Re: The time it takes to emit one photon



Igor Khavkine wrote:

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."


Notice the important adjective "variable". If I fix the number of
particles to N, my Hilbert space is composed of wave functions of N
arguments, psi(x_1,...,x_N). If I allow the number of particles to vary,
my Hilbert space is composed of linear combinations of wave functions
with different numbers of arguments, 1, psi(x), phi(x,y), chi(x,y,z),
...

Many, but fixed, number of particles is not the same as a variable
number of particles. It is only when the number of particles is allowed
to vary that the theory can be made equivalent to a field theory (Fock
space + field operators). The equivalence is through second
quantization. 

So far so good. The only thing is that quantum fields are not necessary
for describing the systems with a variable number of particles in the
Fock space. Such a description can be formulated entirely in the
language of "composite" wavefunctions, where each fixed-particle-number
function 1, psi(x), phi(x,y), chi(x,y,z) enters with its own
coefficient, and the sum of squares of all such coefficients is 1.

Such a description is not fundamentally different from the particle
description in ordinary (fixed particle number) quantum mechanics.
The only difference is that the particle number is not fixed.
In my view, this minor difference does not warrant the complete
change of the paradigm suggested by the field theory.

In my view, quantum fields are fine if you interpret them as
formal technical constructs that aid your calculations.
They are not fine when you start to consider them as
"basic ingredients of the universe" and particles as
"bundles of energy and momentum of the fields" (Weinberg's words).

This is not a purely philosophical debate. Particle picture is
essential to make the "dressing transformation" in QFT and to
eliminate "bare particles" and "ultraviolet infinities" for good.



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.



Again, trajectories arize in the classical limit of QM with a *fixed*
number of particles. Photon number is not conserved. When that happens,
the classical limit does not yield trajectories, it yields fields.


I thought that in a classical theory with a variable number of
particles there should be trajectories that can start and terminate
at some points. I don't see how you can jump from a variable
(but still finite) number of degrees of freedom in the particle theory
to the
plain infinite number of degrees of freedom in the field.

The number of particles (including photons) in the Universe is finite.
Field theories seem to disregard this important fact. They use infinite
number of degrees of freedom to describe even one electron
with its field.


 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.


Take |psi> to be a several electron state. <x,y,...|psi> = psi(x,y,...)
is the corresponding several electron wave function. X = <psi|x|psi>, Y
= <psi|y|psi>, ..., are the "classical" expectation values of the
individual position operators x, y, .... The wave function psi(x,y,...)
satisfies the multi-electron Schroedinger equation. The expectation
values X, Y, ... satisfy Hamilton's equations of motion, this is
Ehrenfest's theorem.

Take |phi> to be a several photon state. <0|e(x)e(y)...|phi> =
phi(x,y,...) is the corresponding several photon wave function, with
e(x), e(y), ... being the field operators (which are also decorated with
polarization indices). E(x) = <psi|e(x)|psi> is the expectation value of
the classical field amplitude. The wave function phi(x,y,...) satisfies
the multi-photon "Maxwell equations". The expectation values E(x)
satisfy Maxwell's equations, in the usual sense of the term, which is
also a consequence of Ehrenfest's theorem. The fact that the single
photon wave equation is the same as the linear part of the classical
field equations is a theorem of second quantization.

Do I understand you right? Are you saying that Maxwell's theory can be
applied to the weak-field regime? I don't think so.
Take the Young's double-slit experiment. Maxwell's wave theory describes
the light intensity on the screen by continuous functions E(x) and
B(x). This is all fine while the intensity of light is high: there are
many photons, and the light intensity appears continuous on the screen.
At low intensities, when we can distinguish individual
photons on the screen, the field description doesn't work anymore.
The light intensity produced by one photon is more like a
delta-function. One can reconsile these two contradicting
descriptions in the tradition
of quantum mechanics. One can say that E(x) and B(x) are "sort of"
photon wave functions, and when the photon reaches the screen these
wavefunctions collapse to produce a single observable dot.
This is my interpretation of Maxwell's theory: the fields E(x) and B(x)
there are some surrogates of multi-photon wavefunctions that remained
after we took the (incomplete) classical limit from QED to the theory in
which electrons are treated classically, while photons (due to their zero mass) are treated in a "sort of" quantum way.

 Eugene.

.