For a calculation of the correlation functions
of the original fields 
and
a(t), we must form the functional derivatives
of (181) with
respect to the sources 
,
divide the result by
Z[0,0], and set the sources equal to zero.
Each pair of differentiations
and 
produces a factor
in the integrand.
The path integral over 
-fields
amounts to calculating the Gaussian
averages of these exponentials. For an
arbitrary functional of , these are
defined by
By Wick's rule, we know that
where 
is the
correlation function
Hence
Note that the t,t'-independent last term
in (191) has dropped out, so that
the correlation function of exponentials 
has a finite limit for 
, in contrast to the
correlation function
of the field 
itself.
With the result (192) it is easy to calculate the correlation function of a boson or a fermion field. From (147), its operator expression is given by
Inserting a sum over all intermediate states 
,
we find
The same result is obtained from
the bosonic generating functional (181).
For the normalization factor Z in (193), this has just been shown.
Let us calculate the
numerator, denoting it by
.
Applying to (181)
the differentiations
,
we obtain its path integral
The second factor is equal to the correlation function
(192).
To evaluate the integral over 
, we write
as a spectral sum
After a quadratic completion, the integral over can be performed
and we obtain precisely the
numerator of (194) of the correlation function.
For more than one pair of
exponential fields 
,
we have to calculate
the expectation of
functionals of the form
where the numbers 
have the values
+ 1 for an incoming boson or fermion, and -1 for an outgoing one.
The numbers may be interpreted as the charges
of the fundamental fields.
After rewriting
we can again apply Wick's rule (190) and find
Inserting the correlation function (191), the right-hand side becomes
Since the external sources 
are differentiated pairwise, the total charge 
vanishes
(charge neutrality),
so that the first exponential is equal to unity, thus ensuring that
the expectation has a finite limit for :
It is useful to study the bosonized form of the theory
in the operator language to understand the structure of the
Hilbert space.
For this it is useful to
consider the simpler situation
of
an infinite time interval (corresponding to a zero-temperature
equilibrium calculation).
Then
the integral over in (181)
can be done trivially yielding unity
and forcing
to be zero.
The
Green function coincides with the
vacuum expectation value of the time-ordered product
and (163) yields
The generating functional is simply
To study this theory in the operator language, we take the free plasmon action
go over to the canonical form
and identify
the Hamiltonian as 
.
After replacing
,
,
which satisfy the canonical
equal-time commutation rule
we obtain the Hamilton operator
of the bosonized
model.
In the Schrödinger representation, the operators
are diagonalized on states
and the functional momentum operator
is represented by the differential
operator
.
The eigenstates of the Hamilton operator
consist
initially of plane waves which are eigenstates
of
with arbitrary real eigenvalues p:
We
are using curly brackets to
distinguish the Hilbert space of the -field
from that of the original 
fields.
The eigenstates (208) have the normalization:
In the operator version, the generating functional (204) reads
where are free field operators.
The time-ordered operator on the right-hand side is
taken between the states of zero-functional momentum.
We can now generate all Green functions of
fundamental particles by forming functional derivatives
with respect to 
. First
Inserting the time evolution operator
the matrix element (211) becomes
But the state 
is an eigenstate
of p with momentum p=1, so that (213) yields
and the Green function (211) becomes
The same result would, of course, have been obtained
for the original fundamental fields 
using the Hamilton operator (143):
Observe that nowhere in the calculation has
the Fermi or Bose
statistics of the operators
and 
been used. This becomes relevant
only for higher Green
functions. Expanding the exponential in (210) to the nth
order gives
The Green function
is obtained by forming the derivative
There are 
contributions due to the product
rule of differentiation, n! of them being
identical thereby canceling the factor 1/n! in
(217). The other correspond, from the point of view
of combinatorics, to all Wick contractions in (217),
each contraction being associated with a factor 
. In addition, the Grassmann nature of
source
fields causes a minus sign to appear if the contractions
deviating by an odd permutation from the natural order
. Denoting a Wick contraction
by a common number on top of a field operator, we obtain for example
where the upper sign holds for bosons, the lower for fermions.
The lower sign enforces the Pauli exclusion principle: If 
the two contributions cancel,
reflecting the fact that no two fermions 
can be created successively on the particle
vacuum. For bosons one may insert again the time translation
operator (212) and complete sets of states
with the result:
where 
has been used. This again agrees with
an operator calculation like (216).
We now understand how the collective quantum field theory works
in this model. Its Hilbert space consists of states
of 
functional momenta 
with p=real. When it comes to calculating
the Green functions of the fundamental fields of the original theory,
however,
only a small portion of this Hilbert space is used. A fermion
can make plasmon transitions back and forth between the
ground
state 
and the momentum one state 
,
due to the anticommutativity of the fermion source fields
.
Bosons, on the other hand, can connect all states
of integer momentum 
.
In either case, the collective-field basis is overcomplete as far as the
description of the
underlying system is concerned. The
source statistics
selects only a small subspace for the
dynamics of the fundamental system.
Note that such a projection
is compatible with unitarity. This is guaranteed
by the conservation
law 
const. In higher dimensions, there have to
be infinitely
many conservation laws (one for every space point)
to achieve unitarity.
