if we defineHphysas the subset of states satisfying 46.12, this can be identified
with the space of wavefunctions of two position variablesq 1 ,q 2. One technical
problem that appears at this point is that the original inner product includes an
integral over theq 3 coordinate, which will diverge since the wavefunction will
be independent of this coordinate.
If the conditionp 3 = 0 is instead imposed before quantization (i.e. on
the phase space coordinates) the phase space will now be five dimensional,
with coordinatesq 1 ,q 2 ,q 3 ,p 1 ,p 2 , and it will no longer have a non-degenerate
symplectic form. It is clear that what we need to do to get a phase space
whose quantization will have state spaceHphysis remove the dependence on
the coordinateq 3.
In general, if we have a groupGacting on a phase spaceM, we can define:
Definition(Symplectic reduction).Given a groupGacting on phase spaceM,
preserving the Poisson bracket, with moment map
M→g∗the symplectic reductionM//Gis the quotient spaceμ−^1 (0)/G.
We will not show this here, but under appropriate conditions the space
M//Gwill have a non-degenerate symplectic form. It can be thought of as the
phase space describing theG-invariant degrees of freedom of the phase spaceM.
What one would like to be true is that “quantization commutes with reduction”:
quantization ofM//Ggives a quantum system with state spaceHphysidentical
to theG-invariant subspace of the state spaceHof the quantization ofM.
Rarely are bothMandM//Gthe sort of linear phase spaces that we know how
to quantize, so this should be thought of as a desirable property for schemes
that allow quantization of more general symplectic manifolds.
For the case of a system invariant under translations in the 3-direction,
M=R^6 ,G=R,g=R) and the moment map takes as value (see equation
15.12) the element ofg∗given byμ(q,p) where
μ(q,p)(a) =ap 3μ−^1 (0) will be the subspace of phase space withp 3 = 0. On this space the
translation group acts by translating the coordinateq 3 , so we can identify
M//R=μ−^1 (0)/R=R^4with the phase space with coordinatesq 1 ,q 2 ,p 1 ,p 2. In this case quantization
will commute with reduction since imposingP 3 |ψ〉= 0 or quantizingM//Rgive
the same space of states (in the Schr ̈odinger representation, the wavefunctions
of position variablesq 1 ,q 2 ).
This same principle can be applied in the infinite dimensional example of
the temporal gauge phase space with coordinates
(A(x),−E(x))