Re: The time it takes to emit one photon



Igor Khavkine wrote:

And the best way to see it explicitly is to use the
"dressing transformation" which eliminates bare and virtual particles and
reduces QFT
to a theory of real particles interacting at a distance.


Non sequitor. The way to see it is psi(x,y,..) = <psi|phi(x)phi(y)...|0>. 

1. I am not sure how your formula challenges what I wrote about the
"dressing transformation". Could you please elaborate?

2. I am not sure where you get this definition of the wave function.
Most likely from Weinberg's eqs. (14.1.4) - (14.1.5), however he
doesn't provide any justification that thus defined psi(x,y,..) is
the probability density amplitude for measuring particle positions
at x,y,...

3. I have a different definition for the position-space wave function
in the Fock space. This involves

a) construction of single particle
operators in each N-particle subspace H_N using the tensor product theorem
(see subsection 8.1.1 in my book). In the case of position-space wave
functions, the Newton-Wigner position operators r_1, r_2, ...
should be used.

b) construction of the common basis of eigenstates of the operators
r_1, r_2, ... in H_N.

c) Repeating step 2) for each N (see subsection 9.1.1, after eq. (9.9))

d) taking the scalar product of the state vector |psi> with the basis
 vectors constructed above.


I see a logical gap in your statement. I think if I take a classical
limit of a quantum theory with variable (but finite) number of
particles, I should obtain a clasical theory with variable (but
finite) number of particles.  Nobody has constructed such a theory (as
far as I know), but this is not a reason to believe that such a theory
cannot exist.



What you see is not a logical gap, but something that you wish were
true. However, once you put wishful thinking and what you think should
be aside, you'll see that the limit of a quantum theory with finitely
many particles (wave functions of as many arguments) has classical
particle dynamics as the classical limit, while quantum field theory
(Fock space with field operators) has classical field theory as the
classical limit.



Now, I fail to see what is the difference between "quantum theory with
finitely many particles (wave functions of as many arguments)" and
"quantum field theory (Fock space with field operators)". I though we
agreed that they are equivalent. Let me remind you:

I wrote: "I agree completely: QFT is equivalent to a quantum theory of a
 variable number of particles."

You wrote: "So far so good."

> This has been known for 70+ years. And when I say "known", I don't
> mean
> in the sense of folklore. The calculations are there for anyone to
> see,
> check any book on QM or QFT.

If you want to substitute discussion by pointing to books,
then I would like to draw your attention to my book
physics/0504062 which tells a different story.



So, in my view, Maxwell's theory is, actually, a hybrid in which
massive charges are treated classically while quantum behavior of
billions of photons is approximated by 2 vector functions E(x,t) and
B(x,t). Maxwell fields are just approximatons (quite successful, I
admit) to multiphoton wavefunctions.


Yes, they are approximations in the classical limit, hbar -> 0. 

I thought that if we take the limit hbar -> 0, then undeterministic
wave functions are replaced by trajectories. In this limit, photons
should be described in terms of Newtonian light rays.
No diffraction, no interference.



In addition to photons, Maxwell
lumped
interparticle forces (Coulomb and Biot-Savart) into his E(x,t) and B(x,t).


Non sequitur. 

?? Do you dispute the fact that in Maxwell's theory fields
E(x,t) and B(x,t) play two roles:

1) they describe the intensity distribution
in the radiation field.
2) they determine the forces acting on charged particles
 (via the Lorentz force law)?


In the approximation that leads from QED to Maxwell's field theory, the
limit
hbar -> 0 does not play any significant role.



So the fact that hbar does not appear in any classical formulas is a
coincidence? How about the fact that hbar's value is so small in
macroscopic (say cgs) units? Rhetirical questions aside, what we see
around us every day is described by the hbar -> 0 limit of the quantum
theories that we believe to be fundamental. This simple observation
gives the hbar -> 0 a very significant role and it's name "the classical
limit".


I think that hbar -> 0 limit does not apply to the description of
light in Maxwell's theory. If you think otherwise, then you should
come to the conclusion that there are two different sources of
the interference effect. One source is purely quantum (as in Feynman's
double-slit experiment), another source is due to "classical waves"
(as in Young's experiment). Can these two contributions to the
interference be distinguished experimentally? I don't think so.
In my view, interference is a quantum effect, and Young observed this
quantum effect 100 years before Planck's quantum theory.

In the true classical limit hbar -> 0 all wave properties of photons
should disappear.

It is not difficult to explain why hbar never appeared in
Maxwell's electrodynamics. Take for example, Einstein's formula
connecting photon energy E with the frequency v of its wavefunction

E = hv

In Maxwell's theory the wavefunctions of a large collection of
photons are modeled, of course, by the fields E(x,t) and B(x,t).
Quantity v is macroscopic and is a part of Maxwell's theory.
However E is a microscopic quantity, and individual photons where
not observed in the 19th century. So, E is not needed in
Maxwell's theory, and neither is h.


True. Large number of photons => hbar -> 0 => Maxwell field theory. 

I agree about "Large number of photons".  I disagree about "hbar ->
0". I think in this limit you should obtain Newton's corpuscular ray
optics, i.e., photons moving along trajectories.



Again, belief is no substitute for calculation. See Sakurai's _Advanced
Quantum Mechanics_. There he explicitly relates the strong field limit
(many photons) to the classical limit (hbar -> 0). Unfortunately, I
don't have the book handy, so I can't give a more precise reference.


Thanks for the reference. I'll check that out. It looks suspicious to me
that in the weak field limit (when individual photons can be discerned)
Maxwell's theory gives continuous predictions incompatible with
experiment. This forces me to believe that Maxwell's fields are
some surrogates for multi-photon wavefunctions, rather than their proper
hbar -> 0 limits.

Eugene.


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