We study the quantization of many-body systems in three dimensions in rotating coordinate frames using a gauge invariant formulation of the dynamics. We consider reference frames defined by linear gauge conditions, and discuss their Gribov ambiguities and commutator algebra. We construct the momentum operators, inner product and Hamiltonian in those gauges, for systems with and without translation invariance. The analogy with the quantization of non-Abelian Yang–Mills theories in non-covariant gauges is emphasized. Our results are applied to quasi-rigid systems in the Eckart frame.
We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space
Hkin 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
GR of frame orientations acting on a tensor factor of
Hkin. 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
Hkin. 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.
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum
j and introduce the transformations between them by upgrading the Euler angles occurring in classical
SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large
j, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.
We study the quantization of three-dimensional many-body systems in rotating coordinate frames defined implicitly by frame conditions. We carry out the elimination of orientational degrees of freedom in general, giving the Hamiltonian for the
N-particle system in a broad class of body frames in terms of frame conditions and internal coordinates. We obtain several forms for the kinetic energy operator and compare them to related expressions in the literature.
We study the quantization of three-dimensional many-body systems in rotating coordinate frames defined implicitly by frame conditions. We carry out the elimination of orientational degrees of freedom in general, giving the Hamiltonian for the
N-particle system in a broad class of body frames in terms of frame conditions and internal coordinates. We obtain several forms for the kinetic energy operator and compare them to related expressions in the literature.
We introduce a way to express compact quantum electrodynamics with dynamical matter on two- and three-dimensional spatial lattices in a gauge redundancy-free manner while preserving translational invariance. By transforming to a rotating frame, where the matter is decoupled from the gauge constraints, we can express the gauge field operators in terms of dual operators. In two space dimensions, the dual representation is completely free of any local constraints. In three space dimensions, local constraints among the dual operators remain but involve only the gauge field degrees of freedom (and not the matter degrees of freedom). These formulations, which reduce the required Hilbert space dimension, could be useful for both numerical (classical) Hamiltonian computations and quantum simulation or computation.