  Gauge-invariant systems in unconstrained configuration and phase spaces,
equivalent to second-class 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.

High Energy Physics - Theory