hilbert space via the inverse of the reduction map, indicated by the arrows pointing from the top and bottom left corners to the center, followed by the application of the reduction map, depicted by the arrows pointing from the center to the top and bottom right corners.
• Figure 5: A change of quantum frame perspective has the same compositional structure as coordinate changes on a manifold. The 'quantum coordinate maps' $\mathcal{R}_A$ and $\mathcal{R}_B$ take as their input the perspective-neutral physics on $\mathcal{H}_{\rm phys}$ and map it to a description relative to the perspective of either quantum reference frame $A$ or $B$. The quantum coordinate maps $\mathcal{R}_A,\mathcal{R}_B$ are maps between Hilbert spaces (quantum reduction maps). Just like coordinates on a manifold, a perspective need not be globally valid (due to the Gribov problem) Vanrietvelde:2018ditVanrietvelde:2018pgbhoehnHowSwitchRelational2018Hoehn:2018whn.

Theorems & Definitions (41)

• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof