Re: Just what is a particle?

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

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.

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

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

> 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').

> 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.  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.

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).

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).

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.

> 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.

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.

7) Others ----

Depending on the particular type of field whose quanta represent
particles of that type there may be a host of other discrete properties
attached to the particles.  Things like particular values of spin,
(hyper)charge, flavor, color, etc. come to mind.

David Bowman
David_Bowmangeorgetowncollege.edu