About UV divergences
Alejandro
to qft for some (long) time.
It seems that the problem with the k going to infinity (for example in
page 24 of ICTP talk) is to localize the nucleons at certain points in
space. This is, it comes because of treating nucleons as point
particles.
But: is this compatible with the quantum nature of the nucleons? isn´t
it reasonable to suppose that there is a density (not a delta one)
giving the wavefunction or distribution of the nucleon in space? then k
will enter as a length parameter for the volume or charge of this
nucleon, as Dirk says later.
What happens with self interactions when the particle is extended and
not punctual?
why is this k not to be included in the theory? (if this is the case)
Dirk - Andre Deckert
First something organizational: Miguel, do you use the google group, too? If
so, I will mail just to the group and not to you separately in the future.
Ok, now to Alejandro's posting:
It could well be that there are no point-like particles at all and one has
to discuss the advantages and disadvantages of using point-particles or
extended charge distributions. Certainly true, on the scale of today's
observations we know that most of the nucleons have a structure like
neutrons and protons -> quarks-> ???
Let me summarize the point of that talk:
The pro's for the extended charge distributions are:
1) it cures the UV problem
2) ???
The contra's are:
1) If it is assumed to be rigid, it violates relativity, as I sketched in
the Appendix.
2) If it is non-rigid then one has an additional arbitrariness in the
theory because the shape and size of the distribution fundamentally enters
in the interaction terms. One way out is to propose a relativistically
invariant law that rules the shape and size in space-time. Maybe some QCD
law that describes how the building blocks of the nucleons hold together.
But then how does one write down this theory? With a point-particle
interaction or again with an extended charge distribution of these building
block? With point we run into a new UV problem, with extended charge
distributions we don't know the shape and size... and we are back to where
we started.
The contra's for point-particles are:
1) The ultraviolet-divergences.
The pro's are:
1) Perfectly relativistically invariant objects.
2) No additional arbitraryness.
I don't want to say that the only possible way to construct an interaction
theory is one about point-particles. I rather mean that for now, using
points, is the simplest approach because points are the most-natural
relativistic objects. Every other object must be chosen carefully from the
representation of the Poicaré group, i.e. Tensorsfields, Spinorfields,
etc... and they are much more complicated to handle not even mentioning that
we would not have a clue how to write down a law that rules there shape and
size correctly.
On the other hand I think it is necessary to have a point-particle theory
even though many objects that we at the moment have in mind may not be
point-like. E.g. comparing the estimates size of the electron (or even the
quarks) with the one of a point, one would guess that a mathematical theory
describing the interaction between these objects properly should be such
robust that it survives taking the limit from the electron radius to a point
without ending up in a mess. Let it is done in classical electromagnetism
the extended charge distribution theory emerges from the point-particle one
by smearing out the points to balls of finite radius. The interaction
principle stays the same.
>> why is this k not to be included in the theory? (if this is the case)
Do you mean, why is k left arbitrary? Then I hope what I said above answers
the question. If you don't mean that please explain the question again to
me.
!!!
Indeed, that question, whether to use points, or distributions or something
else should be one of our first questions to answer before we proceed. For
my part I argue that a point-particle theory if it exists (and I strongly
believe in this from the work of Wheeler and Feynman in the
action-at-a-distance theory) is a good "point" to start with if we aim at a
fundamental understanding of relativistic interaction.
I would be interested what kind of positions you have with respect to
points, distributions, etc... and of course what you think about what I have
just said.
Cheers,
Dirk
Miguel Ballesteros
Hi Dirk and Alexandro.
I sorry for the delay, I haven't answer the mail because right now I
have a lot of work to do; nevertheless, I certainly will find the time
to go further on our discussion, it is important to me.
Dirk, yes I use the google group, so we can write there.
I red your conference in ICTP, it is really interesting. I don't
think that pure particles exists, they are just a way for us to
understand some classical phenomena and they works well in the
macroscopic world (far away from the point particle sources). I also
think that the way in that the coulomb law and lorentz's force were
invented was looking at experiments were the test particles were far
away from the sources and there the test particles did not interact
with their own field. Certainly I that that the point particle point of
view is the simplest one and the most natural one, if we can create a
model with point particles that works it would be the best option, we
don't have to believe that the point particles really exist, they are
only a abstract concept that allow us to predict some phenomena, just
as it does in classical mechanics. If we want to model the nature whit
point particles, we should keep the spirit or the first experiments, it
means that we analyze the behavior of test particles in a
electromagnetic field, there the test particles NEVER interact with
their own field. If we want to create a theory we should keep that in
mind.
However, the Dirac approach of a effective mass and field gibes good
results in QED, I mean renormalization, why? I don't know, maybe it
is true that the interaction changes with the scale and the particles
do interact with their own field.
I have to add another comment, actually there do exist rigid bodies
macroscopic, and they do move in this relativistic world so it should
be possible to define a relativistic rigid body theory with no
contradiction.
One last question, I have a friend that is interested in this
discussion group, he is physicist, and his interest the quantum gravity
he knows about string theory, quantum loop gravity, non commutative
geometry, and all that kin of thinks. Maybe his interests are far from
ours, but maybe it is interesting to have different opinions. So do you
want he to join us?
Regards,
Miguel
Alejandro
Alejandro
I had some problems trying to reply, actually, i replied and the
message was lost somehow. So here i go again.
Miguel, it´s ok by me if someone wants to join the group.
What i said in the lost message was this:
Dirk, you did answer to my questions.
I agree that we should focus in principle on poinlike particles.
I don´t think, though, that the arbitrariness of the parameter k is a
problem, as long as the theory tells you how to measure it.
Relativistic rigid bodies don´t seem to be a very fundamental obstacle
either.
But taking into account the shape of the nucleon would get us into lot
of thecnical difficulties which (might be) non essential.
It seems that Wheeler Feynman´s approach to relativistic interaction
is a good one. Also, as Miguel said, it is not really surprising that
Maxwell-Lorentz is not good, since Lorentz force corresponds to
experiments where the field generated by the (test) particle is not
taken into account.
So by WF the classical source of uv divergences is cured. It would be
interesting to try to push WF ideas to the quantum theory. Dirk, did
you do something in this direction? which are the main ingredients that
a WF-like theory of quantum interaction should have?
Well, hope that i can discuss more frecuently,
Best, Alejandro.
Dirk
Everybody who is really interested in our topics and constructively
takes part in the discussion may of course join and is invited to do
so. However we must keep in mind that nowadays it is hard to find
people that have been educated in field theory, string theory or any
kind of modern theory of gravitation, who are still able to realize the
primitive problems one is confronted when writing down a simple
relativistic interaction theory - classically or quantum mechanically.
At least that way my experience.
>I red your conference in ICTP, it is really interesting. I don't
>think that pure particles exists, they are just a way for us to
>understand some classical phenomena and they works well in the
>macroscopic world (far away from the point particle sources). I also
Only to avoid a misunderstanding. Point particles are not only heavily
in use when talking classical theory but also QFT. That of course does
not effect the question if they are the right concept.
Please write me your opinion about the following thoughts:
(1) The method of second quantization is equivalent with a
multi-particle theory allowing for creation and annihilation of
particles. Meaning: QED is a multi-particle theory, and if you take the
Lagrangian seriously these particles interact point-wise.
(2) The electron is, if not a point, extremely small - even on the
scale of atoms. We are talking 10^-15m as predicted by classical theory
and know that it
must be at least three magnitudes smaller than that from accelerator
experiments. So on the scales of atoms or bigger objects any theory
describing the
electromagnetic interaction I would expect to mathematically survive
the point-particle limit.
(3) Where is the border line between quantum and classical? Consider
an electron evapurating out of the kathode in a TV monitor tube, then
being accelerated and guided by electromagnetic fields until hitting
the TV screen. All perfectly described by classical theory, although
the elctron is "quantum mechanically" small. The texbook answer goes
like this: "The electron interacts so often with molecules in the air
and thus so many degrees of freeedom are involved. This is clearly a
macroscopic, i.e. classical, problem." Though I never computed such a
scenario, my physical common sense, comparing the electron radius and
velocity with the free way length in a gas close to vaccuum, would make
me feel that in a 20cm TV tube there is almost no interaction with the
gas molecules at all.
>think that the way in that the coulomb law and lorentz's force were
>invented was looking at experiments were the test particles were far
>away from the sources and there the test particles did not interact
>with their own field.
That was certainly the case and indeed was the most natural way to
approach electrodynmics. First asking, what kind of fields are
generated by known source trajectories? Then, what kind of trajectories
are taking by test particles in given fields? The combination of the
two scenarios, fields generated by sources which in return force the
source on trajectories is the problem that has never been solved since
150 years.
What do you mean by far away from the sources? The problem in
electrodynamics classically or quantum mechanically is that the
particles interact with their own generated field. Meaning the position
of the source is identical with the position of the interaction. The
only theory that excludes such a kind of interaction is wheeler-feynman
electrodynamics. The equations of motion is in my PDF file. You will
see that the sum over the fields is taken only over (i notequal j).
>Certainly I that that the point particle point of
>view is the simplest one and the most natural one, if we can create a
>model with point particles that works it would be the best option, we
>don't have to believe that the point particles really exist, they are
>only a abstract concept that allow us to predict some phenomena, just
>as it does in classical mechanics.
For my side I have to admit that I am very confused about if
point-particles exist or if the sources are something else. The
arguments I brought up speak a lot for point-particles but on the other
hand they were mostly mathematical arguments. I am open for anything
making sense.
That's why I went to CERN and stayed there for three months in an
experimentalists group. From that time I learned that before and after
the interaction zone they were only thinking in terms of
point-particles. Talking about the interaction zone was barely
possible. If I insisted the concepts got very blurry.
Indeed QFT only gives the asymptotically incoming and outgoin states an
interpretstion. The S-matrix hides all that away that happens in the
interaction zone. I detest this Heisenberg's point of view. Physics
must be able to describe what happens for all times during an
experiment. Otherwise we only have an effective theory.
What kind of concepts do you have in mind if not point particles?
It's probably worth opening up a sperate discussion topic.
>If we want to model the nature whit
>point particles, we should keep the spirit or the first experiments, it
>means that we analyze the behavior of test particles in a
>electromagnetic field, there the test particles NEVER interact with
>their own field. If we want to create a theory we should keep that in
>mind.
If the particles don't interact with themselves there is no problem.
Everything is well-defined in classical as well as qauntum mechanical
theory. Except for a conceptual problems with the unboundedness of the
dirac operator and the zero point divergences which are all home-made.
>However, the Dirac approach of a effective mass and field gibes good
>results in QED, I mean renormalization, why? I don't know, maybe it
>is true that the interaction changes with the scale and the particles
>do interact with their own field.
The effect of radiation damping (synchrotron raditation) for electrons
even at small accelerations proves that even clasically we need to
include a mechanism like self-interaction.
Indeed I think that the way people tried to implement the mechanism of
self-interaction is wrong. E.g. the idea of Wheeler and Feynman is much
better. But the way or another we must take account of it!
>I have to add another comment, actually there do exist rigid bodies
>macroscopic, and they do move in this relativistic world so it should
>be possible to define a relativistic rigid body theory with no
>contradiction.
What kind of rigid bodies do you have in mind since we know that
everything is build out of e.g. quarks and electrons?
To see that rigid bodies violate relativity can be done even without
computation. Imagine a stick of length l. By definition of a rigid body
any force acting on one of the ends is instantaneously transported to
the other end over a length l. So the violation occures on the length
scale of the stick.
The definiton of a rigid body can be refined in a suitable way for
special relativity, e.g. as having the same shape only in the frame
instantaneously at rest but changing the form and size by the lorentz
transformation if we look at it from different inertial frames.
---
So far we overlooked the biggest problem, the uv divergences.
I think the first step is to find an agreement what kind of properties
a propper electrodynamic theory should have (about what shall the
theory be?, about sources and fields?, structure of the sources?, do we
need fields?, what kind of effects should be described?, etc...)
I would suggest to open up the following headline topics:
"About what is electrodynamics?"
Were we can discuss what kind of basic structure a theory should have.
Points and fields, extended charges and fields or only points like in
Wheeler-Feynman, or only fields? It would be great if you present your
ideas about an electrodynamic theory there to start with.
By the way, Miguel, what about you Munich plans? Can I help you with
anything organizational?
All the best,
Dirk
Dirk
>problem, as long as the theory tells you how to measure it.
>Relativistic rigid bodies don´t seem to be a very fundamental obstacle
>either.
Maybe there is a misunderstanding with the meaning of the term rigid
body. Please have a look at my last poting answering to Miguel.
>But taking into account the shape of the nucleon would get us into lot
>of thecnical difficulties which (might be) non essential.
I fully agree.
>So by WF the classical source of uv divergences is cured. It would be
>interesting to try to push WF ideas to the quantum theory. Dirk, did
>you do something in this direction? which are the main ingredients that
>a WF-like theory of quantum interaction should have?
Whenever I have a bit spare time I play with some ideas of a quantum
version but my ideas are very immature.
The word is on the street that Wheeler and Feynman tried to do that
there lifes long. Indeed after Feynman presented the theory in a
seminar infront of people like Einstein and Dirac his PhD supervisor,
Wheeler, announced that in two weeks he will give a follow up seminar
on how to quantize the theory. But that seminar never took place.
However I personally think the only good approach to electrodynamics up
to today is wheeler-feynman electrodynamics and it is valuable in
understanding where the self-interaction problem comes from. In fact it
is the ONLY theory that describes the irreversible effect of radiation.
>Well, hope that i can discuss more frecuently,
The frequency we have is ok. It takes a while to understand the
otherones' ideas and I always need a quite hour to sit myself down and
write down my ideas in return. As long as we keep the spirit up!
I find it very valuable to discuss with you. I am really happy to have
met you, Miguel and Alejandro!
All the best,
Dirk