Re: Albert Einstein

From: Gregory L. Hansen (glhansen_at_steel.ucs.indiana.edu)
Date: 02/08/05 

Date: Tue, 8 Feb 2005 03:41:27 +0000 (UTC)

In article <MOPNd.44084$K7.1630@fe2.news.blueyonder.co.uk>,
Kevin Aylward <salesEXTRACT@anasoft.co.uk> wrote:
>Gregory L. Hansen wrote:
>> In article x3NNd.43361$K7.33499@fe2.news.blueyonder.co.uk,

[...]
>> Your description gave no hint that
>> an additional set of fields are needed, which is not the usual
>> formulation of QM.
>
>What additional set of fields did you have in mind?

http://www.quantum.univie.ac.at/zeilinger/philosop.html

"An opposite position is assumed by the causal or ensemble interpretation
[...]
present work I am referring mainly to Bohm's position. According to Bohm,
the wave-function supplies an additional potential - the
quantum-potential, as he called it. This potential, when inserted into the
Hamilton-Jacobi-equation of classical physics, leads to well determined
trajectories of the individual particles."

[...]
>> It's classical particles following classical trajectories determined
>> by a new kind of classical field.
>
>Nope. Nine. No. No. No.
>
>As I said, you don't understand what the ensemble interpretation
>actually is at all. I know where your coming from, there is classical
>ensemble approach used in the past, but the ensemble described by
>Ballentine, Einstein, has *absolutely* *nothing* to do with that
>version. It certainly does *not* propose any trajectories for particles
>in the slightest. You must be confusing the ensemble with some other
>approach, like Bohmian mechanics.

So it would seem.

>
>The ensemble approach simple says, essentially, that the state vector
>does not apply to an individual system. That's it. Trajectories are just
>as undefined in the ensemble approach as they are in the standard
>approach.

But didn't you just say that the uncertainty principle reflects ignorance
rather than indeterminacy? That an electron really doesn't "sample" an
extended region of a crystal?

[...]
>> You have that backwards. QED was derived from Maxwell's equations.
>
>Maxwell's equations were used as a guide. QED contains more information
>then Maxwell's equations. Sure, QED was *motivated* by Maxwell's
>Equations as a tool, just as many equations are motivated by incorrect
>ideas. What you are claiming is essentially, that the shrodinger
>equation was derived from the Borh model of the hydrogen atom, which of
>course, fails on other atoms.

In a typical textbook derivation we start out by deriving the Dirac
equation for a free particle, and add the electromagnetic interaction by
converting the partial derivative to a gauge covariant derivative. But
where did that gauge covariant derivative come from? It's the canonical
momentum from the Mawell Lagrangian. Not a quantum Maxwell Lagrangian,
there's only one. Not something similar to a Maxwell Lagrangian, not a
corrected version. It's straight out of classical field theory
with momentum in the position representation. But it does successfully
hide the role of Maxwell's equations from the student.

The approach not usually taken is that shown e.g. in Peskin & Schroeder
for the Klein-Gordon equation. They find the conjugate momentum density,
the wave equation, the conserved current, etc. It's really not classical
or quantum theory, it's just field theory, but it's the same wave
equation treated classically in Goldstein. By page 20 they Fourier
transform the field. Still not quantum.

They have an aside on the harmonic oscillator where the field and
momentum are put in terms of creation and annihilation operators. Still
not quantum. It's an interesting problem to solve the classical harmonic
oscillator by that approach. They write the SHO Hamiltonian as w(a^dag a
+ 1/2). That is sort of quantum; the 1/2 doesn't appear in the classical
problem because the classical a and a^dag commute.

Then they define the SHO state |n> = (a^dag)^n |0>. And define the
Klein-Gordon fields and momentum densities as operators with their own
a_p and a_p^dag.

NOW it is truly quantum. The equation of motion still looks exactly the
same, it's quantum because they defined |n> and the action of the field
operators on it.

I haven't seen a similar approach taken with the electromagnetic field,
but every approach takes a Lagrangian density or a canonical momentum
straight from the classical theory. From there, it is quantized by
working on the fields, not on the interaction. Maxwell's equations
weren't a motivation or a guide, they were a postulate.

By the way, from Newtonian mechanics,

  E = p^2/2m + V(x)

Let E->i*hbar*d/dt, p->-i*hbar*d/dx, right-multiply both sides by a
psi(x), there's the undergrad Schroedinger's equation, essentially a
restatement of Newton's second law in a mechanics where the state is in a
Hilbert space rather than a phase space. The Klein-Gordon equation comes
the same way, from E^2 = p^2 + m^2.

Everybody thinks that as soon as you go quantum, everything classical goes
out the window. But it's still there.

[...]
>Look mate, do Maxwell's Equations, as is, explain the photo electric
>effect and black body radiation, or not?
>
>Nothing you say changes these facts.

No, not as written, with the E and B taken as classical fields.

We're probably arguing some semantic point here. But I don't do "as
written". E.g. when I say "Maxwell's equations" I'm perfectly happy to
go with equations written in the four-vector formalism. As far as I'm
concerned, the Maxwell Lagrangian is as good as Maxwell's equations
because you get them when you minimize the action. Same with the Maxwell
Hamiltonian; find the Poisson bracket or the commutator of the field with
the Hamiltonian. All of that and more are just different representations
of the same physical content. When I talk about Maxwell's equations I
don't mean a specific set of equations, but the interaction described by
those equations.

And, well, darnit, the canonical momentum comes straight from the Maxwell
Lagrangian, which comes from Maxwell's equations. So when you say
Maxwell's equations are wrong, but then QED is derived with that canonical
momentum, I say no. QED is just using Maxwell's equations in a different
way. It's the field that is quantized, not the interaction.

>> You'd never know that after taking a course in quantum field theory.
>> Very little of that was probably covered in the classical course,
>
>It was covered in my "advanced" mechanics course, e.g. H. Goldstein.

We did a lot with action-angle variables. I still don't know why. I've
come to feel that the graduate classical courses should essentially be
prep for QFT, but that's not really how it was approached at my school. 

-- 
"Outside the camp you shall have a place set aside to be used as a 
latrine.  You shall keep a trowel in your equipment and with it, when you 
go outside to ease nature, you shall first dig a hole and afterward cover 
up your excrement." -- Deuteronomy 23:13-14

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