Photons, Neutrinos, And Their Anti-Particles

In his popular book "QED" Richard Feynman wrote

  "Every particle in nature has an amplitude to move backwards
   in time, and therefore has an anti-particle...  Photons look
   exactly the same in all respects when they travel backwards
   in time...so they are their own anti-particles."

Now the question is, what does it mean to "look exactly the same"?
Should we consider extrinsic as well as intrinsic properties?  Usually
when thinking about the identity of a particle we restrict ourselves
to the intrinsic properties.  For a trivial example, a Volkswagon in
Miami is considered to be "the same" as a Volkswagon in Baltimore,
even though they occupy very different positions relative to the rest
of the material world.  Thus we "abstract away" spatial translations
to help classify and identify objects. Similarly we tend to "abstract
away" differences in orientation as well as differences in velocity
(both translational and angular).

But what about a relation between an object's angular velocity and
it's translational velocity?  Suppose every basketball we see is both
translating and spinning, with the spin oriented parallel to its
velocity.  We might then say that there are two kinds of basketballs,
those that spin clockwise (when viewed from "the front") and those
that spin counter-clockwise.  On the other hand (so to speak), if we
wished, we could easily abstract this difference away.  It's really
only an extrinsic distinction.  Of course, on some level, every
distinction is "only extrinsic", e.g., it isn't clear how charge or
mass could even be defined without reference to some extrinsic
interactions.  

This shows that the usefulness of abstractions depends not so much on
the intrinsic/extrinsic dichotomy as it does on the _immutability_ of
properties.  A Volkswagon can be moved from Miami to Baltimore, and we
can take any given basketball and spin it any way we like, so we are
inclined to abstract away these differences.  In contrast, it's not so
easy to change the mass of an electron, so mass is a useful parameter
for classifying (and distinguishing between) particles.

Now consider what Eisberg and Resnick say on the subject of particles
and anti-particles:

  "There is an obvious distinction between a particle and and its
   anti-particle if they are charged, because their charges are of
   opposite sign.  The distinction is more subtle if the particle
   and antiparticle are neutral, like the neutrino and antineutrino.
   Nevertheless, there really is a distinction...  the component of
   intrinsic spin angular momentum along the direction of motion is
   always -hbar/2 for a neutrino and +hbar/2 for an antineutrino."

It's not unreasonable to ask if it's useful to make this distinction
between neutrinos and anti-neutrinos.  Is this percieved difference 
in the direction of spin really an invariant, immutable, property?
Notice that it depends on the "direction of motion" of the particle.
But is this "direction" an inherent property of the particle, or
simply a circumstance of the particle?  As Feynman observes with 
regard to a photon, emitted at point A and absorbed at point B, we 
can just as well regard the transaction as an emission from B and 
absorption at A.

   "As far as calculating (and Nature) is concerned, it's all
    the same (and it's all possible), so we simply say a photon
    is 'exchanged'..."

Thus, the "direction of travel" of a photon is, in a sense, ambiguous.
This might be seen as just another way of saying that a photon happens
to be its own anti-particle, but this is related to the fact that
photons "travel" along null spacetime intervals, and it has possible
implications for neutrinos.

Eisberg and Resnick describe the Wu experiment which showed that
parity is not conserved in beta decay.  They go on to say that this
fact is due to the helicity of the antineutrino.  By "helicity" they
mean the "handedness" of the intrinsic spin angular momentum along
the direction of motion, which is always -hbar/2 for a neutrino and
+hbar/2 for an antineutrino.  Moreover, they continue,

   "...it is not possible for an antineutrino, or a neutrino,
    to have a definite helicity...unless its rest mass is zero.
    If it had a non-zero rest mass, it would travel with velocity
    less than c, and we could always find a moving frame of
    reference in which its linear momentum would be reversed
    in direction...   But the Goldhaber experiment shows that    
    antineutrinos and neutrinos do have definite helicities...
    so we can conclude that their rest masses are zero..."

How can this be reconciled with the idea that neutrinos may actually
have non-zero rest mass?  If neutrinos have mass, must we then
conclude that they do not have definite helicity after all?

Of course, any assertion of empirical results should be qualified 
by the phrase "within experimental accuracy".  Some people have
suggested that there is something "weird" about Eisberg and Resnik's 
line of reasoning (quoted from the 2nd Edition of "Quantum Physics"),
but compare their comments with the following remarks taken from 
"Subatomic Physics" by Frauenfelder and Henley:

   "Is the assignment of a lepton number meaningful and correct?
    We first notice that a positive answer defies intuition.
    Altogether four neutrinos exist, electron and muon neutrino
    and their two anti-particles.  Neutrinos have no charge or
    mass; they possess only spin and momentum.  How can such a
    simple particle appear in four versions?  If, on the other
    hand, it turns out that the neutrino and anti-neutrino are
    identical, then the assignment of a lepton number is wrong...

    The results from the neutrino reactions are corroborated
    by other experiments, and the fact has to be faced that
    neutrino and anti-neutrino are different.  The neutrino
    always has its spin opposite to its direction of motion,
    while the anti-neutrino has parallel spin and momentum.
    In other words, the neutrino is a left-handed and the
    anti-neutrino a right-handed particle.  Such a situation
    is compatible with lepton conservation only if the
    neutrinos have no mass.  Massless particles move with 
    the velocity of light, and a right-handed particle remains
    right-handed in any coordinate system.  For a massive
    particle, a Lorentz transformation along the momentum
    can be performed in such a way that the [direction of]
    momentum is reversed in the new coordinate system.  The
    [direction of the] spin, however,...is not changed...
    A massive anti-neutrino would change into a neutrino, and 
    the lepton number would not be conserved."

This seems quite consistent with Eisberg and Resnick.

So, should we regard the lepton number as a meaningful and conserved
quantity?  If the only distinction between the neutrino (L=+1) and 
the anti-neutrino (L=-1) is their helicity, and if this is not
Lorentz-invariant, then it seems to follow that lepton number is 
not conserved, and the absolute distinction between neutrino 
and anti-neutrino disappears.  Is this a necessary conclusion
if it should turn out that neutrinos have mass?

Georg Kreyerhoff says that if neutrinos are massive, we can't 
assign lepton numbers according to their helicities, and in this 
case helicity is not the only distinction between neutrinos and 
anti-neutrinos.  He goes on to outline two possiblities for
massive neutrinos:

 1) The neutrino is a Dirac fermion, which means a fermion described 
    by the Dirac equation. It would be on the same footing as the 
    electron or the muon, which also are Dirac fermions, which have
    a mass and lepton number, two possible helicities and an anti- 
    particle, which also has two possible helicities, but opposite 
    charge and lepton number. Lepton number is conserved in this 
    scenario.

 2) The neutrino is a Majorana fermion. For such a fermion the charge 
    conjugate state ( the antiparticle ) is (up to a possible phase 
    factor) equal to the parity transformed state, so the neutrino can 
    be considered to be its own antiparticle.  Such a neutrino would 
    indeed violate lepton number conservation and the search for 
    lepton number violating processes is actually a matter of current 
    experiments. The process searched for is the neutrinoless double 
    beta decay ( N(Z) -> N(Z+2) + e^- + e^- ) which violates lepton 
    number by two and involves a massive Majorana-neutrino as an 
    intermediate virtual particle. [N(Z) means a nucleus of charge Z.]