Science and Technology news... :: View topic - Query about Range of validity of field equations in Quantum

Jay R. Yablon · Guest

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@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

Igor Khavkine · Guest

On Nov 16, 3:06 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

[quote]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.
[/quote]
Jay, as I>ve pointed out in previous discussions, the action of a
system is not a meaningful physical observable. The only remnant of
the action relevant to physical observables are the equations of
motion.

The question of validity of the equations of motion is, at least in
some sense, answered by Ehrenfest>s theorem.

The significance of the field propagator or inverse propagator in the
path integral is best understood by considering finite dimensional
examples of Gaussian integrals.

Igor

Jay R. Yablon · Guest

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:f09481e1-8383-442a-8c69-2dcbd4f9d6da@c36g2000prc.googlegroups.com...
[quote]On Nov 16, 3:06 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

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.

Jay, as I>ve pointed out in previous discussions, the action of a
system is not a meaningful physical observable. The only remnant of
the action relevant to physical observables are the equations of
motion.
[/quote]
Igor, if that is the case, then what would we observe in the equation of
motion that could indicate to us whether the action S >> h-bar or action
S ~ h-bar? If there is no observable that would tell us the magnitude
of an action directly or by inference, then aren>t we discussing how
many angels fit on the head of a pin? So let>s focus the question: what
might we observe that would tell us whether the action whether action S
[quote]h-bar or action S ~ h-bar for a photon, or an electron, or any
system? . . .[/quote]

[quote]The question of validity of the equations of motion is, at least in
some sense, answered by Ehrenfest>s theorem.
[/quote]
.. . . Toward that end, I will also note that
http://farside.ph.utexas.edu/teaching/qm/lectures/node31.html concludes
by saying that:

". . .This is known as Ehrenfest>s theorem. When written in terms of
expectation values, this result is independent of whether we are using
the Heisenberg or Schrödinger picture. In contrast, the operator
equation (264) only holds if and are understood to be Heisenberg
dynamical variables. Note that Eq. (265) has no dependence on h-bar. In
fact, it guarantees to us that the centre of a wave-packet always moves
like a classical particle."

See also http://en.wikipedia.org/wiki/Ehrenfest_theorem.

Does this perhaps tell us that the magnitude of the action impacts how
much the the "not centre" of a system moves like, or deviates from, a
classical particle?

[quote]The significance of the field propagator or inverse propagator in the
path integral is best understood by considering finite dimensional
examples of Gaussian integrals.
[/quote]
Yes, I understand that too, and better than I did 18 months ago thanks
to your help. But I>d still like to be able to look at Maxwell>s
equations, or the inverse of Maxwell>s equations (propagator time
current density) and make some direct statement about how the validity
of that equation is affected by the magnitude of the action, and its
seems rather troubling to not be able to answer such a question except
very indirectly.

Thanks for your reply.

Jay.

[quote]
Igor
[/quote]

Jay R. Yablon · Guest

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:f09481e1-8383-442a-8c69-2dcbd4f9d6da@c36g2000prc.googlegroups.com...
[quote]On Nov 16, 3:06 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

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.

Jay, as I>ve pointed out in previous discussions, the action of a
system is not a meaningful physical observable. The only remnant of
the action relevant to physical observables are the equations of
motion.

The question of validity of the equations of motion is, at least in
some sense, answered by Ehrenfest>s theorem.

The significance of the field propagator or inverse propagator in the
path integral is best understood by considering finite dimensional
examples of Gaussian integrals.

Igor

Igor,[/quote]

I already sent one reply, so I do not know which will post first. But
let me consider another line of inquiry.

It seems pretty clear that Maxwell>s equation J^v=d_vF^uv and its
inverse A^v=(Q^-1)^vu J_u will only apply in a classical limit --
however one articulates that limit. Yes?

If Ehrenfest teaches that the center of motion remains classical even
for a non-classical physical system, my question becomes, how much of
Maxwell>s equation and its inverse can we retain even for quantum
systems? For example, can we use equations such as:

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

to relate the expectation value of A^v and J_u? Or even,

A^v=(Q^-1)^vu <J_u> (2)

where we use an expectation value only for the current <J_u>?
Fundamentally, how much of these classical relationships can we retain,
when we move into the quantum world, i.e., what is the best statement
that can be made, in the quantum world, about these classical
relationships?

A different question also arises. The classical *magnetic* equation
written in duality notation reads (I won>t get into how this arises
identically from the Abelian gauge potential or from the geometric fact
that the exterior derivative of an exterior derivative is zero since we
should all know this):

d_v*F^uv = P^v = 0 (3)

What happens to this equation in the quantum world? Do we now say
merely that its is "expected" that there are no magnetic four-currents
P^v but that this can be statistically violated in some instances
because we are only dealing with probabilities and not deterministic
certainties and that P^v is really a <P^v> in the quantum world?
Differently, what is the correspondence principle for magnetic currents?
And, what does this do to the differential geometry of exterior
derivatives? Does that become statistical also?

Thanks,

Jay.

Igor Khavkine · Guest

On Nov 19, 5:47 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
[quote]"Igor Khavkine" <igor...@gmail.com> wrote in message
news:f09481e1-8383-442a-8c69-2dcbd4f9d6da@c36g2000prc.googlegroups.com...

Jay, as I>ve pointed out in previous discussions, the action of a
system is not a meaningful physical observable. The only remnant of
the action relevant to physical observables are the equations of
motion.

Igor, if that is the case, then what would we observe in the equation of
motion that could indicate to us whether the action S >> h-bar or action
S ~ h-bar?
[/quote]
Nothing.

[quote]If there is no observable that would tell us the magnitude
of an action directly or by inference, then aren>t we discussing how
many angels fit on the head of a pin?
[/quote]
Exactly.

[quote]So let>s focus the question: what
might we observe that would tell us whether the action whether action
S >> h-bar or action S ~ h-bar for a photon, or an electron, or any
system? . . .
[/quote]
Nothing.

The reason is very simple. One can always add a constant to the action
without changing the equations of motion.

Igor

Jay R. Yablon · Guest

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:mt2.1-9365-1227237759@argon.astro.indiana.edu...
[quote]On Nov 19, 5:47 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
"Igor Khavkine" <igor...@gmail.com> wrote in message
news:f09481e1-8383-442a-8c69-2dcbd4f9d6da@c36g2000prc.googlegroups.com...

.. . .

So let>s focus the question: what
might we observe that would tell us whether the action whether action
S >> h-bar or action S ~ h-bar for a photon, or an electron, or any
system? . . .

Nothing.

The reason is very simple. One can always add a constant to the action
without changing the equations of motion.

Igor
[/quote]
Would it be fair to say that even if the action is itself not an
observable, it is still an important physical construct insofar as the
inverse of a portion of the action is useful for calculating the
propagator which is the inverse portion of the path integral? And,
because that propagator does contain observables?

Jay.

Igor Khavkine · Guest

On Nov 21, 7:26 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

[quote]Would it be fair to say that even if the action is itself not an
observable, it is still an important physical construct insofar as the
inverse of a portion of the action is useful for calculating the
propagator which is the inverse portion of the path integral? And,
because that propagator does contain observables?
[/quote]
Yes, that much is true, exactly for the reason you gave.

Igor

Jay R. Yablon · Guest

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:mt2.1-9365-1227237759@argon.astro.indiana.edu...
[quote]On Nov 19, 5:47 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
"Igor Khavkine" <igor...@gmail.com> wrote in message
news:f09481e1-8383-442a-8c69-2dcbd4f9d6da@c36g2000prc.googlegroups.com...

Jay, as I>ve pointed out in previous discussions, the action of a
system is not a meaningful physical observable. The only remnant of
the action relevant to physical observables are the equations of
motion.

Igor, if that is the case, then what would we observe in the equation
of
motion that could indicate to us whether the action S >> h-bar or
action
S ~ h-bar?

Nothing.

If there is no observable that would tell us the magnitude
of an action directly or by inference, then aren>t we discussing how
many angels fit on the head of a pin?

Exactly.

So let>s focus the question: what
might we observe that would tell us whether the action whether action
S >> h-bar or action S ~ h-bar for a photon, or an electron, or any
system? . . .

Nothing.

The reason is very simple. One can always add a constant to the action
without changing the equations of motion.

Igor
[/quote]
Igor,

This gets back to my original question, over which I am still somewhat
perplexed, which is what exactly to make of the statement by Zee on the
first page (19) of
http://jayryablon.files.wordpress.com/2008/11/zee-field-equations-and-inverses-markup.pdf
where he says "in the limit where h-bar is much smaller than the
relevant action . . ." we can use the Euler Lagrange equation. But if
one can always "add a constant to the action without changing the
equations of motion" and if asking about how large the action is, is an
unanswerable question akin to medieval religious discourse, then how do
I know when the Euler Lagrange (EL) variational procedure is and is not
applicable?

This seems a slippery slope. Say I have an action that is close to
h-bar so I cannot use EL according to Zee. So, I add a large >>>h-bar
constant to the action, change nothing observable, and now I can apply
EL. Or vice versa, I have what I call a "classical" situation where EL
applies, I subtract a large >>>h-bar constant to obtain and action
~h-bar, nothing observable changes, and now I cannot use EL.

So, is it EVER possible to use EL and so does, e.g., Maxwell>s equations
EVER have a meaning? Either one can never apply EL, one can always
apply EL, or one can sometimes apply EL. And if the answer is
"sometimes," then how do I know if this time is or is not one of those
sometimes? And if the answer is "sometimes": that you can use EL for a
"classical" problem but not for a "non-classical" problem, what
observable objective circumstance tells me that my problem is classical
or is non-classical?

Jay.

Igor Khavkine · Guest

On Nov 23, 3:19 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

[quote]This gets back to my original question, over which I am still somewhat
perplexed, which is what exactly to make of the statement by Zee on the
first page (19) ofhttp://jayryablon.files.wordpress.com/2008/11/zee-field-equations-and...
where he says "in the limit where h-bar is much smaller than the
relevant action . . ." we can use the Euler Lagrange equation. But if
one can always "add a constant to the action without changing the
equations of motion" and if asking about how large the action is, is an
unanswerable question akin to medieval religious discourse, then how do
I know when the Euler Lagrange (EL) variational procedure is and is not
applicable?
[/quote]
Jay, I hope you realize that this is an odd question to ask. It is
like asking whether addition between real numbers still works
depending on the physical situation. The Euler-Lagrange variational
proceduce is a mathematical technique, which is applicable whenever
the relevant mathematical hypotheses are satisfied, independent of the
physical situation.

Do you per chance refer to the equations of motion for the physical
degrees of freedom (which may have been obtained from the
Euler-Lagrange method of minimizing an action functional) when you use
the abbreviation EL? If so, please use the terminology "equations of
motion" or "Euler-Lagrange equations". I know this may be a pedantic
point, but I do not want to be answering the wrong question.

[quote]This seems a slippery slope. Say I have an action that is close to
h-bar so I cannot use EL according to Zee. So, I add a large >>>h-bar
constant to the action, change nothing observable, and now I can apply
EL. Or vice versa, I have what I call a "classical" situation where EL
applies, I subtract a large >>>h-bar constant to obtain and action
~h-bar, nothing observable changes, and now I cannot use EL.
[/quote]
Jay, here you have aptly demonstrated exactly why Zee>s statement, at
face value, is either wrong on over simplification of a different more
precise statement. See below for a more detailed explanation.

[quote]So, is it EVER possible to use EL and so does, e.g., Maxwell>s equations
EVER have a meaning? Either one can never apply EL, one can always
apply EL, or one can sometimes apply EL. And if the answer is
"sometimes," then how do I know if this time is or is not one of those
sometimes? And if the answer is "sometimes": that you can use EL for a
"classical" problem but not for a "non-classical" problem, what
observable objective circumstance tells me that my problem is classical
or is non-classical?
[/quote]
I may have already answered this question in this thread, but here it
is again. The equations of motion (aka Euler-Lagrange equations) are
valid (1) as expectation values (Ehrenfest>s theorem) or (2) as
operator equations (Heisenberg equations of motion).

OK, Zee>s statement, which I>ll paraphrase as "classicality applies when
the relevant action is >> h-bar", is not completely wrong. Instead it is
an oversimplification or physicists' code. It>s a shorthand for one of
the following approximations: (1) the saddle point approximation for
evaluating oscillatory or exponentially damped integrals (including path
integrals); (2) the WKB (aka semiclassical) approximation for solving
the Schroedinger equation. Each of these approximations is applicable
under precise mathematical circumstances. I encourage you study each of
these approxmations to learn the details. You>ll find multiple
references both online, as well as in books on QFT, QM, some EM books,
mathematical methods, PDEs, asymptotic expansions, etc.

Igor


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