Re: The time it takes to emit one photon



Igor Khavkine wrote:

Thank you for acknowledging that my particle-based approach is
equivalent to the traditional field-based aproach.



Before you go giddy with joy, let me point out that both approaches are
taditional, that even the equivalence between them is traditional, and
that you have no priority claim to either of them. The Hilbert space
consisting of (anti)symmetrized wave functions with a variable number of
arguments lies at the very core of second quantization. It is explicitly
used, for example, in F.A. Berezin, _Method of Second Quantization_
(1966), and many other places.


Unfortunately, I do not have this book with me. Does this book have
an explicit expression for the interacting Hamiltonian in the Fock
space through creation
and annihilation operators of real particles, where each term is
finite, and this Hamiltonian can be used for calculations of both the
S-matrix and the time evolution without encountering divergent
integrals? If it does, then I certainly missed a lot in my education.
If it doesn't, then you and I are talking about two different
"particle-based approaches".



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.


Yes it is. The ultraviolet infinities have been eliminated long
before your philosophy or "dressing transformation" existed. Old
news. We've been there.


Feynman-Schwinger-Tomonaga theory "swept infinities under the rug".
True, one can have a completely finite formulation in terms of
Glazek-Wilson "similarity renormalization". However, this approach
requires unphysical "bare particles". The only approach to QFT that
can be formulated from the beginning to the end without encountering a
single divergent integral or bare particles is RQD.



Are you not forgetting something? Previous discussion has made it clear
that you use the same renormalization procedures, that you so deplore,
to construct the coefficients in the Hamiltonian of your theory. This
voids any claims of superiority that your theory can make. I also seem
to recall that this point has been made on half a dozen separate
occasions. Do you intend to repeat the above claim again?


Sure I do, because this claim is correct. You are right that I still
need to use the renormalization procedure in order to write down the
"dressed particle" Hamiltonian. But this doesn't invalidate my claim
that my approach is free of divergences "from the beginning to the end".
The point is to understand where is "the beginning" of a physical
theory. This is not a hairsplitting, this is a very important point.

In my view, the beginning of a realistic theory is when you constructed
the Fock space and defined 10 generators of the Poincare group there.
At this point you completely fixed the physical content of the theory
and interactions acting between particles. From this point you should
be able to get
physical results and predictions by a mechanical application of
mathematical rules. From this point on you do not expect to see any
surprises, like divergences. All calculations must be straightforward.

Everything happening before this point I call pre-theory. I don't care
much which weird speculations or tricks you may employ in the pre-theory
phase. You can just pull the Fock space and 10 generators out of your
pocket. That's fine too.

It is good to recall the major pre-theory steps involved in the
standard QFT:

1. Gauge invariance principle
2. Action and Lagrangian
3. Canonical quantization
4. Noether theorems
5. Renormalization counterterms

After these steps are done, the standard QFT finally has a definition
of the Fock space and 10 Poincare generators there. That's the point
which I mark as the "beginning". Now, let's see what do we have at this
starting point. There are quite a few problems:

a) The Fock space is built out of bare particles rather than real ones
b) The Hamiltonian is infinite (masses and coupling constants are
   infinite)
c) If you exercise extreme care in cancelling all divergences you can
   get good results for the S-matrix in this theory, but
d) you cannot calculate the time evolution. At least, the standard
   formula exp(iHt) doesn't work at all.

Glazek-Wilson approach alleviates problems b) and c), but it doesn't
solve a) and d).

Now, let's see what are the pre-theory step in RQD. I make the same
steps 1.-5. as above (you are right that step 5. involves divergences,
but we haven't reached the beginning of the theory yet, so no harm is done). In addition, I perform one more step:

6. Unitary dressing transformation.

Now I reached the beginning of my theory which has:

A) Fock space built out of real particles
B) Explicit finite expressions for the Hamiltonian and 9 other
   Poincare generators as functions of creation/annihilation
   operators for real particles
C) The S-matrix and all related properties can be calculated
   from here by standard quantum-mechanical formulas without
   encountering any divergences
D) The same is true for calculations of the time evolution by
   means of the standard formula exp(iHt)


Does a particle based approach exist? Yes. Does it always exist? No.
The cases where it fails have already been discussed in the thread
news:TmTje.9720$Db6.6575@okepread05 . 

Could you please remind me when the particle based approach does not
exist? I am not so proficient in newsgroups. How can I reach the
thread you quoted?


What I cannot understand is how the switch is ossured (physically, not
formally) between quantum and classical mechanisms when we simply
change the light intensity (the number of photons) without changing
anything else.



In either case, the underlying description is quantum. One way to obtain
the classical limit is to look at a particular set of states that
reproduce the classical results, through expectation values, to high
videlity. These are so called coherent states.

Let N be the particle number operator. The particle description is
appropriate when its expectation value has a small variance,
<N^2>-<N>^2. When this quantity is large, the field description is more
appropriate. For coherent states <N^2>-<N>^2 ~ <N>. On the other hand,
field intensity is proportional to <N>. So if you change the intensity
from high to low, you are changing <N> from high to low, and hence
changing <N^2>-<N>^2 from high to low. In other words, you are smoothly
going from the (classical) field description to the (classical) particle
description. The underlying quantum formalism does not change.

Now, you got me totally confused.
You are saying that at high <N> photons
are described by classical field (Maxwell's) theory. I disagree with you, because in my view Maxwell's theory belongs to the quantum domain.
This disagreement is fine, at least I understand your point.
Now you said that at low <N> we get a classical particle description.
That I cannot understand at all.
How classical particle mechanics can explain
diffraction and interference? I hope you are not disputing the fact that
even when few photons are present, the diffraction and interference
effects are still there.


and I am not sure that Maxwell's
theory will folow in its entirety as a "classical" limit of the more
general QED.


Maxwell's equations do follow from QED. This has been known (read
demonstrated) since the time of Pauli, Dirac and Fermi. 

Maxwell's equations may follow from QED at a formal superficial level.
However, I doubt that classical electrodynamics is a proper
classical limit of QED. The Pauli-Dirac-Fermi QED predicted
infinite scattering cross-sections. If your statement is correct,
then Maxwell's theory should do the same, which is not true.

I am also not sure that Maxwell's theory is a proper classical limit of
the Tomonaga-Schwinger-Feynman QED. In this theory, the mass and charge
of the electron are infinite, which is not true in Maxwell's
electrodynamics. I doubt that these discrepancies can be somehow
fixed by
taking the classical limit. Apparently, we attach a different meaning to
the word "demonstrated".

Eugene.

.

Relevant Pages

  • Re: How real are the "Virtual" partticles?
    ... where applies is when we consider a theory of free particles. ... the representation of the Poincare group in the Fock space ... Another situation can be met in interacting theory when (wave packets ...
    (sci.physics.research)
  • Re: On the wave function of the photon
    ... My impression is that the answer is no, because I don't see how to accommodate virtual particles in Fock space. ... It's described in textbooks as a way to make a relativistic quantum theory that allows for the creation and annihilation of particles, but all the successful relativistic quantum theories are field theories, which don't have particles in the first place. ... Most QFT textbooks successfully hide this fact by placing too much emphasis on fields. ... It seems more reasonable to represent photons by Wigner's unitary irreducible representations of the Poincare group. ...
    (sci.physics.research)
  • Re: please help my confusion about particles and irreps.
    ... > two-fold cover of the Poincare group. ... fundamental systems ("elementary particles or whatever you want to call ... Infraparticles are non-trivial representations which can't ... In the Fock space one can define ...
    (sci.physics.research)
  • Re: On the wave function of the photon
    ... How is Fock space actually related to QFT, ... annihilation operators on a given element G of Fock space, ... For the creation operator b^+, you form all products permuting the ... > creation and annihilation of particles, ...
    (sci.physics.research)
  • Re: Download a new book on quantum mechanics and relativity.
    ... >>I think that physics is a science about measurements. ... >> physical system consisting of one or more particles. ... The Hamiltonian I am using has infinite number of terms with clear ... Hamiltonian by the dressing transformation. ...
    (sci.physics.relativity)
Quantcast