About what is electrodynamics?
Dirk
as I said in one of my last postings I want to open up a more refined
topic:
"About what is electrodynamics?"
Here I would like to invite you to disucss about what an
electrodynamics theory should be.
Point-sources and fields, extended charges and fields, only
point-particles or only fields?
It would be great if you explain your physical ideas of an
electrodynamic theory without any need to be mathematically profound.
This way we might be able to agree on one or more certain ontologies
that we should investigate in depth.
I hope you guys have more sun that I do. It's cold like sh... in
Munich!
All the best,
Dirk
Miguel Ballesteros
Hi Dirk and Alejandro.
First, Dirk, thank you very much for your question about my plans in Munich. I think that is will be organized al the end of my PhD, now I have no plans, firs I have to do some good work with mi supervisor.
About electrodynamics I have no answer but some ideas.
The discussion between pint particles or extended ones is already done and I think that we all want to keep the idea of point particles. I want to say only one last comment:
the idea of rigid body is off course unacceptable. Alejandro and me where thinking in a different concept, I think that the idea that we had about a rigid body was an object that looks like rigid body only in his system of reference and the shape of this thing changes buy changing the reference system. The idea I had to introduce extended bodies was something wit a charge density or a probability in the charge (not Dirac delta) with a behavior in congruence with relativity (the shape of the density changes whit the coordinate system).
After the discussion, I think that point particles is a good beginning. So point particles and fields is the most intuitive idea to me. I would like to think that the particles is what we see, I think that they have a clear and concrete interpretation. The interpretation of the fields (instead of particles) is very obscure to me. So why do we have to think in fields instead of particles? Because there are some advantages to use fields. One that that I think is important is the following:
If we want to quantize, then we need some function that represents the probability of the free particle, that's what that free Shroedinger equation do. But the problem is the causality, one can change this equation and one can think that it "looks" like a relativistic equation (I am thinking about the Dirac and Klein-Gordon equations) but the big problem (the causality) apparently can't be solved with this approach. I think that it is a very hard problem to put together the causality wit the Shroedinger approach (I believe that it is maybe impossible). The fields solve that problem, both the Klein-Gordon and Dirac fields. I think that if we don't have another idea to put together causality and the uncertainly principle this is a strong reason to think about fields and fields. Nevertheless I really want to work with point particles and fields because is much more natural to me, so I hope we could have some different idea to overcome the problem of causality after the quantization. I know that right now we are in classical mechanics, so take this only as a comment.
I have a question to both. Where exactly is that the point particle appears in quantum fields?, I know from the work of Dirk that the shape of the divergences looks like the same, but it is seen after calculating the eigenvalues, I would like to know when that concept appears form the basic physical principles of the theory. The same I would like to know in quantum mechanics; of course in quantum mechanics the point source appears in the potential (something very strange because the point sources are classical ones, with no uncertainly principle), but is it the only place where that concept appears?.
The whether in Mexico city is very nice, warm all the time, the problem we have is the pollution. Also the problem we have now is the repression form to government to the people of Oaxaca, who don't want to have the corrupt, killer and fascists govern that they have. I hope you don't have this kinds of problems there.
Regards,
Miguel.
Alejandro
This topic about what ¨is¨ or ¨should be¨ a theory of
electrodynamical interaction is very interesting. Moreover, i think it
is fundamental if we want to go further since it seems the roots of QFT
problems go deep even up to this point.
I always give some thought to this kind of problems about what
¨understanding¨ physics means. So here are some ideas in that
direction.
Before i tell you my (very) modest ideas, i wanted to give you the
reference of a very good paper i found:
arXiv:physics/0603149 v1.
There, the author exposes some ideas of Einstein who, apparently, did
not like his theory of General Relativity (!) because it was only about
the field and did not tell you anything about matter. So, following
Einstein, QFT would be just nonsense.
What i think is the following:
First, it seems that, as a first approach, treating charges or currents
(or mass in GR) which are the sources of the fields independently from
the fields themselves is pragmatically usefull. This is: one can
measure the forces involved in (classical) electromagnetic
interactions; then define the fields at one point as being
(proportional to) the force that a (test) charge or current would feel
because of the presence of a fixed charge distribution (CD). So, once
the field is defined in this way, you go back to the experimentaly
obtained laws and you get the equations which determine the fields from
the CD, i.e., Maxwell equations. This has shown to be, as we discussed
before, a perfect approach to understand innumerable physical
situations, most of them (i don´t dare to say everyone, but...)
involving fixed CDs (that is, situations in which we know from the
begining the dynamics of charges and/or currents that generate the
fields) and we want to find the field in some other part of space,
other than the part where this CD is. A very simple example: a point
charge being at some fixed point p of space. Then the corresponding
electric field is defined everywhere but in p which the position of the
charge. Following the argument i was giving, this is perfectly fine
because the field tells us how this charge would interact with another
charge or charge density, so we do not need to know the ¨field¨ at p,
another charge would never be in the very same point as the first one.
(So, field based self interactions??).
One last comment about this: suppose you have a CD and you find the
field by solving Maxwell equations. Suppose that you then perform an
experiment to actually measure the electric and/or magnetic field. You
usually do this by putting some other known ¨patern¨ or test charge
or current in the field and, knowing how the interaction with the field
is (eg. Lorentz force, etc.), by watching the dynamics of this patern
charge, you measure the field. Every experiment of this kind i did as
an undergraduate in the lab yield as (spectacular) result that the
theoretical calculations where right, even when this calculations
include huge simplifications. But one simplification which was implicit
in every calculation is that the field generated by the patern or test
cd is small enough to not be taken into account. Although, actually,
this patern cd do generates a field which interact with the original CD
and thus altering the field you wanted to measure in the first place.
Still you get good results. Why is this so? it seems that the theory is
faithfull with nature´s behaviour in the sense that there do are small
fields that can be neglected.
Posed in another way, if you replace your patern or test charge
distribution by other one which is also ¨small¨ you get the same or
similar result. Since all these results are similar (or ones better
than the others as the charge gets smaller) you can speak about
measuring ¨the¨ field generated by the original CD. So these fields
are actual physical entities in the same sense as one says that forces
in Newtonian mechanics are.
But: as in Dirk´s paper, if we want to use this ¨currents (direct
sum) fields¨ approach in situations where you have more than one
charge distribution interacting with each other through Lorentz law,
you get in trouble. The complication is predictable: the field
generated by the particle i modifies the dynamics of the particle j,
then modifying the field j which afects the particle i and then the
field i.
I couldn´t get into the core of the ideas behind Wheeler-Feynman
electrodynamics; but, speaking from a very abstract (and losse) point
of view, it seem that this Maxwell-Lorentz= ¨sources (direct sum)
fields¨ way of thinking is not very suitable for describing
interaction dynamics. This ¨direct sum¨ phylosofy is directly
reflected in the fact that you have two coupled set of equations: one
for the fields and one for the sources.
Maybe, one could think of a more adapted approach to the problem which
unifyies or ¨glue¨ in some sense what in the usual approach we
understood separatelly as charges and fields. Right now, I have no idea
how to do this. But one thing which is certain, whatever model one
proposes, it should be empirically constructed by considering
experiments in which the effects of different sources interacting with
one another are important.
Aside: it seems that Einstein tried to construct a unificating theory
for both, fields and matter, but failed. So: big challenge (and little
hope, ja!).
One last thing about ¨understanding¨: i would like to use classical
Newtonian mechanics as a toy model to ilustrate my idea. When you first
see Newton´s laws, you notice that they involve the motion of point
particles. The second law ma=F blames the Force F, which we don´t know
what it is, for the motion of this particle. But Newton was a genious:
he added two other laws which specify the nature of this F and, also,
the theory tells you how, by performing experiments interpreting points
as giving the position of a body in space, you can ¨measure¨ the
typical forces involved in some kind of interaction (eg. gravitational
interaction and force) and predict the result of new experiments;
capturing the escence of nature´s behaviour. Also, you can relate in
principle different kind of facts: an apple falling and a planet
orbiting the sun!
One can ask, how does an extended body move? Newton´s laws has no
answer. You can think of an extended body as a ¨limit¨ of an infinite
number of particles and put a mathematical background for this limiting
procedure (Riemman sums, etc), but you cannot be certain that this will
actually model nature until you make predictions and perform the
corresponding experiments. As we know, this limit-way-of-thinking leads
to correct predictions. So one can add this limiting idea as a
principle: extended bodies behave like systems of many Newtonian
particles. This fact is not usually stated as a principle, but i think
it is important to see it in this way and not as a ¨theoretical
derivation¨ from Newton´s Laws because it could well had been the
case (before the experimental verification) that extended bodies do not
move as predicted.
So one can say that ¨understands¨ extended bodies motion because: one
¨understands¨ point particle Newtonian mechanics and also
¨understands¨ the above (abstract) limiting procedure. Note that the
above is purelly conceptual: if you ¨really¨ want to think that an
extended body is made of many particles (say atoms), then you find that
the laws for mechanics of atoms are different from Newton´s and you
would get into a deep trouble: how to explain macroscopic mechanics
from quantum microscopic one. So the many point particles picture has
this bitter non-physical flavour but still is a way of
¨understanding¨.
Finally, my point is that, at least sometimes, understanding something
(not only mathematically, but conceptually) might consist of modeling
it with some understood pieces (eg. point particle mechanics) and
introducing a new understandable ingredient (eg. taking the infinite
particles limit) in order to adapt the whole to the desired new case.
This is what we do when we model gasses or solids as being many atoms
interacting and take thermodynamic limits and do statistics. In these
cases, another important consequence of this ¨understanding¨ is that
it allows you to make predictions about very different kind of
experiments: you have a microscopic model for temperature, preasure,
termal and electric conductivity, etcs.
This last feature is something that you certainly do not have in QFT,
all the theory is adapted to accelerator experiments and to obtain the
S matrix, which is like the only thing we hope to know about a black
box.
Well, this reply its getting lengthy, i stop here because, otherwise,
you will get too bored (if you are not too bored now!). The above is
some phylosophical nonsense, but may be of some use for our pourposses.
So, summing up, i am open to a source-field unifying vision of
electrodynamics.
Miguel, i think that the point particle interpretation in QFT enters
because the fields (as the potential you mentioned in Schodinger
equation) depend on each point in spacetime and not in a region. So,
interactions are produced pointwise and then particles (which are
responsible for this interactions) are thought to be points.
Unfortunatelly, that political situation you described in Mexico seems
to be common to all Latin America, including Argentina. I always
implicitly hope things will get better, but saddly reality shows that
they are going to get even worse.
Anyway, i´m very happy to be able to discuss these things with both of
you! Best, Alejandro.
> 2006/11/1, Dirk <dirk.deck...@gmx.de>:
>
>
>
>
>
> > Dear Friends,
>
> > as I said in one of my last postings I want to open up a more refined
> > topic:
>
> > "About what is electrodynamics?"
>
> > Here I would like to invite you to disucss about what an
> > electrodynamics theory should be.
> > Point-sources and fields, extended charges and fields, only
> > point-particles or only fields?
>
> > It would be great if you explain your physical ideas of an
> > electrodynamic theory without any need to be mathematically profound.
> > This way we might be able to agree on one or more certain ontologies
> > that we should investigate in depth.
>
> > I hope you guys have more sun that I do. It's cold like sh... in
> > Munich!
> > All the best,
>
> > Dirk- Hide quoted text -- Show quoted text -
Dirk
I am very sorry that I took so much time answering this time. We had
been very busy at the department here. The time before Christmas always
get crazy...
>only one last comment:
>the idea of rigid body is off course unacceptable. Alejandro and me where >thinking in a different concept, I think that the idea that we had about a >rigid body was an object that looks like rigid body only in his system of >reference and the shape of this thing changes buy changing the reference >system. The idea I had to introduce extended bodies was something wit a >charge density or a probability in the charge (not Dirac delta) with a >behavior in congruence with relativity (the shape of the density changes >whit the coordinate system).
Please forgive me another last comment. The way I used the word rigid
was not in the sense: "rigid in every Lorentz frame" but "rigid only in
the frame which is instantaneously at rest" - in other frames it'll
deform according to the Lorentz transformation. My point was that even
those bodies being rigid in the latter sense violate relativity.
>point particles and fields is the most intuitive idea to me. I would like >to think that the particles is what we see, I think that they have a clear >and concrete interpretation. The interpretation of the fields (instead of
I agree.
>particles) is very obscure to me. So why do we have to think in fields >instead of particles? Because there are some advantages to use fields.
Just as a remark. There is one reasonable way to interpret features of
a field as particles, which is to give the field the meaning of a
mass-density (like one wanted to think about the wave function in QM -
if I remember it correctly the first one who advertised this idea was
Schroedinger himself). One could now look for soliton like solutions
and interpret their peaks as particle positions. There was a time in
which people worked hard on this idea on the mathematical side.
>One that that I think is important is the following:
>If we want to quantize, then we need some function that represents the >probability of the free particle, that's what that free Shroedinger >equation do. But the problem is the causality, one can change this equation >and one can think that it "looks" like a relativistic equation (I am >thinking about the Dirac and Klein-Gordon equations) but the big problem
I don't quite get what you mean by that. First it would help to know
about what notion of causality you are talking. Physicists are
unfortunately very confused about that in quantum mechanics. This is
indeed a very important point to clarify! Every relativistic wave
equation, e.g. Maxwell, Klein-Gordon, Dirac, etc... has solutions
running backwards in time. We decide, recall your undergrad. lecture
about classical electrodynamics, that from the solutions to these
equations we choose to only take the retarded ones to be the physical
one because we have some idea of causality. In clearer words, the
physical laws are almost all time-symmetric, and we break this symmetry
by a choice at the very fundament of the theory. If we take a look at
statistical mechanics, there the causality comes from the physics not
from the physicists by the special initial conditions, which lead to
the second law of thermodynamics. Wheeler-Feynman Electrodynamics is
for example completely time-symmetric, but the symmetry is broken by
the special type of initial conditions of our universe - and therefore
is till today the only theory which explains the irreversible effect of
radiation. Do we really need to put in "causality" in the equations of
motion of the fundamental physical laws? Taking a look at statistical
mechanics I would say no. For special initial conditions the causality
we experience emerges in a thermodynamic limit even of completely
time-symmetric laws, like Newton's equation.
>(the causality) apparently can't be solved with this approach. I think that >it is a very hard problem to put together the causality wit the Shroedinger >approach (I believe that it is maybe impossible). The fields solve that >problem, both the Klein-Gordon and Dirac fields. I think that if we don't
>have another idea to put together causality and the uncertainly principle >this is a strong reason to think about fields and fields. Nevertheless I
The uncertainty principle is a corollary that pops out of the
definitions of the operators x and p and the generalized Fourier
transform. This corollary tells us how to relate ignorance of the
random variable x to the one of p - not more. One has to be careful
with physical conclusions for the momentum because p is not the real
momentum of some particle shortly before measurement of p at time t.
Instead it is only the asymptotical momentum, which is the "real"
momentum one particle would have if at time t the interaction would be
switched off and we would wait asymptotically long, t->infinity. If I
am too brief let's discuss this topic separately.
>I have a question to both. Where exactly is that the point particle >appears in quantum fields?, I know from the work of Dirk that the shape of >the divergences looks like the same, but it is seen after calculating the >eigenvalues, I would like to know when that concept appears form the basic >physical principles of the theory. The same I would like to know in quantum >mechanics; of course in quantum mechanics the point source appears in the >potential (something very strange because the point sources are classical >ones, with no uncertainly principle), but is it the only place where that >concept appears?.
One can show that any field theory on the Fock space is equivalent to a
multi-particle theory of a varying total number of particles (e.g.
Schwabl, quantum mechanics II). Otherwise consider the example:
Take psi(x) to be the annihilation field operator of some scalar field
that shall describe the particles, then an interaction with an external
potential is modelled by H_I=integral(dx Psi^\dagger(x)V(x)psi(x),
where V(x) is some nice function of x. Take a generalized n-particle
state |x1,...,xn>=const_n psi\dagger(x1)... psi\dagger(xn) and compute
H_I|x1,...,xn>=sumover(i)of(V(x_i)|x1,...,x2>). Now usually the V(x)
generated by sources x1,...xn in QED is not a nice function of x but
has divergences exactly where the x1,...,xn are.
In quantum mechanics with external potential like the computation of
the hydrogen atom the coulomb potential of the proton is classical
because we fix the protons position in space to the origin => meaning
the proton position is classical: deltax deltap = 0. If the proton
wasn't treated classically of course the field would have some
uncertainty, too. On the other hand, the second quantization of the
fields is just a book keeping tool, where many fields are treated
simultaneously for different configuration of sources (suppose e.g. you
have a state which is a superposition of one particle with its field
and two particles with their field). The Fock space so to say keeps
track of the fields of all possible configurations of sources.
------
In the end I like to emphasize that before we move on we have to
clarify all this physical folklore like uncertainty relation, first and
second quantization, etc... These might seem dull questions and every
physicist seems to have understood them but in fact these are the
building blocks of the physical picture. And in fact I doubt that many
physicists using something like the Copenhagen, Decoherent history or
Many-worlds interpretation have a clear picture of them.
I hope you can understand what I mean in my answers because I wrote in
a kind of a hurry.
All the best,
Dirk
Dirk
Alejandro, I want to add some comments to your idea:
1) Are you addressing a geometrization of the electromagnetic forces
like what has been done for the gravitation?
It is true that gravitation is another problem but I try to leave my
hands off other interactions than the electromagnetic ones because
intrinsically the same problem we discussed so far remains but is
shifted to different locations within the theory. I have only very
basic understanding of GRT but I would expect if you solve an
interacting two particle system in GRT you'll get the same divergences
not in the force but in the equation of motion of your metric of
space-time. I might be wrong since I never thought too much about it. I
only expect that physicist so far only considered GRT space-time of
some big (external) stars that curves space-time and one test-particle
moving there on a geodesic. Correct me if I am wrong.
2) You said we never have to evaluate these fields you describe at the
position of the source?
You might have the same in mind that young Feynman had when he first
got in contact with Wheeler. If you would e.g. take Maxwell-Lorentz
dynamics and exclude the evaluation of the field at the position of its
source, meaning in the sum over the source terms you omit the i=j term,
then the theory will not describe radiation damping anymore.
Wheeler-Feynman Electrodynamics solves this problem by introducing the
advanced fields (the solutions to the Maxwell equations physicists
usually forget), which act backwards in time on the source to account
to radiation damping.
3) Today there are mathematical proofs for Newton's mechanics that
extended body can be treated like point particles as long they maintain
a certain distance with respect to each other. I worry not too much
about building things like tables and chairs out of point particles as
long as it is possible to describe the interactions of these particles
in a mathematical well-defined way. In the end let's only consider a
theory which describes electron and photons and analyze it on the scale
from typical QED scattering up to the electron flying in a classical TV
tube.
4) You mentioned Newton's theory of gravitation. There can solve the
coupled problem of e.g. two stars orbiting each other but there you
also neglect the effect of radiation damping because you omit the
self-force. You never evaluate the force law on the position of the
source of the gravitational field.
Alejandro, could you give a concrete example of two particles, maybe in
the Newtonian setup, of what you have in mind. I mean not with exact
computations or solutions but, say, some roughly defining equations of
motions? Then we could discuss your idea more concrete. Don't worry
about similarity to qed for now, the toy model should just have the
basic features you described.
All the best and in the future I promise I won't let you wait so long
again,
Dirk
Miguel Ballesteros
Miguel Ballesteros
Dirk - André Deckert
Dear friends,
I answer to Miguel’s comments:
>What I mean about causality is that if we start with some function with compact support ( the particles are localized in some region) and we evolve the particles with the Klein Gordon or Dirac >free equation, then after any time, the function evolved won't have compact support. That means that some particles traveled faster than the speed of light. I know that the stationary phase will travel like the >classical
I would think the question, since we have these relativistic wave equations at hand, is what their solutions mean physically. A mass density like discussed earlier would be ruled out regarding such equations of motion since parts of the matter move faster than light as Miguel says. I am a bit confused because on the one hand I know that the classical Klein-Gordon equations has a Greens function vanishing everywhere outside the light-cone for mass equal zero (Maxwell equation) and this property on a plainer cone for bigger masses but on the other hand I remember from classes that this is not the case for the quantum mechanical Dirac or Klein-Gordon propagator. Do you happen to know why the quantum propagator loses this property? Only by imposing the commutation relations?
Embarrassingly I never did the computation for the Dirac propagator before. I’ll catch up on it for our further discussion. Usually physicists argue in terms of commutators. They vanish in qft outside the light-cone when every particle has its anti-particle. But that is obviously not satisfactory for a one particle theory.
From my Bohmian point of view I regard the solution to a wave equation as a generator of a velocity field which guides particles on their trajectories. Psi and even the velocity field are only mathematical entities. How their support behave is not so important to physics since there is no interpretation for them. However the velocities should behave in a physically reasonable way such that the particles do.
Another point is, why should we not allow single particles to move faster than light as long as the interpretation of the things we see does not violate special relativity. Feynman had a long time the picture in mind that e.g. an anti-particle is a particle that moves back-wards in time. On the other hand why should such a simple symmetry like the one of special relativity hold only for complicated macroscopic objects?
You see, I am very confused about these things. Maybe we could focus on that for a while?!
>About the uncertainty principle I have another opinion than Dirk's. I know that it is a corollary of the definitions of momentum and position by the Fourier transform. My perspective is the contrary: the >Fourier transform should be a "corollary" of the uncertainty principle. If the uncertainty principle wouldn't exists, then the Fourier transform wouldn't work.
>…
>There is something that I don't understand about what you said on the interpretation of momentum. You said that the momentum given by the wave function is not the physical one so we need to turn the >interaction off and wait to have the real momentum given by the wave function, but if we turn the interaction off and we wait, the particles will separate as free particles.
What I mean is the following. Take two operators A,B on L^2. Say both operators have a continuous set of generalized eigenfunctions {f_a}_a resp. {g_b}_b. So using Dirac notation
A|f_a>=a|f_a> and B|g_b>=b|g_b>
Then one may define a generalized Fourier transform on the intersection of the subspace spanned by {f_a}_a and the one spanned by {g_b} by
<g_b|psi>=int(da <g_b|f_a> <f_a|psi>)
Together with its inverse
<f_a|psi>=int(db <f_a|g_b> <g_b|psi>)
An expectation value of an operator shall be denoted by <A>=<psi|A|psi>:=int(da a|<psi|f_a>|^2) where psi is some Hilbertspace element in a suitable domain. Let <A>=<B>=0 for simplicity then
<AB> <= sqrt(<A^2><B^2>)
by Schwartz. This can be rewritten to
|<AB-BA>| = |<[A,B]>| <= |<AB>|+|<BA>| <= 2 sqrt(<A^2><B^2>)
if one prefers. Now x and p are operators like A and B described above so for A=x and B=p we yield
dx dp = <x^2><p^2> >= 1/4 |<[x,p]>|=1/4
which is a mathematical relation connecting the variance of one operator with the one of another but nothing more.
Now what is relevant to ask is what physical entities do x and p encode? If these are not the position and momentum at an instantaneous time then why should one expect [x,p]=0?
Say x is the actual position then what is p?
Let us give the probability of the asymptotic (t->infinity) velocity of a particle being in some set C of velocities.
lim(t->infinity)P_psi(x/t in A) = lim(t->infinity)P_psi(x in tA) = lim(t->infinity)<chi_tA(x/t)>
here chi_A(x) is the indicator function 1 for x in A and zero otherwise.
…= lim(t->infinity)int(d^3x |psi(x)|^2 chi_tA(x/t))
But asymptotically (large t and no potential) psi(x) behaves like 1/(it)^(3/2) exp(I x^2/2t) (F psi)(x/t,0), where
psi(x,0) is the initial value for the solution to the Schroedinger equation and F the standard Fourier transform. So
…= lim(t->infinity)int(d^3x/t^3 |(Fpsi)(x/t,0)|^2 chi_tA(x/t))
= int(d^3k |(Fpsi)(k,0)| chi_A(k))
The random variable x/t converges in probability to a random variable, say v_inf, which is |Fpsi|^2 distributed. Again, this is the asymptotic velocity of a particle. Not the instantaneous one! Transforming v_inf back to position space by Fourier transformation yields p=-i m d/dx.
Therefore one should not regard [x,p]<>0 as surprising. It is only a relation between the position of a particle and its asymptotic velocity. The instantaneous position and velocity will however commute like e.g. in Bohmian Mechanics.
>I really enjoy this discussion, and I find is very constructive to me. I learn with you a lot of new things. It would be great that we could keep the interest and enthusiasm for a long time despite of the delayed >answers.
The same holds for me. Better we take our time in writing up our arguments than spoiling it with corrections of them. To me the modus we have at the moment is fast enough. It’s sufficient to be productive, i.e. to learn things from each other, and not so time consuming that one has to delay his primary work.
So keep the spirit up and talk to you soon,
Dirk
Miguel Ballesteros
I red carefully the last mail of Dirk, I think that we should clarify
the problem of causality and and relativistic quantum mechanics. What
I think is that the solutions of the Klein-Gordon Equation delocalize
the support of functions when the evolution is applied, but I don't
have a proof right now. Also I hear from a man in the ICMP that it may
be impossible to put together the logic in quantum mechanic (Hilbert
spaces, orthogonal Projections and so on) with causality. I also know
that causality is solved in QFT with the use of some commutations
relations that actually use again que klein-Gordon and Dirac
equations. I am a bit confused in this aspect I suggest to clarify
this before going on, what do you think?.
I am afraid that this mail is very short, and it is because right now
and the (last 2 weeks) I have a lot of work to do. I am solving one
problem that involves algebraic topology and differential geometry and
I am not an expert on that areas, so I have to spend a lot of extra
time to learn such areas. So I still need some time to give a better
answer.
Sorry for the delay. Nevertheless, I still keep the spirit up and I
hope you too.
Best,
Miguel Ballesteros.
2007/1/16, Dirk - André Deckert <dirk.d...@mathematik.uni-muenchen.de>:
Alejandro
I am back at work now, after some vacations. In the time between my
last posting and this, i was thinking of a way to answer Dirk´s
request about giving a concrete example about the "generalized
interaction picture" i had in mind, may be in the simple case of two
particles.
What i had in mind was some vague idea towards describing the
interaction without using "intermediary" fields. Maybe, writing some
kind of functional involving the particle´s positions whose minimal
configuration gave the physical trajectories. I was quite sure that
one could give, at least, an effective model may be only valid in some
special regime like slow motions, etc, which avoids the use of fields
and gives electromagnetic interaction another descriptive perspective.
But, honestly, i could not go any further towards something less vague
and more explicit.
Luckily for me, i am not the first (nor the brightest, je!) who wanted
to go in this direccion. I realized about this fact when i was finally
able to download Wheeler-Feynman´s paper about absorvers giving the
explanation for radiation damping effect (i was guided to this paper
by Dirk´s Trieste talk slides and ideas). Indeed, it seems that many
people went in this non-intermediaries direction, starting from Newton
and Gauss themselves and ending with the action-at-distance theory for
describing the electromagnetic interaction. This is a concrete theory
which goes in the direction that i naively wanted to go. I am still
far from understanding it, but i´m working on it. Also, i found many
recent papers on the classical relativistic interaction between two
charged particles, it seems to be a nontrivial subject!
There are some ideas present in the introduction of W-F´s paper that i
think are interesting for us:
-One is the idea that both Maxwell-field theory and action-at-distance
theory contribute to the understanding of the EM interaction. We are
used (or at least, i do) to the way of thinking in which one and only
one theory or description must be "the right one". But W-F explicitly
say that they consider that this two descriptions are complementary,
because one gives better explanations of phenomena that look more
obscure in the other description. To be precise, in Maxwell-field
description, radiation damping is not a clear subject (at least, that
´s what they say). Dirac gave a "hand trick" derivation of the correct
potential but this does not give a satisfactory physical explanation,
but just its quantitative description of it in terms of fields.
Lorentz proposed a model in which the electron was not a point but has
a finite length (this also goes in the direction of our previous
discussion on point vs extended particles!) but the theory runs
mathematically complicated and still (consequently) does not give a
good explanation. So in this paper, W-F give a physical explanation
for the radiation damping phenomena, which was a true open problem for
theoretical physics because it is an experimentally well-tested
phenomena without explanation at that time, using the ingredients of
action-at-distance description of em interaction.
So, here, two different conceptual frameworks or descriptions of EM
interaction are used "simultaneously" to "cover" the whole variety of
EM phenomena. At some points of this "phenomena variety" one
description gives better descriptive tools than the other. An
important fact, which comes from the work of Feynman and others, is
that the two are shown to agree or be equivalent in the "patching
intersection": i.e. that when a system is describable by both, this
two descriptions are equivalent (in some definite sense) and yield the
same predictions. I shall upload other papers i found in which they
propose an unifiying picture for classical EM.
I found another recent paper (i shall upload it) in which they use
elements in QED theory, like the interchange of virtual photons, to
enlighten Classical EM. This is certainly a way in which we wouldn´t
like to go, since from the very beginning we don´t like QFT techniques
and, more preciselly, is "virtual space like photon interchange" a
good physical explanation of some phenomena or better than, say,
action-at-distance theory explanations? it seems that this paper
focuses more on calculations and math derivations of formulas than in
concepts.
-The other ideas are less precise and more involved; and, since this
paper is from the 40´s, i do not know if someone did something with
them.
The ones i find more interesting are the ones related with the
prescindibility/"not being enough aspect" of fields and about the way
one looks at "interaction" itself. For example, they quote a guy named
Tetrode (among others) who seems to have introduced the concept that
"radiation" is not to be taken as a physical process by itself; but
only as a part of a transference between a source and an absorver. A
body like an electron which is alone in the universe would never
radiate, it radiates "because" there is an absorver. This is a key
ingredient in W-F´s solution to the radiation damping problem.
In that way of thought, I dare to do the following suggestion for
quantum interaction description, just to see what turns out of it. May
be the same happens with EM and other interactions, in the sense that
we are, "may be" again, giving fields and intermediary particles a
conceptually fundamental role that they do not deserve. So one could
try a description in which they are not so fundamental. For example, a
process in which an electron emits a photon is "nothing" by itself, it
aquires sense only one other particle absorves it, this is the new
"quanta" for the interaction description: the whole emission-absortion
process which, in turn, could be describable in terms of other
elements than fields. As in classical EM, this would give just another
description of the same phenomena, but to be of some use, it should
enlight (at least one of) the dark aspects of QFT. I do not know how
to even begin to construct such a theory. I found a paper (which i
shall upload) which gives an expression for a Hamiltonian underlying
action-at-distance theory, the lack of this Hamiltonian seems to be
one of the reasons for the fact that W-F theory did not go Quantum at
its time (40´s?).
Another interesting quote on W-F´s paper is that Tetrode believed that
quantum EM interactions should be described by forces more complicated
than those given by electromagnetic (field) theory. May be the
problems in QFT arise from trying to impose a wrong kind of
description for the interactions.
Coming back to the concrete example issue, and again luckily for me, i
found a recent paper (which i shall upload) giving a model for two
charged classical particles which encodes action-at-distance and a
geometrical perspective too! this is far more than i could have done.
Nevertheless, this paper goes in the direction of trying to integrate
a Dirac equation for two body interaction which is not what we wanted
to deal with, so i think it is not of much use.
I would like to create a new discussion item for us to put there
references we found and seem interesting and also comments about them,
what do you think?
Finally, i would like (if you agree) to propose that we focus on some
concrete subject, may be a simple one to start. This might seem
strategically wrong, since we still haven´t defined the directions we
would like to take in our discussions, but may be it has the advantage
of putting the quantitative discussion-mechanisms to work and general
ideas shall arise later more easily. So my proposal is to do that in a
separate discussion item and continue with this one if you want. Since
Dirk is working in a close area (what are you working on now Miguel?),
may be you Dirk can propose something for us to discuss, may be some
concrete qft calculation with dark interpretation or whatever you
want.
Well, i hope as always that we keep the spirit up on this discussion!
Best wishes for both, Alejandro.
PD: do you know about any conference where we can physically converge?
PD2: about the causality problem in Dirac´s equation, i do not know
the details, but my point of view is this: Dirac equation is very good
in some aspects but has problems in others; so may be one is not to
take very seriously some delicate aspect conclusion as the probability
density interpretation conflicting with causality.
Dirk - André Deckert
I'm still in a hurry. I just finished renovating my old apartment because I
moved out and today is the first day back at work again. Nevertheless I
can't hesitate to make some remarks.
First, I like very much what you have written, Alejandro, and I am looking
forward to read the papers you have mentioned. There is a long list of names
involved in WF starting with Gauss continuing with Fokker, Tetrode, Schild,
Driver, Narlika, etc... where we could also find good hints for our ideas.
Second, the absorber theory is something we have to fully understand. I find
it amazing that one can write down a fully time-symmetric equation like the
WF law and by the choice of special initial conditions (the absorber) one
can derive an effective theory (Lorentz-Dirac force law) where it seems that
only the retarded solutions play a role (in fact the Lorentz-Dirac EM theory
is again time symmetric but the retarded fields seem to play an more
important role). In a sense this is similar to what people have done in
statistical mechanics - the microcanonical laws are ruled by time symmetric
classical mechanics while in the thermodynamics limit (many particles and
special initial conditions) this time symmetry is broken as the entropy
always increases.
I have a lot faith in a WF like interaction even there is no quantum version
of it which is basically due to the fact that there are no trajectories in
quantum mechanics (unless one uses Bohmian mechanics). But that's a
different topic.
Third, we could start looking at the only analytical examples one has found
for WF - the Schild solutions (phys. Rev. 131, 2762, 1963). These are
basically trajectories of two particles orbiting each other. We may try to
do something similar for a two-particle wave function, i.e. finding an
interaction term for some Dirac/Klein-Gordon type evolution equation.
Forth, I've looked at what Miguel said more closely. If you start with a
wave function of compact support at some time t then this support spreads
ruled by the Klein-Gordon/Dirac wave equation not faster than the speed of
light - the slower the bigger the mass is. You can read this behaviour of
the Green's function for this type of initial conditions. It is only the
Schroedinger equation (i.e. the Laplace) that immediately spreads the
support up to infinity - but Schrodinger's equation was never meant to be
relativistic invariant.
Ok, I'll need a bit more time and will write something proper soon. As we
said before, no one of us has enough time at the moment to contribute very
frequently but as long as we keep the spirit up it's all good :) I haven't
forgotten about our discussion it was just a lot happening here.
By the way has anyone invited the guys that have asked for an invitation to
the group?
All the best and till soon,
Dirk
-----Ursprüngliche Nachricht-----
Von: qft-dis...@googlegroups.com
[mailto:qft-dis...@googlegroups.com] Im Auftrag von Alejandro
Gesendet: Dienstag, 27. März 2007 01:50
An: QFT discussion
Betreff: Re: About what is electrodynamics?
Miguel Ballesteros
It is good to hear from you again.
First:
I think that we have a miss understanding about causality and I would
like to clarify that before go further.
This is what I meant when I said that relativistic Quantum mechanics
(Dirac equation) violates the causality principle:
Suppose we use a continuous positive function H = H(p) (p = momentum)
as a Hamiltonian. And we define in some reasonable way (no matters
how) a position
operator. We start with some state localized in some sub-region B of
the three dimensional Euclidean space. Suppose that B is compact.
suppose that s > 0 is any number, and we choose some open set B' as
far as we want from B. Then there is a time t (0 < t < s) such that
the probability of finding the particle at time t in B' is different
from cero. This theorem has been proved in the theorem 1.6, page 29 of
"Thaller, Bernd The Dirac equation. Texts and Monographs in Physics.
Springer-Verlag, Berlin, 1992."
Obviously this theorem does not apply to the Dirac and Klein-Gordon
free Hamiltonians, because their are not positive. Nevertheless; if we
restrict our selves to states of positive energy, the above theorem
applies (Section 1.8.4 of the same Book). This Oviously means that
there is the possibility for the particles to travel as fast as we
want (Section 1.8.2 of the same Book).
In quantum fields the concept of causality is different because it is
constructed from some commutator and it means that the particles are
not correlated if they are causally disconnected, there the green
function plays a important role; maybe thats what you meant Dirk.
Nevertheless I do not understand quite well why is that causality
can't be solved in relativistic Quantum mechanics and it can be solved
in QFT. I mean what's the basic conceptual difference that allows one
to have causality. Could you make some comments in this direction?.
About the people that want to belong to our discussion I don't know
anything. I have no problem if you want to let him go in.
Now I answer the question of Alejandro about my work.
Right now I am working with the Schroedinger equation in exterior
domains. The problem I have is an inverse problem that allows us to
recover the magnetic and electric potentials from the high energy
limit of the Scattering operator (I also give error bounds in terms of
energy (velocity)). Also I can recover some fluxes in the complement
of the exterior domain: the obstacle (Aharonov-Bohm effect). Before
this I studied the same problem with the Klein-Gordon equation.
Basically I use functional Analysis and mathematical physics. The way
I do research is in the spirit of a mathematical physicist. I think
that I am more a mathematician than physicist although I did both
bachelors.
It is a pleasure to talk to you again. Even though we left a lot of
time the discussion group, I keep the spirit high. I understand that
all of us have a lot of things to do and it is difficult for us to
take a lot of time in this discussion. I also understand that we
(Alejandro and I) are not working right now in QFT. But I believe that
this discussion group is very useful (at least for me) and maybe in
the future it can bring to us some recompense in our work. At least I
am sure that all of we have the same interests, so we should keep the
spirit high.
Best whishes for holydays (if you have any),
Miguel Ballesteros.
007/3/29, Dirk - André Deckert <dirk.d...@gmx.de>: