In the abelian case, the Green functions
of the initial bosons or fermions
did not involve the
full Hilbert space
of the bosonized theory.
The same thing is true in the nonabelian case.
The initial particles are represented only by a subset
of the wave functions of the spinning top.
This is seen by calculating the two-point
correlation function, obtained from the
functional derivatives 
of the generating functional
.
In the operator form (223) of the generating functional, the two-point correlation function is given by the expectation value
for which we easily calculate
where 
is the energy
difference between a state carrying one
boson or fermion
and the vacuum state 
:
In the bosonized theory we differentiate (251) and find
with U(t) short for
.
As in the abelian case, we
evaluate the bosonized expression (268)
in the
operator language using the Schrödinger representation.
Due to the presence of the correction factor
in the measure
of the path integral (268), the Hamilton operator
associated with the action in
(268)
is proportional to the Laplace-Beltrami operator
where 
is the inverse of the metric
defined by the kinetic term in the classical Lagrangian
having the form
In our model
The Hamilton operator contains no extra term proportional to the curvature scalar, and coincides with the one arising from quantizing the generators of the rotation group in the classical expression
leading to the well-known operator
This was shown in Ref. [9].
The eigenfunctions are
with the energies
In this Schrödinger representation, the correlation function (268) is given by the expectation value
where we have replaced the matrices
by the spin-1/2
representation matrices 
of Eq. (235), and written
them short as
, as we did with U(t).
The vacuum state
has the Schrödinger wave function
,
and an energy
Inserting the time evolution operator, we write
with 
of (272) and find a phase
where is the energy difference between the
boson wave function
and the ground state 
. Its value is the same as
in the operator calculation (267).
Then (275) reduces to the integral
Using the unitarity property
of the rotation functions 
we can rewrite this as
which is, of course, the same as in (266).
In this expression we observe a nonabelian version of the projective
properties of the bosonized theory in the Hilbert space
of all rotational wave functions.
At the level of spin 1/2,
there are four rotational wave functions
.
The correlation function (281), however, contains
one contracted index
which makes the angle 
disappear. The same happens
in all higher-point correlation functions.
Thus, the correlation functions
of the bosonized theory make use only
of a
subspace
of the total Hilbert space of the spinning top
in which the Euler angle is absent.
The correlation function
(281) looks as though the wave function
of a spin-1/2 particle were
.
These are orthogonal and complete
in the scalar product defined by
This subspace of top
wave functions is
equivalent to the space
of spherical harmonics 
.
Except for the presence of half-integer spins,
the spectrum corresponds to that of
a particle on the surface of a three-dimensional sphere,
where the energy eigenvalues
appear only (2j+1)-times rather than 
-times in the spinning top.
This is the selection mechanism reducing the partition function
of the spinning top (264)
to the smaller sum (238) over the initial states.
If the initial fundamental particles
are fermions, the orthogonality relation of the rotation functions
together with the Grassmann
algebra ensure that the bosonized theory represents
properly
the
anticommutation rules of the original fermion operators.
If one wants bosonized particles to cover a Hilbert space that is completely equivalent to the spinning top, one must start with twice as many bosons as before. The appropriate Lagrangian is then
and the Hamilton operator
This can be written as
where
are two independent sets of angular momentum operators with the commutation rules
The Hilbert space consists of the states
If we consider only the states with an equal number of a and b particles,
the Hilbert space is equivalent to that of the spinning top. To enforce (289), we have to extend the Lagrangian (283) by a Lagrange multiplier
It is worth pointing out, that a free-oscillator version of the Lagrangian (283) with the constraint (290),
arises from a nonholonomic transformation
of the path integral of the hydrogen atom (see Chapter 13 in [9]).
Thus,
the path integral
of the
hydrogen atom
could, in principle, also be solved
by
a Duru-Kleinert transformation to that
of a spinning top containing an extra
energy term proportional to 
.
