Re: The time it takes to emit one photon

On 2005-08-21, Eugene Stefanovich <eugenev@xxxxxxxxxxxx> wrote:
>
>
> 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?

This formula is a direct reversal of the procedure of second
quantization, which can be checked by direct calculation. It is
mentioned not only in Weinberg, but in other books as well, mostly in
more old fashioned treatments of second quantization. Dirac is a
prominent example. Also, this formula is used in disguise in the Green
function or correlation function formalism. What I challenge is your
introduction of unnecessary steps into the equivalence, such as
"dressing". QFT is applicable to many theories, some of which don't
require renormalization.

> 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.

This is an important, but perhaps subtle difference.

> > 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.

My comment was regarding the construction of the classical limit. I've
outlined this construction several times previously. If a citation is
not sufficient, you're welcome to ask specific questions. As to book
thumping, one would have to establish the credibility of the author
before doing so.

> 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.

> ?? 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)?

The E(x,t) and B(x,t) fields play a *single* role, to determine the
force on a test charge at any point in space and time. The fact that
they carry energy and momentum (radiation) stems from their equations of
motion and their coupling to matter. Radiation and the Lorentz force law
are two manifestations of the same phenomenon.

> 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.

Yes, there are and yes they can. But Feynman's double slit example does
not describe a quantum effect here. What we call interference fringes
are a generic phenomenon common to all linear wave equations. This
includes both Schroedinger's and Maxwell's equations. So what's quantum
about it? A state with one electron is described by a wave function (of
one argument) obeying the Schroedinger equation. When we consider the
hbar -> 0 limit, only a single particle trajectory remains. The single
argument of the wave function becomes a dynamical variable of the
particle and no interference is seen. This limit can be reversed by
simultaneous consideration of many trajectories for the same particle
and by assigning phases to these trajectories. This is Feynman's
description of the double slit experiment, it shows how to recover the
wave function and the interference fringes with it.

For the sake of argument, suppose we want to apply the same treatment to
a state with a single photon. Everything is fine, the same argument
applies, with a few differences. We have to use Maxwell's equations
instead of Schroedinger's, and we have to use a multi-component wave
function. There will be one distinct kind of photon for each independent
polarization. Each independent polarization of the wave function will
describe the probability amplitude for the corresponding photon
particle. This implies that the values of the photon wave function are
not the same as the electric and magnetic field (or rather the vector
potential) amplitudes.

But that is not what we see when it comes to light. In most situations,
the photon number is very large. This implies that we have to use the
classical hbar -> 0 limit. But unlike the electron case, we do not
recover trajectories, rather we recover the electric and magnetic
fields. The big surprize is that they satisfy the same Maxwell equations
as the single photon wave function! Coincidence? No, it is a consequence
of second quantization. Note however, that the single particle (photon)
equation is always linear, while the classical field equation may be
non-linear. Any field non-linearities get translated into into
interactions when multi-particle (multi-photon) wave functions are
considered. So, simply because Maxwell's equations are linear, we
automatically get interference fringes completely within the classical
regime.

But what if we increas hbar, do we get any actual quantum effects? The
answer is yes, but the detection is more subtle. When we increase hbar,
we now have to consider multiple *field configurations* at the same time
and assign phases to each of them (cf. "multiple particle trajectories
at the same time"). One example of this is a cavity containing a QED
state of the form a|1 photon> + b|2 photons>. Each state |n photons>
(more or less) describes a classical field mode, with the number n
parametrizing the amplitude of the classical field. Depending on the
geometry of the cavity, the classical field mode may already exhibit
interference fringes. But at the same time, quantum mechanically the
state is in a superposition of two different classical field
configurations (different intensities). Wherever we see superposition,
we will see interference. However, in this case, it will not be as
visual as in the electron double slit experiment. So, yes, there are two
kinds of detectable interference here.

>> 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.

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.

Igor

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