Reference: References: Robert D. Klauber, Student Friendly Quantum Field Theory, (Sandtrove Press, 2013) – Chapter 4.
The total spin of a multiparticle Dirac state is a bit trickier to calculate than the total momentum. For the momentum, the result turned out to be
which is just the sum of the momenta of the particles and antiparticles. This works because a multiparticle state is an eigenstate of all the number operators, with the eigenvalues being just the number of particles in each state with spin and momentum , and itself is just a 3-vector which multiplies the result. For example, we can operate on a multiparticle state to get the total momentum like this:
If we tried something analogous for the spin component we would get
As with the momentum, a multiparticle state is still an eigenstate of the number operators, but is a matrix operator which can operate only on single particle states, that is, states containing a single 4-component spinor. That is, the operation
is not defined, so 3 is not a well-defined quantity.
The solution turns out to be defining the total spin operator as
where the in the integrand is the usual Dirac spin operator, and and are the general solutions to the Dirac equation
Klauber evaluates the integral in his section 4.9.1 for the case of and containing only and operators. The integration uses the usual property of such integrals that any term in the integrand containing an exponential goes to zero because of the boundary conditions. We are left with
If we apply this operator to some state then because the operator (an annihiliation operator) is the first one to operate on the state, only operators with and will produce a non-zero result. In those cases, the particle is annihilated and then replaced with a particle because of the creation operator . That is, the sum over collapses to a single term where and the sum over is eliminated:
Now suppose we apply this operator to a 2-particle state . We’ll get a term like 11 for each particle, with the result
Thus in general, because only the terms in the sum over in 10 that correspond to the momenta of the particles in the many particle state will be non-zero when this operator is applied to that many particle state, and the sum over also collapses, we can write 10 as
[Klauber’s equation 4-113 is a bit sloppy since he applies 10 to a state which he calls which contains the two summation variables and , when in fact this state should refer to a specific spin and momentum and not be part of the sum over and . Likewise, he retains the sum over in 4-114 even though the annihilation operator removes all but one momentum. The final result 4-115 does appear to be correct however.]
The expectation value of the operator between two multiparticle states and is therefore
Because the term is the product of a row vector (), a matrix () and a column vector (), the result is just a scalar, that is, a number. Also, because the spinors are eigenspinors of the operator, we have and . Finally, the inner product . So applying 14 to the state , for example, and calculating the expectation value in that state, we get
The expectation value of between any two different states produces zero because different multiparticle states are orthogonal.