Re: Bosonization

From: Igor Khavkine (igor.kh_at_gmail.com)
Date: 06/28/04 

Date: 28 Jun 2004 13:10:44 -0400

Igor Khavkine <k_igor_k@lycos.com> wrote in message news:<pan.2004.05.25.21.43.57.237197@lycos.com>...
> I am trying to come to grips with bosonization. This is an equivalence
> between some fermionic and bosonic field theories in 1+1 dimensions.
> A precise example from relativistic field theory is the Thirring model,
> which is equivalent to the sine-Gordon model. Another example from
> condensed matter theory is the Luttinger model (a fudged up 1D electron
> gas), which is equivalent to the 1D bosonic field.
>
> I think I've got a fair understanding of the concepts and the
> tricks involved in the calculation, but I would like to clear up some
> technical issues.

Since I last posted, I think I've managed to clear up most of the
questions I had about bosonization by following up references.
In particular, cond-mat/9805275 has been very helpful, where an
explicit operator and Fock space equivalence between the fermionic
and bosonic theories is established. However, some murky issues still
remain.

> I'm also confused about the issue of normal ordering. I see that it is
> necessary to remove singularities that otherwise appear in products of
> operators at nearby spacetime points. I would be happy if normal
> ordering appeared only in the definition of the constructed fermion
> field operators. However, it seems that all operators built out of the
> fermions need to be further normal ordered. I would imagine that if I
> could identify the ground states of the fermion and boson theories, as
> well as build fermion operators with the right anticommutation relations,
> then I would be able to construct the fermion Fock space and re-express
> all operators in the fermion theory using bosons. So how does normal
> ordering fit into this picture?

First, part of my confusion in the above has been aleviated. The reason
normal ordering is necessary is to regularize and eliminate some
divergences that arise in some ground state expectation values. The
simplest example is the infinite charge density or energy contributed
to the vacuum by the "Dirac sea" when the Dirac field is quantized.

Now, I've seen many claims about normal ordering in the literature, and
it is not obvious to me that they are equivalent or follow one another.
Here are the normal ordering prescriptions that I've run into:

  A,B,C,... -- field creation and annihilation operators

Operator commutation:
  : ABC... : = same product but with all creation (annihilation)
                   operators manualy moved to the left (right)

Subtracting the ground state expectation value:
  : ABC... : = ABC... - <ABC...>
                 where <...> denotes the ground state expectation value

Point splitting:
  : A(x)B(x) : = lim{a->0} A(x+a)B(x) - <A(x+a)B(x)>
           this prescription seems to be related operator product
           expansions, but I have not yet been able to penetrate
           all the hype surrounding them

One claim is that the vacuum expectation value of a normal ordered operator
is 0. This is easy to accept given any of the above prescriptions. Another
claim is that ALL matrix elements of normal ordered operators are finite.
Moreover, it is claimed that the normal ordered operator :A(x)B(y)C(z)...:
is well enough behaved when some of the coordinates coincide so that
we can take its derivatives.

It is not clear to me how the last two claims follow from the definition
of normal ordering. Any insight would be appreciated.

Thanks in advance.

Igor