From: mcirvin@husc8.harvard.edu (Mcirvin)
Newsgroups: sci.physics
Subject: Re: Scalar spin
Date: 8 Oct 92 17:44:07 GMT
Article-I.D.: husc8.mcirvin.718566247

jonathan_scott@VNET.IBM.COM (Jonathan Scott) writes:

>What is known about the scalar component of spin?

>This is what I have worked out for myself so far:

>If the spin operator either for the Dirac equation or non-relativistic
>spin-1/2 equation is systematically extended to include a scalar part,
>like a four-vector, the scalar spin operator S_0 comes out as hbar/2
>times the identity matrix but with an overall minus sign for the
>negative-energy (positron) solutions.  This means that its eigenvalues
>for spin-1/2 functions are +hbar/2 for particles and -hbar/2 for
>antiparticles.  The four-vector describing mean values of spin
>observations is then equal to the four-vector particle probability
>current minus the four-vector antiparticle probability current.

The Dirac equation already contains this, really: there's no
reason to add anything else, and the Dirac equation describes
things in a proper Lorentz-invariant way.  In the most convenient
notation the three Pauli matrices get replaced by four gamma
matrices, and the one which is "rotationally invariant" (i.e. which
is sandwiched between fields and states to make rotationally invariant
matrix elements) in the rest frame is just what you've described,
a couple of identities with a minus sign for the antiparticle part.
It's the same matrix which is called beta in Dirac's original
notation.

You can think of it as a kind of particle-number operator; your
speculations about quantum numbers are basically right.  In the
Dirac formalism it doesn't have a factor of 1/2 (none of the gamma
matrices do; the 1/2's come in when you take the nonrelativistic
limit), and I don't know about factors of hbar since I habitually
set those to 1.  So it has an eigenvalue of +1 for particles
and -1 for antiparticles; if the theory is one of electrons, it's
just minus the electric charge, in units of e.  It's a quantum
number that the particles carry; depending on the rest of the
theory it may or may not be conserved.

You can think of the identity as a "fourth Pauli matrix" if you
like; I've seen it called sigma-zero.  But the nice thing about
using the gamma matrices instead is that the corresponding
operators (sandwiched between fields and states) really do transform
like a Lorentz four-vector, instead of some random thing with four
components.  Of course, in the Dirac equation the spinor has
four components instead of two, so that such things as this
gamma-zero or beta matrix can be written as a single matrix.

The quantity to which you refer was originally thought to
be the proper generalization of the probability density
in nonrelativistic QM, which was confusing since it came
out negative for antiparticles!  The suggestion that it
be interpreted as a charge density instead (in those days
they were just trying to handle electrons), combined with
the apparently negative energy of these states, led eventually
to Dirac's hole theory, the precursor of the modern field-
theoretic approach to relativisitic QM of fermions. 

A good place to read all about this is Sakurai's red book,
"Advanced Quantum Mechanics."  It goes into detail about what
all of the various relativistic matrices mean.  The only
problem is that the notation is out of date.  Sakurai uses
the old ict notation for time components in special relativity,
and his gamma anticommutation relations are correspondingly
different (he even calls the convention commonly used today
"deplorable" in a footnote).  Nevertheless, the discussions of
the matrices' physical significance are illuminating.  

Modern discussions tend to have so much distaste for hole theory
that they never treat these matrices as operators similar to 
nonrelativistic QM ones, that you just put between wave
functions.  They are instead justified on the basis of arcane
reasoning about Lorentz group representation theory, which is
logically fine but opaque for beginners.  It's interesting to
see how things developed in a semi-historical way even if some
of the arguments are now seen to be inadequate.
-- 
Matt McIrvin, posting nonsense again 
Newsgroups: sci.physics