## [7Bf 7 [VBi 6j47   *For a discussion of the d functions, see Rose (1957). will use this result in the next chapter when we construct the most general solution to the Dirac wave equation.

To complete the discussion of the homogeneous Lorentz group, we need only to find the representation of the time reversal transformation S{T). This will be postponed until Chapter 8, where time reversal will be discussed in some detail.

### 5.10 BILINEAR COVARIANTS

In this section we discuss the construction matrix elements in Dirac space and the matrix operators from which these matrix elements are constructed. It is convenient to express the most general matrix operator in terms of elementary operators which have definite transformation properties under the homogeneous Lorentz group. These elementary operators are referred to as the bilinear covariants.

The study of the covariance of Dirac matrix elements begins with the observation that 5(A) is not unitary in every case. This follows from the fact that a, and 75a1 are Hermitian operators, but only the generators of rotations have a factor of i in the exponent. Hence S(R) are unitary, while S{B) are not. In fact

Hence the density ip^tp is not Lorentz invariant.

To find a Lorentz invariant density, consider a density of the form p(x) = ip* (x)6ip{x) , (5.123)

.vhere 9 plays the role of a "metric" tensor. Invariance gives f/{x') = P(x) = ii>'i(:rW(z')

Since this must hold for any ip, we have the requirement

Expanding S in a power series quickly shows that (5.125) is equivalent to the requirement that the "metric" 9 commute with the generators of the rotations but anticommute with the generators of the boosts,

These conditions are satisfied if 9 — and then, for all A

The requirement of covariance has thus led to the introduction of an indefinite metric, similar to the one we encountered in the two-component KG theory (the operator 70 plays a role analogous to r3). Because this metric must occur in all matrix elements with well-defined covariance properties, it is convenient to introduce the Dirac adjoint as follows:

This is always a row vector, and a Dirac matrix element will be formed by multiplying from the left by the adjoint spinor ip(x) and from the right by the normal spinor ip(x). Then i>'(x') = S(A)iP(x) i)'{x') = i,{x)S-\A)

and p{x) = 4>{x)i>(x) = 1p'(x')lp'(x') = p'(x')

is a Lorentz invariant scalar density.

All Dirac matrix elements will now be written in the form

■4>{x)Ti){x) , where T is a 4 x 4 complex matrix. The most general such matrix can always be expanded in terms of 16 independent 4x4 matrices multiplied by complex coefficients. In short, the matrices T can be regarded as a 16-dimensional complex vector space spanned by 16 matrices.

It is convenient to choose the 16 basis matrices, 1 „ so that they have well-defined transformation properties under LT's. Since the 7M's have such properties, we are led to choose the following 16 matrices for this basis:

757M

757M

 # matrices scalar 1 vector 4 antisymmetric tensor 6 axial vector 4 pseudoscalar 1 It can be seen by inspection that all of these matrices are linearly independent. Furthermore, their properties under Lorentz transformations are suggested by their labeling. For example p'^ix') = fi(x')^tp'(x') = tp(x)S~l (A)^ S(A)ip(x) = A\p"{x) , the correct transformation law for a vector field. Note that the 75 defined above is identical to the one previously introduced in Eq. (5.106) and that an alternative form is This way of writing 75 is useful for proving that 75 transforms as a pseudoscalar (see Prob. 5.4). In particular, one can show that
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