Re: Various questions

On 2007-04-11, Greg McLac <some@xxxxxxxxxxx> wrote:
Igor Khavkine:
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.

I do, because it isn't the definition of a size. According to you, a marble
in a opaque box, then with an unknown position, has the size of the box.
You may have a different definition of size, but it won't be the same as
everyone.

If the box is totally opaque then I don't even know if there's a marble
inside. To put it less prosaically, both definitions of "size" I gave
are amenable to experimental measurement. It also so happens that I've
got a theory that allows me to compute the same quantity (which can then
be compared to the measured one). In your idealized situation of a
marble in a box, you're literally telling me that I can't perform any
measurements to determine what's inside the box. Then there's little
point in discussing what theoretically would be in it or how big the It
would be.

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 ...))). :-)

What is the colour of an electron? An electron have no colour and no size,
because there's no orbiting electrons, and because there's no characteristic
parameter with the dimension of length in its equation. Why is that so
difficult to understand? What's the taste of an electron? Does it taste
like chicken?

For some reason, since childhood, I've thought that electrons are
yellow, protons are orange, and neutrons are green. But that's just me.
:-) More realistically, perceived color is determined by the response of
the cones of the human eye to electromagnetic radiation. Schematically,
the eye has three photoreceptors, each with different frequency response
curves, peacked at red, green, and blue respectively. Every color the
we perceive is a combination of signals from each of the three
photoreceptors. Once the spectrum of incoming light is known, it is not
difficult to estimate the rods' response to it. Since a moving electron
can emit light, it's spectrum can be examined (theoretically and
experimentally) and its perceived color determined. So, a moving
electron has color, and its color depends on its motion. And if the
emitted light has no appreciable overlap with the frequencies
perceptible to the human eye, then it's black. Now, the taste of an
electron is a more intriguing question... I wonder if a single electron
can have a noticible effect on our taste buds. If not, it's got to be
tasteless.

The lesson is that if a question can be reduced to something measurable,
then it usually can be answered. Even though the question might sound
silly. BTW, there is a fundamental length associated to an electron: the
Compton length = h/mc ~ 2.4*10^-12 m. Some consider it the lower bound
on the size of an electron. No, you don't have to be one of them.

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.

Berkeley textbook about electromagnetism.

Hmm, not a QFT text. Don't see how a (possibly questionable) heuristic
calculation from that book could be a point against QFT.

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?

Divergences pop up like mushrooms in a warm spring afternoon. The cupboards
swarm with skeletons.

Divergences pop up in many places. In QFT, in particular, they are
handled with the following steps: regularization, renormalization. Both
have technical definitions and are on sound mathematical footing, as
long as one is concerned in computing things perturbatively. If you are
interested in understanding the technical details, I can explain the
basic principles and give references for further reading. As neither
cupboards nor skeletons have technical meanigs that I am aware of I
can't comment on their presence.

The "heuristic" explanation also give a realistic
value, a numerical coincidence isn't a proof a theory. All that only
reflect that all the various theories try and explain the same body of data.

Theories exist to fit existing experimental data and to predict the
outcomes of future experiments. They cannot be proved, but they can be
disproved. There is no law that says that only a single theory shall fit
the data. What is the objection?

"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.

"unspecified quantities, that are wholeheartedly called creation operator".
Surely you know the specified expression of this operator for a bosonic HO.
Which is it for the "fermionic HO"?

The creation annihilation operators for the usual HO are a bit hard to
write out explicitly, as they are infinite dimensional. But for a
fermionic HO it's actually doable. First, algebraically:

H - Hilbert space of states, 2-dimensional
b*, b - creation annihilation operators
{b,b*} = b b* + b* b = 1 - canonical anticommutation relations
K = w b* b - Hamiltonian (sorry already used H)
Heisenberg equations of motion:
db/dt = -i [b,K] = -i w ({b,b} b* - b {b,b*}) = i w b
b(t) = b(0) exp( i w t)
b*(t) = b*(0) exp(-i w t), similarly
|0>, |1> - ground and excited states
b|0> = 0 , b*|0> = |1>
b|1> = |0>, b*|1> = 0
K|0> = 0,
K|1> = w

And now, in matrix form:

|0> = [ 1 ], |1> = [ 0 ],
[ 0 ] [ 1 ]

b(t) = [ 0 1 ] exp(i w t), b* = [ 0 0 ] exp(-i w t), K = [ 0 0 ].
[ 0 0 ] [ 1 0 ] [ 0 w ]

You can check that these matrices satisfy all the above given
properties. Is this specific enough?

"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.

It says that you can't quantize a fermion with commutators. Much ado for
nothing. We already empirically know that half-integer spin particles obey
the Fermi-Dirac statistics. What does this theorem learn us?

Let me repeat in a play by play. Suppose we have three theoretical
hypotheses: (A) elementary particles are described by 4-dimensional
relativistically invariant field theory, (B) elementary particles are
described by 4-dimensional non-relativistic field theory, (C)
elementary particles are described by field theory with 2 or 3
dimensions.

Ignore for the moment any prodiction of either hypothesis beside the
connection between spin and statistics. In answer to the question "Can
there exist bosons of half-integral spin or fermions of integral spin?"
each hypothesis answers: (A) No, (B) Yes, (C) Yes.

Let's look at the empirical data: So far, neither half-integral spin
bosons nor integral spin fermions have been observed.

How does each hypothesis fare in the face of empirical data?
(A) Consistent, (B) Consistent, (C) Consistent.

Hmm, how about some more empirical data: "So far" means over a long
period of time and over a wide energy range.

The theoretical picture is now somewhat different.
(A) Consistent, (B) Consistent, but highly unlikely, (C) Consistent, but
highly unlikely.

And finally, if ever we do in deed discover an elementary particle that
is either a half-integral spin boson or an integral spin fermion, then
we get: (A) Inconsistent, (B) Consistent, (C) Consistent.

In other words, the connection between spin and statistics is a firm
prediction of some theoretical hypotheses. The empirical connection
between spin and statistics allows us to falsify some of these
hypotheses and to classify others as likely or unlikely. That's science.

Toy models are fun, but space-time is actually 3+1 dimensional.

In deed. However, if you think that all 2+1 dimensional models are toys,
the Nobel prizes handed for the quantum Hall effect say otherwise.

Finally, let me repeat and rephrase a statement I made earlier: the
anticommutation relations for fermionic field operators are a direct
consequence, through second quantization, of indistinguishability of
fermionic particles and their statistics. You said that it was vague.
Now, what part(s) of it do you think vague? If I know, I may be able to
correct that impression.

Igor

.

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