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Igor Khavkine
#2
Jun19-07, 05:00 AM
P: n/a
On 2007-06-17, Jay R. Yablon <jyablon@nycap.rr.com> wrote:
> I have a question about Gaussian integrals and path integrals. In the
> below, I use $ to designate "integral from negative infinity to positive
> infinity." I am concerned here only with J=0.
>
> The basic Gaussian integral used in QFT, for J=0, is:
>
> $ dx e^(-.5iax^2) = (2pi i/a)^.5 (2)
>
> When a is promoted to a matrix A, this factor becomes:
>
> $ dx e^(-.5iAx^2) = (2pi i/det A)^.5 (2)
>
> This, as I understand it, works with discretized spacetime. In the
> continuum limit, one passes over to the path integral, which, in this
> J=0 special case, is:
>
> Z(J=0) = C = Ce^iW(J=0) (3)
>
> This factor C, which corresponds with the determinant in (2), is often
> omitted. I would like to know:
>
> Is this factor, once we pass to the continuum, also proportional to
> 1/det A? Or, is it proportional to the actual inverse of A, that is,
> A^-1 (what becomes the propagator)?


It remains the determinant of A. There is no reason why in the infinite
dimensional limit, det A should suddenly become A^-1. More over, one is
a scalar, while the other is a matrix. They are not interchangeable.

> Is there any reference that explicitly shows this factor, for, say, a
> free scalar field psi with, say:
>
> Z = $ Dpsi e^[i $ d^4x (-.5)psi(d^2+m^2)psi] (4)
>
> Main question: is this a
>
> (d^2+m^2)^-1 = D, i.e., a propagator?
>
> or, is this a
>
> 1/det(d^2+m^2)?


It's definitely the latter. The reason this factor is dropped most of
the time is that it merely corresponds to an overall normalization.
Since we are only interested in the quantity Z(J)/Z(0), it gets
cancelled. Another way to ignore this factor is to invoke the relation
Z(0) = <vac|vac> = 1, which holds if we work with a normalized ground
state.

I can't give a reference off hand to where this determinant is handled
explicitly, but some authors do take this approach. Google for
"functional determinant".

Igor