On Nov 5, 1:19 pm, Mike <mj...@sirus.com> wrote:
> We are told that as we manually integrate the path integral in the
> region of far flung paths, contributions of the integral cancel out
> with other parts of far flung paths, leaving only the classical path
> that contributes most to the final result. My question is if the
> functional derivative of the Action integral does not have a zero for
> any path, no classical path, then is it still possible to evaluate the
> path integral at all? Or will everything cancel out? I wonder if there
> is any mathematical proof that the solvability of the path integral
> requires a zero variation of the action. Thanks.
To this end I consider that in QM typically the path integral is
developed by inserting the identity operator an infinite number of
time. i.e.:
<x'|x"> = S dx1 <x'|x1><x1|x">
where S dx1 is the integral with respect to the variable x1, and where
S dx1|x1><x1| is the identity which can be applied an infinite number
of times:
<x'|x"> = S dx1dx2...dxn <x'|x1><x1|x2><x2|...|xn><xn|x">
Then each of the <xi|xj> is recognized to be a Dirac delta function,
<xi|xj> = D(xi-xj).
When this is substituted into the integral above,
<x'|x"> = D(x'-x") = S dx1dx2...dxn D(x'-x1)D(x1-x2)...D(xn-x")
This interesting because the delta function has the same recursive
property as the identity above so that an infinite dimensional
integral can be formed as is done with the identity.
So if we choose the gaussian form of the Dirac delta function that has
a complex exponent, the path integral for a free particle can be shown
to result. See
http://hook.sirus.com/users/mjake/delta_physics.htm
So if I generalize the exponent to be a functional (an action integral
in this case) and ask where might the requirement come from that the
functional derivative of the action be zero, I consider what happens
if I take the functional derivative of the entire path integral. Is
this passed on to taking the functional derivative of the exponent
(the action) in the process? Since the path integral is equal to a
Dirac delta, my question is what is the functional derivative of a
Dirac delta function? Thanks.
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