Rotating frames and gauge invariance in three-dimensional many-body quantum systems

Abstract

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 noncovariant gauges is emphasized. Our results are applied to quasi-rigid systems in the Eckart frame.

Many-body Hamiltonians in implicitly defined frames

“…We obtain several forms for the kinetic energy operator and compare them to related expressions in the literature, showing how the coefficient functions are fixed by the frame convention, through frame conditions, and internal coordinates. In the case of linear frames and body-frame coordinates our results reduce to those previously obtained in [1].…”

“…We do not impose any restriction on the form of frame conditions, provided they fix the frame uniquely. The particular case of frame conditions depending linearly on the particles coordinates was considered in [1] from the point of view of gauge invariance.…”

“…Typically, {R α = 0} is a singular configuration. Furthermore, for the two sets of conditions in (1) to be compatible the rotational conditions G a must be translation invariant, G a ({r α + v}) = G a ({r α }) for any v. This is satisfied by all usual frame conditions (see, e.g., [1,5]). In what follows, however, it will be enough to assume only the weaker form,…”

“…The body frame, specified by the frame conditions (1), fixes relations of the form R α = R α ({t a }) such that the conditions (1) are satisfied identically when evaluated on R α ({t a }). Thus, the functions R α ({t a }) are a parametric solution to the conditions (1). Through the relation inverse to (11), r α = U † ({θ a })R α ({t b }) + r cm , the internal coordinates {t b } together with the orientational and translational ones, {θ a } and r cm resp., give a set of 3N local coordinates in configurations space.…”

“…A simple example is given in section 6.1. The derivation of the Hamiltonian operator and Hilbert-space inner product in terms of linear body-frame coordinates and their conjugate momenta has been discussed in detail in [1,7]. In this paper we confine ourselves to a dynamical description in terms of Euclidean-invariant internal coordinates {t a }.…”