Query about Range of validity of field equations in Quantum Field Theory

Dear Friends:

I have been on a "vacation" from physics for the past few months, other than to moderate SPF posts a couple of times a day, while overloaded with other matters. Now, as I seek among other things to break my recent addiction to following the US elections, ;-) I would like to return to some questions that I was exploring at sci.physics.foundations and sci.physics.research about 18 months ago.

For this discussion, I shall refer to several pages from Zee's book
"Quantum Field Theory in a Nutshell," which I have posted below:

http://jayryablon.files.wordpress.com/2008/11/zee-field-equations-and-inverses-markup.pdf

In this posted excerpt, I have put highlight boxes around two passages I
wish to consider, one on page 19, the other on page 168.

On page 19, we start with a path integral which includes the action
S=$Ld^4x, there $ is an integral, and L is the Lagrangian density. We
"recover the classical field equation" using the Euler-Lagrange equation
in the circumstance "with h-bar much smaller than the action we are
considering." While this example uses a Lagrangian for a *scalar field
theory*, let us think about this same line of development using the
Maxwell Lagrangian in which case the classical field equation is
Maxwell's equation J^v=d_vF^uv, or even using a non-Abelian Yang-Mills
Lagrangian or the Proca Lagrangian for a massive vector boson.

Now, we move to the highlighted excerpt on page 168. The context is
that of having to fix the gauge because the Q_uv defined just after
equation (3) on page 167 has no inverse, but what I am interested in is
how one should interpret the inverse equation:

A^v = (Q^-1)^vu J_u (1)

in the highlighted section on page 168. Clearly, the inverse (Q^-1)^vu
is directly related to the photon propagator, and (I believe?) is
completely valid under all known circumstances, i.e., this inverse can
be used directly in equation (2) on page 167 in the place of K^-1.

But what is the range of validity of of the inverse Maxwell equation
(1)? By the discussion on page 19, above, am I correct in concluding
that (1) above is only a "classical" solution which applies only "with
h-bar much smaller than the action we are considering"?

If so, how would one describe the physics of the range of situations
where (1) above is a valid solution? Yes, I know that this would be
physics situations in which "the action we are considering" is much
larger that h-bar, and that this is in the nature of a "classical /
quantum correspondence" principal, but what, exactly, does that mean in
terms of the physics of the situation being considered? For a single
photon, what would it mean to have an action much larger than h-bar?
For a single electron? What does it means when the action for *any*
system is much larger than h-bar? Action is of course dimensioned in
angular momentum, but I don't think that this has anything to do, at
least directly, with the angular momentum of the system or particle.
What I am really looking for is a simple, direct, intuitive, physical
understanding of what it means for a particle or a system to have an
action much larger than h-bar and thus of what it means for equation (1)
above to be applicable and / or not applicable.

Thanks,

Jay.
____________________________
Jay R. Yablon
Email: jyablon@xxxxxxxxxxxx
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

.

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