Field quantization, plane wave and normal oscillator
il...@abv.bg
Hi!
I have a problem with the energy of the plane wave as used in the
method of second quantization of a field (but prior to quantization).
In the method of second quantization a scalar field (or any other) ‘f’
is represented as a sum (integral) over plane waves with all possible
wave vectors k (and –k). Than it is shown using Fourier
transformations that the energy of the field is equal to the energy of
the normal oscillators.
I am wondering how could this be applied to the energy of a field
composed by just one plane wave. Acos(wt+kx).
1. I think that the energy of the plane wave is infinite.
Each plane wave can be replaced by an infinite system of oscillators
which are bound to each other. If we imagine that the wave was
created in one point in the infinite past from an oscillator which has
delevered energy to his neighbours, it apparently has to consume
continuously energy from a source in order to set in motion the
infinite chain of oscillators.
2. Now this plane wave is represented in the quantization procedure as
one oscillator (in k-space) with that same frequency ‘w’. Than the
energy of the field (the lone plane wave in the case) should be equal
to the energy of that oscillator.
But it is finite.
I hope someone who understands deep the process of second quantization
does know how to explain this.
Than you in advance.
Ilian
Cl.Massé
news:62a26941-7604-490a...@x9g2000yqk.googlegroups.com...
Just take an infinitesimal value for the amplitude of each plane wave. In
the case of a localized field, the spectrum is continuous, so that the
amplitude is associated to an infinitesimal interval of k, which amounts to
the same.
--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability.
Peter
Let me just remark, that this sounds like you are going to treat the
spatial and temporal components differently (as in the normal mode
quantization), a severe drawback, cf. W. P. Schleich, Quantum Optics
in Phase Space, Berlin: Wiley-VCH 2000, Ch.10
Good luck,
Peter
il...@abv.bg
Yes, maybe.
But i am reading QFT (Zuber) and this what is in.
Does this problem not stand in Schleich, Quantum Optics?
Ilian
il...@abv.bg
> <il...@abv.bg> a écrit dans le message denews:62a26941-7604-490a...@x9g2000yqk.googlegroups.com...
Why should I take infinitesimal value for A.
I am considering just one wave and the amplitude can be chosen as one
wishes.
If I consider EM field there are solutions for the vector-potential
with arbitrary amplitudes.
Then in free space there can exist just one wave of arbitrary
amplitude. So in the case I am envisioning the amplitude is given and
fixed.
Ilian
Cl.Massé
> Why should I take infinitesimal value for A.
> I am considering just one wave and the amplitude can be chosen as one
> wishes.
> If I consider EM field there are solutions for the vector-potential
> with arbitrary amplitudes.
> Then in free space there can exist just one wave of arbitrary
> amplitude. So in the case I am envisioning the amplitude is given and
> fixed.
You mean "one *plane* wave". It has an infinite energy, and so isn't
physical, even if it is a solution of the wave equation. In a closed
system, you'll also see that a wave with an energy greater than a certain
value can't exist. In physics, the wave equation isn't the only constraint,
other equation must be fulfilled.
il...@abv.bg
> <il...@abv.bg> a écrit dans le message denews:020dea4e-3d92-42d6...@j8g2000yql.googlegroups.com...
Thank you very much for the considerations. At least they show to me
that I am not completely wrong to find a problem which is not
indicated in any textbook.
Maybe you are right about the shape of the wavefunction in QM - there
is a requirement for it to be going to zero at infinity - in fact
integral of f.f_* must be 1 - for one particle or N for N particles.
But when we have a free of sources equation the number of particles
associated with the wave is indetermined.
What I am concerned about is:
1.Math
The theorem about oscillators doesn’t work in a case (one plane wave)
– so it is wrong at all
2.Physics
>From the plane waves (which are unphysical) the concept of QED
photons is directly introduced. So the particles are implanted in the
theory on an unphysical basis.
For me this is a very strong drawback for the physical meaning of QED.
Regards: Ilian
Oh No
mathematical justification for the usual method of quantisation which
arbitrarily replaces scalar quantities with operator quantities. One
either simply does it, as a purely mechanical procedure ignoring the
kind of issue which you raise, or one must go to an entirely different
way of approaching quantisation.
I refer you to
http://www.teleconnection.info/rqg/RelativisticQuantumTheory
http://www.teleconnection.info/rqg/QuantumElectrodynamics
The method for second quantisation (which is also described in Scharf,
Finite Quantum Electrodynamics) is as follows:
After introducing quantum theory for single particles, construct the
Fock space for an indefinite number of particles. Define creation and
annihilation operators on Fock space. Define the field operators
in terms of creation and annihilation operators.
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)
il...@abv.bg
>
> The method for second quantisation (which is also described in Scharf,
> Finite Quantum Electrodynamics) is as follows:
>
> After introducing quantum theory for single particles, construct the
> Fock space for an indefinite number of particles. Define creation and
> annihilation operators on Fock space. Define the field operators
> in terms of creation and annihilation operators.
>
> Regards
>
> --
> Charles Francis
> moderator sci.physics.foundations.
> charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
> braces)
>
> http://www.teleconnection.info/rqg/MainIndex
Thank you very much for this reply.
I have posted that question on spr too but there is absolutely no
answer more than a week fom now.
Are you sure am not running in an elementary mathematical error? I can
not believe that such a fundamental theory may rest on unjustified
mathematical and physical basis.
Hadn't anyone pointed this issue out 60 years ago? I think it was Born
who had introduced second quantization then.
Can not anyone point this out and then when writing books on QED to
explain that the procedure is justified just by coincidence. Do you
know some book which says it?
Ilian
Oh No
>On 13 ___, 13:15, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
>> Thus spake il...@abv.bg
>>
>>
>>
>>
lectures when I first came across it as an undergraduate, because it
signalled the abandonment of mathematics. The attitude of the lecturer,
who told us that if we tried to think about it we would fail our exams,
was the clincher. The people who actually took the course seemed to be
those whose didn't care about whether it was right, but only that the
exam questions were easy, so that is what you also find when you ask
people who you think should know better. Eventually I realised it was
what I wanted to study, so I did it my own way during, and since, my
doctoral thesis, by studying foundations, not by studying the way it is
usually approached.
You do find a minority who approach it more sensibly. From that point of
view I would recommend you approach qm from its axioms, e.g. Jauch
Foundations of Quantum Mechanics, Von Neumann Mathematical foundations
of Quantum Mechanics.
Scharf, Finite Quantum Electrodynamics, is the only mathematical book of
qed of which I know. It's not an easy book, and not perfect, but it
undermines a huge amount of what is usually done in qft. I hope too you
may find my website useful.
Unfortunately, for practical purposes, you have to know what everyone
else does, so there is no way out but to study the standard books too.
You will find there are some who are aware that a more mathematical
approach is possible, and who probably think that this makes
quantisation ok, but mostly there is a kind of conspiracy of silence.