Equivalence of approaches to relational quantum dynamics in relativistic settings

Abstract

We have previously shown (arXiv:1912.00033) that three approaches to relational quantum dynamics -- relational Dirac observables, the Page-Wootters formalism and quantum deparametrizations -- are equivalent. Here we show that this `trinity' of relational quantum dynamics holds in relativistic settings per frequency superselection sector. We ascribe the time according to the clock subsystem to a POVM which is covariant with respect to its (quadratic) Hamiltonian. This differs from the usual choice of a self-adjoint clock observable conjugate to the clock momentum. It also resolves Kuchař's criticism that the Page-Wootters formalism yields incorrect localization probabilities for the relativistic particle when conditioning on a Minkowski time operator. We show that conditioning instead on the covariant clock POVM results in a Newton-Wigner type localization probability commonly used in relativistic quantum mechanics. By establishing the equivalence mentioned above, we also assign a consistent conditional-probability interpretation to relational observables and deparametrizations. Finally, we expand a recent method of changing temporal reference frames, and show how to transform states and observables frequency-sector-wise. We use this method to discuss an indirect clock self-reference effect and explore the state and temporal frame-dependence of the task of comparing and synchronizing different quantum clocks.

Figures (3)

• Figure 1: Depicted are the surfaces $\mathcal{C}_+$ (red) and $\mathcal{C}_-$ (green) defined by $C_+ = 0$ and $C_- = 0$, respectively. The union of these surfaces is the constraint surface $\mathcal{C} = \mathcal{C}_+ \cup \mathcal{C}_- \subset \mathcal{P}_{\rm kin}$, while their intersection $\mathcal{C}_+ \cap \mathcal{C}_-$ is characterized by $p_t = H_S = 0$, and is depicted by the thick black line. We have assumed that $H_S$ is not degenerate (see Fig. 1 of Hoehn:2018whn for a similar depiction when $H_S$ is doubly degenerate).
• Figure 2: A summary of the Page-Wootters reduction maps and their inverses. The analogous state of affairs holds for the quantum symmetry reduction maps and their inverses.
• Figure 3: Schematic representation of a temporal frame change, as defined through Eq. \ref{['TFC']}. The figure encompasses both the relational Schrödinger picture of the Page-Wotters formalism and the relational Heisenberg picture of the quantum deparametrization, as well as their mixtures, since $I,J\in\{\rm{PW},\rm{QR}\}$. Viewing the reduction maps $\mathcal{R}_I^{+_i}(\tau_i)$ as quantum coordinate maps, any such temporal frame change takes the form of a quantum coordinate transformation from the description relative to clock $C_1$ to the one relative to clock $C_2$. Just as coordinate transformation pass through the reference-frame-neutral manifold, the quantum coordinate transformations pass through the clock-neutral physical Hilbert space in line with the general discussion of the clock-neutral structure and quantum general covariance in Sec. \ref{['sec_cRDOs2']}. Due to the double superselection rule, the quantum coordinate transformations have to preserve the overlaps of the frequency sectors of $C_1$ and $C_2$. Here we illustrate the example of the overlap of the positive frequency sectors of both clocks, so that the corresponding frame transformation passes through $\mathcal{H}_{+_1,+_2}$ (cf. Eq. \ref{['overlap']}).

Theorems & Definitions (35)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof