Underlying Gauge Symmetries of Second-Class Constraints Systems

Abstract

Gauge-invariant systems in unconstrained configuration and phase spaces, equivalent to secondclass constraints systems upon a gauge-fixing, are discussed. A mathematical pendulum on an n − 1-dimensional sphere S n−1 as an example of a mechanical second-class constraints system and the O(n) non-linear sigma model as an example of a field theory under second-class constraints are discussed in details and quantized using the existence of underlying dilatation gauge symmetry and by solving the constraint equations explicitly. The underlying gauge symmetries involve, in general, velocity dependent gauge transformations and new auxiliary variables in extended configuration space. Systems under second-class holonomic constraints have gauge-invariant counterparts within original configuration and phase spaces. The Dirac's supplementary conditions for wave functions of first-class constraints systems are formulated in terms of the Wigner functions which admit, as we show, a broad set of physically equivalent supplementary conditions. Their concrete form depends on the manner the Wigner functions are extrapolated from the constraint submanifolds into the whole phase space.

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

“…From other hand, the quantization of constraint holonomic systems leads to the conclusion that the dynamics is determined by the induced metric tensor only [40,43]. The limiting procedure and imposing the constraints are not equivalent schemes of the quantization.…”

“…The skew-gradient projection method is found to be useful to formulate classical and quantum constraint dynamics [14,15,37,38,39,40]. In Sect.…”

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

“…The Stratonovich version [7] of the Weyl's quantization and dequantization is discussed in Refs. [14,16,17,18,19,20]. Wigner functions have found numerous applications in quantum many-body physics, kinetic theory [21,22], collision theory, and quantum chemistry [23,24].…”

“…The notion of phase-space trajectories arises naturally in the deformation quantization through the Weyl's transform of Heisenberg operators of canonical coordinates and momenta [19,34,42,50]. These trajectories obey the Hamilton's equations in the quantum form and play the role of quantum characteristics in terms of which the time-dependent symbols of operators are expressed.…”

“…In this work, we report the semiclassical expansion of quantum characteristics for many-body scattering problem, based on the method [19,34,42,50], and provide tools for calculation of average values of time-dependent physical observables and scattering cross sections.…”