Re: Various questions

On 2007-04-09, Greg McLac <some@xxxxxxxxxxx> wrote:
Me:
I'm surprised by this theory that allows a changing size for an
elementary particle. Is it no longer elementary? Have you a reference
on the topic please?

Igor Khavkine:
I was referring to Quantum Field Theory (specifically, Quantum
Electrodynamics, QED) as the best treatment of elementary particles that
we currently have.

Ah, I should have guessed. QM is sufficient, let's keep simple. The
extension of the wave function, the standard deviation of the probability
density to be less vague, has nothing to do with the size of the particle.
Indeed, in the Dirac equation there is no parameter of the dimension of a
length, so the word "size" for an electron is meaningless. People say the
electron is structureless, and it's already a dubious enough term.

The wave function point of view is valid, but it has some drawbacks
(probability interpretation, anti-particle states, multi-particle
states). All of these things can be handled and interpreted using wave
functions, however they can be handled much more naturally in the field
language. There is a one-to-one correspondence between the wave function
and field languages. So, once something is understood in the field
language, it can be translated into wave function language (and
sometimes vice versa). Fields are there to help and make things
simpler, not to shy away from.

About the "size" of an electron. Personally, I think that the standard
deviation of the expectation value of its charge density is as good a
measure of "size" as any. You may disagree. Alternative reasonable
measure of "size" is the electron's scattering cross section for a
particular type of scattering experiment (this quantity also state
dependent). You may disagree again. You may even prefer to say that the
concept of "size" does not apply to the electron (as you do above). This
is as good a position to hold as any, as long as you can articulate it.
Unfortunately, when someone asks "What is the size of an electron?" you
are stuck explaining how something may have no size (meaning that it's
not a point particle (meaning that it doesn't have a finite size
(meaning it doesn't have an infinite size either ...))). :-)

Another problem, in QED that time, is when the electron is treated in
position eigenstates. In order that it has not an infinite energy, it's
surrounded by "a cloud of virtual particles", whatever the meaning of that
phrase.

That's precisely why I don't use that phrase and also discourage others
from doing so. Oh, and an electron in a position eigenstate does have
infinite energy, whether you tave "virtual particles" into account or
not. It's for the same reason that a delta-function is not a
normalizable state.

A characteristic size then appears which is the size of the cloud.
It can be calculated in an intuitive way by equating the self-energy of a
distribution of charge with the electron mass. By giving it an angular
momentum corresponding to its spin, and making the relativistic corrections
due to the speed of its equator, the quantum gyromagnetic factor 2 is
roughly found again. But all that should more or less be regarded as mere
numerical coincidence.

All that is awfully technical, so the short answer is: "maybe". According
to the version number of the theory, the answer may vary, and we don't even
know whether we have got a bugfree version.

That's an awfully complicated way to calculate size and spin. I've
certainly never done it this way, nor have I seen it done this way in
textbooks, except maybe as a heuristic explanation. Not sure about what
you mean by "bug free theory". QFT certainly has some drawbacks, both
mathematical and physical, but it's "bug free" enough to describe the
spin and charge density of an electron, has been so for over 50 years.
Care to share your reservations?

Save that for fermions, i.e. material particles like electrons, the
simple harmonic oscillator doesn't apply. But the explanation remains
about the same. In modern theories, the electrons are
indistinguishable. So they haven't the same mass, they are the same
particles with one mass.

I'm curious as to what you actually mean in this last paragraph. What is
special about fermions that makes the simple harmonic oscillator apply
to them, but not to other particles?

I think it's grammatically clear I said the simple harmonic oscillator model
doesn't works for fermions, and the Pauli's exclusion principle anyway
prevents that more than one energy level exist. The excitations of a HO are
always bosons, and the fermion case is introduced very formally and
mysteriously, as is their mass: anticommutators are postulated out of thin
air between unspecified quantities, that are wholeheartedly called field
operator, and the whole is promoted to the pompous status of spin-statistic
theorem.

I'm sorry I misunderstood you originally. I now see what you meant.
However, I'm going to have to take you to task on some of the things you
said in the last paragraph.

"The excitations of a HO are always bosons":
Technically, the simple harmonic oscillator is a system with one
degree of freedom. If you think of it as describing a particle
confined to a one-dimensional harmonic potential well, then there's
nothing to prevent that particle from being either a boson or fermion.
Bosonic or fermionic statistics come in only when you have multiple
particles. On the other hand, the quantum SHO could be thought as
describing a system with a variable number of particles confined to a
point. Then yes, these particles would be bosons. However, this is a
somewhat esoteric interpretation and I doubt that's what you meant.
But I could have misunderstood again.

"the fermion case is introduced very formally and mysteriously":
The introduction of canonical anticommutation relations (instead of
the more familiar canonical commutation relations) can be mysterious
at first. However, there is a very good reason for introducing them.
The answer lies in the correspondence between the many-particle and
field theory formalisms through second quantization. In the
many-particle formalism, (anti)symmetrized multi-particle wave
functions represent states with multiple bosons or fermions. The
(anti)commutation relations of particle creation-annihilation
operators in the field theoretic formalism is nothing but a reflection
of that fact.

"unspecified quantities, that are wholeheartedly called field operator":
When dealing with wave functions, a state with n+1 particles is
created by multiplying a wave function with n particles by a
1-particle wave function and (anti)symmetrizing. In field theory, a
state with n+1 particles is created by multiplying an n-particle state
by a field operator. This is consistent with the promotion of wave
functions to operators in second quantization.

"the whole is promoted to the pompous status of spin-statistic theorem":
There is a theorem called the Spin-Statistics Theorem, but I'm not
sure how it can be pompous. It is applicable to relativistically
invariant field theories in 4 dimensions. Once you know that
quantizing (fermions) bosons requires the introduction of
(anti)commutation relations, it says that you cannot quantize an
integer spin field as a fermion, nor a half-integer spin field as a
boson. However, once you relax the hypotheses, the theorem no longer
applies. For example, non-relativistic theories allow fermions to have
integral spin. Also, when you go down some dimensions, bosons and
fermions can have any spin. For instance, spinless fermions are used
often in toy models of condensed matter theory, where models with only
1 or 2 spatial dimensions are common.

In my lurking period, I noticed that you often describe questions
as "vague", while your answers are not always clearer.

You're probably right. I'm not always a model of clarity, nor am I
always right. However, if you think that something I said is vague or
you wish to understand in more detail, I'm happy to discuss it further.

Igor

.

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