Path Integral Relationship to Classical Equations of Motion?

[Jay R. Yablon]

To our SPR cousins:

I'd like to ask a question I have been thinking about a lot lately and
see what others have to say.

Would it be fair to say that the "path integral" plays a role in quantum
field theory which is identical to the role played by the "equation of
motion" (geodesic equation in GR, Lorentz force law in EM) in classical
theory? If yes, then has anyone developed a "correspondence" principle
where one starts with the path integral for, say, a large number of
electrons, superimposes all of them, and ends up with the Lorentz force
law and or the geodesic equation in a very direct manner? If so, I
would love to see that connection. Put slightly differently, isn't the
path integral approach really nothing more or less than the equation of
motion of a single particle / field quantum, which motion happens to be
non-classical but should become classical for large numbers of field
quanta assembled in current densities?

Thanks,

Jay
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Web site: http://home.nycap.rr.com/jry/FermionMass.htm
Co-moderator, sci.physics.foundations (Come visit out new group)

[Arnold Neumaier]

Jay R. Yablon schrieb:
>
> I'd like to ask a question I have been thinking about a lot lately and
> see what others have to say.
>
> Would it be fair to say that the "path integral" plays a role in quantum
> field theory which is identical to the role played by the "equation of
> motion" (geodesic equation in GR, Lorentz force law in EM) in classical
> theory?

No. The classical analogue of a path integral is again a path integral,
except that in the classical case, the path integral is Euclidean and
describes a stochastic process.

> Put slightly differently, isn't the
> path integral approach really nothing more or less than the equation of
> motion of a single particle / field quantum, which motion happens to be
> non-classical but should become classical for large numbers of field
> quanta assembled in current densities?

There are a few papers on third quantization that try to do something
like this. Try scholar.google.com for finding these papers.

Arnold Neumaier

[Igor Khavkine]

On 2007-02-22, Jay R. Yablon <jyablon@nycap.rr.com> wrote:
> To our SPR cousins:
>
> I'd like to ask a question I have been thinking about a lot lately and
> see what others have to say.
>
> Would it be fair to say that the "path integral" plays a role in quantum
> field theory which is identical to the role played by the "equation of
> motion" (geodesic equation in GR, Lorentz force law in EM) in classical
> theory? If yes, then has anyone developed a "correspondence" principle
> where one starts with the path integral for, say, a large number of
> electrons, superimposes all of them, and ends up with the Lorentz force
> law and or the geodesic equation in a very direct manner? If so, I
> would love to see that connection. Put slightly differently, isn't the
> path integral approach really nothing more or less than the equation of
> motion of a single particle / field quantum, which motion happens to be
> non-classical but should become classical for large numbers of field
> quanta assembled in current densities?

What every student of field theory should learn is that a path integral
is secretly just a formula for exp(-iHt), where H is the Hamiltonian and
t a time interval. The path integral form is obtained by writing this
time evolution operator as a product of a bunch of ones that look like
exp(-iHdt), with dt << t, and then inserting a bunch of resolutions of
identity between these factors. This description is of course highly
simplified. But, keeping it in mind may help you plow through the
detailed derivation given in a QFT book like Weinberg (vol.1, Ch.9)
or Peskin & Schroeder (Ch.9) [1].

There are many other quantities that are easily expressible in terms of
path integrals, the main ones of interest are vacuum expectation values
of time ordered products of field operators. But, essentially, anything
expressible in the state/operator formalism is expressible in terms of
path integrals, and vice versa.

However, different expressions lend themselves to different
approximation techniques. The path integral is famous for justifying the
least action principle. The field or particle configurations that
contribute to the path integral the most are those that extremize the
action. That's the usual connection made between path integrals and
classical mechanics.

Hope this helps,

Igor

[1] It's a funny coincidence that both of these books introduce path
integrals in chapter 9. Maybe it's tradition. :-)