The Rarita-Schwinger Equation

Start with the Klein-Gordon equation, which gives us the proper relativistic description of the relation among mass, linear momentum, and total energy of a particle;

(Eq’n 1)

We can rewrite that as

(Eq’n 2)

Unfortunately, the square root does not give us a proper operator (we need it
to be linear), one that will extract an eigenvalue from the state function and
produce a partial probability density that we can use in Born’s theorem. Instead
of factoring Equation 1 into a pair of square roots, we can assume that the
components of the momentum four-vector are each multiplied by a matrix
μ^{i}
such that μ^{2}=I
and that the rest-mass term is multiplied by I,
the identity matrix. In vector algebra we have the identity

(Eq’n 3)

because **A**x(**B**x**A**)=0 (**B**x**A** standing
perpendicular to the plane defined by **A** and **B**). With that
assumption and identity we can rewrite Equation 2 as

(Eq’n 4)

in which we employ the Einstein convention of summing over doubled indices from 1 to 4. We now have a pair of properly linear operators.

Either one of those operators in Equation 4 can zero out the state function, so we can pick either one as the source of our fundamental equation. We have then

(Eq’n 5)

That’s a general version of Dirac’s equation, the relativistically covariant equation that describes particles carrying one-half of a Dirac unit of spin. The ì-matrices that multiply the components of the linear four-momentum vector represent the components of an inherent angular momentum, a spin, carried by the particle. Equation 5 will describe particles carrying different amounts of spin, depending on the form of the μ-matrices. Now we can replace the four-momentum with the equivalent differential operator,

(Eq’n 6)

in which

(Eq’n 7)

(I was tempted to put the square root of minus one into the four-momentum in Equations 4 and 5, but the quantity under the radical in Equation 2 will always be greater than zero, so the operator must be real until we make the differential substitution.) Thus we have Equation 5 as

(Eq’n 8)

Determining the explicit form of the spin matrices starts
with an enumeration of the spin eigenstates of the particle. Adjacent
eigenstates must differ from one another by exactly one Dirac unit of angular
momentum, so for spin-1/2 particles we have two eigenstates; spin up (S_{z}=+S/2
relative to an arbitrarily defined z-axis) and spin down (S_{z}=-S/2).
For spin-3/2 particles we have four eigenstates; S_{z}=+3S/2,
S_{z}=+S/2,
S_{z}=-S/2,
and S_{z}=-3S/2.

Using Dirac’s bra-ket notation for convenience, with |+> for spin up and |-> for spin down, we have the spin matrix for the z-component of the spin of a spin-1/2 particle as

(Eq’n 9)

Applying the spin operator to the ket extracts the eigenvalue of the spin, then applying the bra to that result gets us the expectation value of the spin in the z-direction. By definition, the eigenvectors (the kets) lie orthogonal to each other, so we have necessarily <-|+>=<+|->=0.

In the derivation of Equation 9 the spin operator does not alter the ket: it merely extracts an eigenvalue from it. But that statement won’t be true of the other two spin operators. The mathematical operators equivalent to measuring the x- and/or y-components of the particle’s spin will change the ket to which they are applied, either transforming it into the other ket or zeroing it out. We can specify the basic properties of two ladder operators through four simple equations:

(Eq’ns 10)

We have simply assumed that such operators exist. Now we want to relate them to the other spin operators.

Equations 10 show the ladder operators changing the eigenvectors of the particle’s spin state and at the same time extracting a description of the change in the eigenvalues, one Dirac unit of spin. If we go up the ladder on a spin-down state and then go down the ladder, we get

(Eq’n 11)

There exists yet another way in which we can extract the square of aitch-bar from the spin-down eigenvector. We start with

(Eq’ns 12)

Combining those equations and replacing the z-component with its associated commutator, we get

(Eq’n 13)

From that result we infer that

(Eq’ns 14)

In that derivation I used the first power of the z-component of the spin
because I can use it, through the commutation relations, to complete the square
for factoring. Thus I obtain the ladder operators in terms of the S_{x}
and S_{y} operators. I could have obtained the same result by using |+>
instead of |->.

We can invert Equations 14 to get

(Eq’ns 15)

With those equations we derive the matrices that correspond to S_{x}
and S_{y}. We get

(Eq’ns 16)

Thus we have all three of the Pauli spin matrices.

Those spin matrices apply to the non-relativistic theory of spin. For a fully relativistic theory we need to use 4x4 matrices to accommodate the matter and antimatter states of the particle. Thus we must use the gamma matrices,

(Eq’ns 17)

in which each entry in those matrices is itself a 2x2 matrix and the sigmas
represent the Pauli matrices. Those matrices act on 4x1 column matrices that
consist of smaller column matrices
ϕ_{1} (which represents
matter states) and ϕ_{2}
(which represents antimatter states).

For spin-3/2 particles we have four eigenstates with respect to orientation relative to the z-axis:

(Eq’ns 18)

Those equations give us the associated spin matrix immediately;

(Eq’n 19)

To get the full matrix description of the other two spin operators we need to
look anew at the ladder operators. We recall that each ladder operator extracts
the appropriate eigenvalue from the eigenvector while shifting that eigenvector
to a neighboring state (for example, S_{+}|–>=
S|->),
as in Equations 10. Thus we have

(Eq’n 20)

and

(Eq’n 21)

So now we have the Dirac matrices (Equations 19, 20, and 21) for a spin-3/2 particle. But those are for the classical (non-relativistic) case, analogous to the Pauli matrices for a spin-1/2 particle. For a proper relativistic case we need a description that accommodates both the matter states and the antimatter states of the particle.

In this case each of the spinors
ϕ_{1}
and ϕ_{2}
represents a four-element column matrix, with each element representing one spin
eigenstate of the particle. To accommodate both of those spinors we assemble
them into an eight-element column matrix and use 8x8 matrices as the spin
operators. In setting up the Dirac equation we used the Dirac matrices,

(Eq’ns 22)

in which the Pauli matrices serve as 2x2 elements of each matrix. The spatial matrices relate the two sub-spinors to each other, in essence connecting the matter states and the antimatter states to each other. For the spin-3/2 particle we have, then,

(Eq’n 23)

with the gamma-null matrix being the same as in Equations 22, but with 4x4 identity matrices instead of 2x2 identity matrices. Equation 8 thus takes the form

(Eq’n 24)

If we carry out the indicated multiplication, we get

(Eq’ns 25)

from which we obtain

(Eq’ns 26)

Referring to Equations 19, 20, and 21, we then devise

(Eq’n 27)

Now we can finish up our solution and write out the state
function. We start with the eigenvalue spinor, and eight-element column vector
in which one element equals one and the others equal zero. It looks like
ϕ_{1}
and ϕ_{2}
stacked one on top of the other. We assume an eigenvalue and apply the spinor to
the right side of Equations 26. If, for example, we take our assumed eigenvalue
as making the spin-minus state of the matter part equal to one, we get

(Eq’n 28)

In that equation N represents a normalization factor.

Equation 8 does not give us the form in which the Rarita-Schwinger equation was originally presented. Rarita and Schwinger gave us

(Eq’n 29)

That is, of course, equivalent to Equation 5, which I prefer to use because it’s so clearly related to Dirac’s equation, which we expect.

Reference:

Lovas, I.; K. Sailer; and W. Greiner, "Generalized Rarita-Schwinger Equations", Heavy Ion Physics, Vol 5 (1997), pg 85.

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