The Rarita-Schwinger Equation

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    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地 1)

We can rewrite that as

(Eq地 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痴 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 μi such that μ2=I and that the rest-mass term is multiplied by I, the identity matrix. In vector algebra we have the identity

(Eq地 3)

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

(Eq地 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地 5)

That痴 a general version of Dirac痴 equation, the relativistically covariant equation that describes particles carrying one-half of a Dirac unit of spin. The -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地 6)

in which

(Eq地 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地 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 (Sz=+S/2 relative to an arbitrarily defined z-axis) and spin down (Sz=-S/2). For spin-3/2 particles we have four eigenstates; Sz=+3S/2, Sz=+S/2, Sz=-S/2, and Sz=-3S/2.

    Using Dirac痴 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地 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稚 be true of the other two spin operators. The mathematical operators equivalent to measuring the x- and/or y-components of the particle痴 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地s 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痴 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地 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地s 12)

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

(Eq地 13)

From that result we infer that

(Eq地s 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 Sx and Sy operators. I could have obtained the same result by using |+> instead of |->.

    We can invert Equations 14 to get

(Eq地s 15)

With those equations we derive the matrices that correspond to Sx and Sy. We get

(Eq地s 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地s 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 ϕ1 (which represents matter states) and ϕ2 (which represents antimatter states).

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

(Eq地s 18)

Those equations give us the associated spin matrix immediately;

(Eq地 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地 20)

and

(Eq地 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 ϕ1 and ϕ2 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地s 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地 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地 24)

If we carry out the indicated multiplication, we get

(Eq地s 25)

from which we obtain

(Eq地s 26)

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

(Eq地 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 ϕ1 and ϕ2 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地 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地 29)

That is, of course, equivalent to Equation 5, which I prefer to use because it痴 so clearly related to Dirac痴 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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