Trace Properties of Pauli and Dirac Matrices

Here, we have used the properties of cyclic permutation of gammamatrices.

Spin Density Matrix

Consider that a beam of spin particles is produced from the interaction withanother system. When the interaction is over, the wavefunction of the wholesystem can be written as a sum of the products of wavefunctions of the othersystems in free state:

where

and represents the wavefunction of the states with momentum p; helicityrepresents the wavefunction of the other system and are constants. Thewavefunctions and are normalized by the conditions:

Operator acts on the spin variable. The mean value of the operator isobtained as

Using Eq. (C.2), can be obtained as

Using Eq. (C.2) and (C.6) and the normalization condition for from Eq. (C.4),the denominator of Eq. (C.5) can be written as:

with

Similarly, the numerator of Eq. (C.5) can be obtained as

Using Eqs. (C.9) and (C.7) in Eq. (C.5), we have

With

This matrix ρ(p) is called the spindensity matrix.

Some properties of the spin density matrix

1. Hermiticity

2. Normalization

Now, in order to obtain the expression for the spin density matrix, we expandρ(p) in terms of bilinearcovariants as

Using the properties of the spin density matrix, we can evaluate theconstants a to e in Eq. (C.19).

To evaluate constant a, we take the trace of Eq. (C.19)

Using the trace properties of gamma matrices as given in Section C.1, weget

Using the normalization condition forρ(p) matrices as given in Eq.(C.15), the constant a can be evaluated as

To calculate, we multiply Eq. (C.19) by from the right side and then thetrace of the resulting equation can be calculated as:

Using Eq. (C.18) in the aforementioned expression, we get

where

and the anti- commutation relation of matrices have been used.