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.

But 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 axis oriented parallel to its
velocity (relative to the ground).  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,
we might also abstract this difference away, especially since the 
direction of travel is relative, and we could transform a clockwise
to a counter-clockwise basketball by changing the observer's state
of motion. Thus the distinction between the two kinds of basketballs
is really (in this sense) just extrinsic.  We might even argue that, 
on some level, every distinction is "only extrinsic", because (for
example) 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, even though it
is in some sense an extrinsic property.

Now consider what Eisberg and Resnick say on the subject of particles
and anti-particles (written when neutrinos were still presumed to be
massless):

   There is an obvious distinction between a particle 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?
Just as with our spinning basketballs, 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?  If neutrinos
were massless and (therefore) moved at the speed of light, we could
not reverse their direction of motion by any change in our frame of
reference, so the "handedness" would be an immutable property. However,
if (as seems to be the case) neutrino's have mass and move at speeds 
below the speed of light, their handedness depends on the frame of
reference.

As an aside, if we consider virtual photons, the direction of travel
is ambiguous due to temporal symmetry. 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'...

Eisberg and Resnick describe the Wu experiment which showed that
parity is not conserved in beta decay.  Consistent with the comments
above, 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 (as noted)
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 more recent determination that 
neutrinos actually have non-zero rest mass?  If neutrinos have mass, 
must we then conclude that they do not have definite helicity after 
all?  If so, how is this reconciled with the Goldhaber experiment, which
was once taken to have shown that antineutrinos and neutrinos do have 
definite helicities?

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 (again, written prior
to the discovery of non-zero neutrino mass):

   "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.  [It is now known that there 
    are actually SIX kinds of neutrinos, consisting of the electron,
    muon, and tau neutrinos and their anti-particles.] Neutrinos 
    have no charge or mass [sic]; they possess only spin and momentum.
    How can such a simple particle appear in four [six] 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."

Overlooking the outdated enumeration of the different kinds of
neutrinos, this assessment seems 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, given
that (as recent experimental evidence seems to indicate) 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.]