Interacting quantum reference frames

Abstract

We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space $\mathcal{H}_{\rm kin}$ by group averaging. We assume the gauge group $G$ is unimodular, and we define a quantum reference frame as a system of coherent states associated with a group $G_R$ of frame orientations acting on a tensor factor of $\mathcal{H}_{\rm kin}$. In direct analogy with the Faddeev-Popov procedure in path integrals, we show that the physical inner product can be written in terms of a choice of frame orientation with an appropriate operator measure, but that the physical Hilbert space is independent of this choice of frame. This yields a generalization of the Page-Wootters reduction map used to define conditional states and conditional probabilities of measurement outcomes given a frame orientation. These probabilities correspond to the expectation values of relational Dirac observables obtained from generalizations of previously considered relativization maps. Our main result is that the map that implements a change of frame is unitary, even when $G$ does not act as a tensor-product representation across a factorization of $\mathcal{H}_{\rm kin}$. The formalism therefore accommodates interactions between reference frames and the systems described relative to them. We apply this formalism to temporal frames, showing that a unitary Schrödinger equation emerges for an interacting clock and system, and that a previously reported time-nonlocal Schrödinger equation is recovered in an appropriate limit. With this, problems of non-unitarity are avoided by the use of the gauge-fixing measure in the physical inner product. We also discuss the case of interacting spatial frames, where the known correspondence between coherence in one frame and entanglement in another is shown to depend on the coupling.