sci.physics.research

On Aug 13, 4:38 pm, "Jay R. Yablon" wrote:

> Just after equation (8), if I may still use "Lagrangian" in place of
> "action" ("functional" in Igor's terms), Zee says "In the limit with
> h-bar much smaller than the relevant Lagrangian we are considering . .
> ." In this phrase, Zee is making clear that the Lagrangian, which has
> dimensions of angular momentum same as h-bar, is itself a number, and
> that this number has a measurable magnitude in comparison to h-bar.

No, you cannot substitute "Lagrangian" for "action functional". You
may, though, substitute "Lagrangian" for "Lagrangian density", when
there is no confusion possible. If Zee uses the first substitution, I
believe it is a bit careless.

The path integral by itself is not observable. The individual pieces
that make up the path integral are also not observable (which includes
the action in its integrand). What is observable are field correlation
functions (if you are talking about field theory) and other
expectation values that can be computed with its help.

All the talk about the magnitude of the action in the exponent of the
path integral amplitude is none other than a reference to the method
of stationary phase for evaluating integrals of the form
int exp(i*f(x)/h) dx, in the limit that h -> 0. However, it is not the
magnitude of f(x) compared to h that is important, rather it is the
magnitude of df/dx, which is what induces larger phase fluctuations and
hence cancellation under the integral sign. The method of stationary
phase uses some analysis to show that the leading asymptotic of this
integral can be obtained from the knowledge of f(x) only around the
values of x where df/dx is least (that is d^2 f/dx^2 = 0, or the phase of
the integrand is stationary), even if these values lie in the complex
plane.

In the case of a path integral, x corresponds to a particle path/field
history and f(x) to the action functional. The method of sationary phase
the states that in the h -> 0 limit, the paths that minimize the action
(classical paths) contribute the most. A good question: when the is
the h -> 0 limit applicable? Recall that observable quantities in
quantum mechanics are expectation values of operators and that path
integrals are means of evaluating those. For example

= int dx_psi O(x) exp(i*S(x)/h) dx

is the expectation value of operator O for state |psi>, dx_psi
represents integration over classical histories and S(x) the action
functional. Notice that the integration measure dx_psi depends on the
state |psi>. And it is the state |psi> that determines how close the
system is to a classical state. So, it is the integration measure dx_psi
that determines which classical histories are integrated over and it
equally determines when the conditions for the stationary phase
approximations are applicable. If they are, then according to previous
discussion, the expectation value should be close to its
classical value, up to corrections small in powers of h. Note that
nowhere in this calculation has S(x) turned out to be observable, only
is.

Igor