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Re: Just what is a particle?

Well, now, I've got problems with all this. Largely, I take
exception to confusing a methematical representation with the stuff that
it represents. See comments below.

Adam was by constitution and proclivity a scientist; I was the same, and
we loved to call ourselves by that great name...Our first memorable
scientific discovery was the law that water and like fluids run downhill,
not up.
Mark Twain, <Extract from Eve's Autobiography>

On Fri, 18 Feb 2000, David Bowman wrote:

Regarding Cliff Parker's request:

Hugh Haskell wrote in an earlier post -- Photons are particles (not like
electrons or protons, but particles nevertheless)

I would like some discussion on this point as I try to clarify my thinking.
What characteristics are necessarily present in order to call something a
particle?

That depends on what kind of particle concept you had in mind. At the
purely classical level a particle is a(n effectively) point mass
completely characterized by its world line in spacetime. IOW, at each
instant of time it's only degrees of freedom are the coordinates of its
location. At the quantum level the concept of a particle is much more
rich.
It need not be inconsistent to define a classical point particle
that way, but I would always use the modifier "point". That's because
there's no general agreement on the subject.


At the quantum level I would call a particle an elementary excitation
(i.e. a single quantum of excitation) of a quantum field.

You can call it that if you want, but don't be surprised if nobody
comes. That definition would definitely not help any of my
experimentalist
friends.

Eugene Wigner has defined a particle as an irreducible continuous unitary
representation of the Poincare' group.

Really? Where did he do that? Im not saying you're wrong, but
he's not referred to in my copy of Naimark.
I tried to do that once, when I was very young and teaching
a graduate course - I forget the name of the course. I came away
with great notes, and so did some of the students. Basically, though,
I managed to lose everyone the first week of the course.

I have listed a few characteristics particles often seem to have and
thoughts about how each may apply to photons. Comments,
clarifications,
disagreements, and instructions are hereby solicited.

1) Charge - No. Photons like neutrons and many other "particles" have no
charge.

Correct, (but the only uncharged *elementary* particles are, AFAWK,
photons, neutrinos, gluons, Z^0s, and *possibly* X-bosons, supersymetric
partners of other uncharged particles, and gravitons).

2) Mass - No. I guess photons are massless since they travel at the speed of
light. I don't really understand what this means however especially when
momentum and energy are considered.

Correct. I believe John Denker already explained this in terms of the
relativistic relationship between energy and momentum for both particles
with nonzero and with zero mass. (He tries to keep the old nomenclature by
qualifying the term 'mass' with the modifier 'rest' to prevent any
confusion by someone who had, unfortunately been led to believe by their
training that 'mass' means something other than 'rest mass').

Don't call it "rest" mass. Call it "invariant" mass, and be brave
in the knowledge that you are being redundant. Redundancy, however,
avoids confusion.
With high school students you can take a slightly different
tack. Tell them that photons have zero mass (as distinct from not having
mass- which is the way many of them interpret "zero mass").
The occasional student who is willing to push you on this
deserves a better answer: mc^2=E^2 - (pc)^2 = 0 for photons. Each term
on the right is measurable. Momentum p can be measured from the recoil
momentum of an emitting atom - that's the basis of the optical glue
that's used to trap atoms. Energy is determined from E=hnu, which refers
back to the photo effect. Alternatively, you can refer to energy-momentum
conservation in Compton scattering.

3) Momentum - Yes. I understand that photons do have momentum. Exactly what
this means however is unclear to me. It must not mean p = mv since photons have
no mass.

It's probably unclear because you don't know what momentum really is. It
is *not* characterized by the Newtonian formula m*v in general. Momentum
is the generator of infinitesimal virtual displacements of the state of a
physical system in space.

I think not. There is a group-theoretical operator, often called
the "momentum-operator", which is the generator of spatial displacements
in a certain space - often a Hilbert space. I think it unhalpful to
confuse this mathematical abstraction with the momentum of a particle.
I would rather say that momentum is the property of a particle that gives
a kick to another particle during a collision. Then, when you show a
student a picture of a bunch of tracks coming out of a point in a modern
detector, there will be a sharp picture of the violence wrought by the
momentum of the incident particle.
Momentum plays a role in most sports. The momentum that occurs
in quantum mechanics is no different.


What this means for a classical system is
different than what it means for a quantum system, because in the two
cases the representation of just what a state is is different, and the
quantities that generate infinitesimal dispacements of these different
things act differently on them accordingly.

See above.

In the case of classical E&M the momentum of the electromagnetic field
ends up being the integral over all space of [epsilon_0]*(E X B).

Again, this misses the physical point. The momentume of sunlight,
that can propel a spacecraft, can be calculated (assuming properties of
the sail) on purely classical grounds from the change in flux direction
at the sail.

In the case of a classical relativistic (free) particle its momentum is
related to its energy according to |p|^2 = (E/c)^2 -(m*c)^2 where the
direction of the momentum is along the direction of motion. In terms of
the particle's velocity v we have p = m*v/sqrt(1 - |v/c|^2).

For a mass*less* particle the momentum/energy relationship is |p|*c = E
with, again, the direction of p being along the motion (velocity
direction) but in this case the magnitude of p has nothing to do with the
speed |v| because in this case always |v| = c completely independent of
the actual value of |p|.

In the quantum case the momentum of a quantum particle is the Hermitian
observable which infinitesimally translates the wave function (i.e. the
-i*h_bar*gradient operator) and whose measured values are the eigenvalues
of this operator which also happen to represent the wave function's
spatial frequency/wave number (when this number is well-defined for the
wave function).

This confuses "momentum", a physical quantity, with the procedure
for representing momentum in a particular language (a particular
representation of quantum mechanics).

In the case of a quantum field the momentum observable is typically the
spatial integral of the -i*h_bar*gradient operator sandwiched between
a field destruction operator on the left and the field creation operator
on the right so this sandwich is taken as a normally ordered product.
The resulting operator operates on the Fock (Hilbert) space of states.

*Only* in the case of a classical *and* Newtonian particle is its
momentum *ever* given by m*v.

4) Inertia - I am really baffled on this one. No mass means no inertia but
photons obey Newton's First Law. How can that be?

Actually there are multiple meanings of the term 'inertia'. No mass
*can* mean no inertia, but it doesn't have to mean this depending on just
which notions of inertia and mass are meant. Both clauses of Newton's
first law don't fully apply to photons since photons never exist in a
state of rest they can not remain at rest.

A photon has a mass value. The mass value is zero. It has
inertia. Newton's laws, which were proposed in the context of Galilean,
not Einsteinian, relativity, must me modified for photons. The correct
statement is that a photon's momentum will remain unchanged in an inertial
frame unless the photon collides with something, or something like that.

5) Energy - Yes, they can cause change I suppose. I used to be more sure of
this, but that was when I thought I understood what energy was. After
considering Leigh's thoughts and those of others I'm not so sure anymore.

True, particle's *do* possess energy. There is no such thing as a free
particle with zero energy.

I don't think that this is responsive, but no matter.

6) Something that is quantized - Perhaps a particle can be considered anything
that is quantized.

Certainly in quantum field theory particle states are those states that
have nonzero excitation quanta. Only the vacuum state is the one that
possesses no particle quanta.

Quantum Field Theory is a lousy guide here because
electromagnetism makes it impossible to define asymptotic states.
"Quantized" is a pun here, used both in its literal sense - broken up into
discrete parts - and in the sense of a second-quantized field theory. I
think the question refers to the first usage; the second usage is
unhelpful.
The answer to the question, taken literally, is an unambiguous
"yes, it can be."
c
7) Others ----
J.J. Thompson demonstrated that cathode rays were composed of
particles when he succeeded in deducing a unique value of e/m for the
constituents. In more recent times, it has been possible, by a
complicated series of inferences, to deduce mass values for the light
quarks (u,d,s). These value are so small that Wilczek has been able
to refer to a proton as an object that has "mass without mass" (Nature
Magazine, a few weeks ago). Renormalization theory teaches us that
a the invariant mass value of a particle depends upon the energy scale
at which it is measured. So you can see that it is possible to make
the discussion very complicated.
But why would anyone want to?
I have already suggested a simple answer to a high school student
who asks about the mass of a photon. See above.
Regards,
Jack