We study the quantization of many-body systems in two dimensions in rotating coordinate frames using a gauge-invariant formulation of the dynamics. We consider reference frames defined by linear and quadratic gauge conditions. In both cases we discuss their Gribov ambiguities and commutator algebra. We construct the momentum operators, inner product and Hamiltonian in both types of gauges, for systems with and without translation invariance. The analogy with the quantization of QED 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.
The separation of the dynamics of the quantum many-body systems into the overall rotation and internal motions leads to their quantization in rotating frames. Whereas the rotating body frame is unique for a rigid body, there is a great degree of freedom for choosing a body frame for a general rotating and vibrating N-particle system. This, in turn, implies the need to study many-body systems for a class of body frames.The concept of vibrational coordinates and of rotational coordinates is particularly clear for the case of a triatomic molecule: three particles form a single plane, in which the molecule vibrates, and at the same time the molecular plane rotates in the space-fixed frame of reference. The overall molecular rotation is made explicit by means of an orthogonal transformation, which brings the space-fixed into the body-fixed axes. The direction perpendicular to the molecular plane uniquely determines one of the three axes of the frame. Rotation about this direction will change the position of the axes in the molecular plane, in fact the axis system. It is already intuitively clear that this axis transformation will not affect the vibrational portion of the kinetic energy operator. The other parts of the kinetic energy, the rotational part and the rotation-vibration coupling part, will then need to adjust themselves to the new definition of the body frame. In this fashion, we may address the problem of the interaction of rotations and vibrations, which is one of the most important problems in molecular studies.The variation of the rovibrational kinetic energy operator following a change of orientation of the axis system is investigated in the present work from the physical and mathematical points of view. Various types of axis embedding are analyzed in a systematic fashion for triatomic molecules of arbitrary structure, including the bisector, principal axis, and Eckart frames. We also consider the bisector axis embedding for tetratomic molecules and the local axis embedding for pentatomic molecules. Different axis conventions lead sometimes to a complex entanglement of rotational and vibrational motions, which may pose serious obstacles to numerically exact approaches.