The Hermitian Conjugate

Now that we have complex scalars to work with, we need to determine how to start to determine what the imaginary portion of the algebra physically represents.

To do this, we will introduce the Hermitian conjugate. The Hermitian conjugate is designated by a superscripted dagger, like A^†.

When the Hermitian conjugate acts on a scalar quantity, it has the effect of a standard complex conjugate.

a^† = a^*

When the Hermitian conjugate acts on a product of algebraic elements, it reverses the order of the elements, similar to the action of the algebraic conjugate

(AB)^† = B^† A^†

If the Hermitian conjugate does nothing when it acts on an element, that element is called "Hermitian" or real. Anti-hermitian, or imaginary elements change sign when acted on by the Hermitian conjugate. The basis elements are assumed to be real elements. Thus, in order to take the Hermitian conjugate of a given element, we merely take the complex conjugate of the components

A^† = (a^μ)^*e_μ

Using the algebraic conjugate we were able to split an algebraic element into a scalar and vector portion, which we physically identify with the time and space portions. We can make a similar split using the Hermitian conjugate into real and imaginary portions.

<A>_R = (1/2) (A + A^†)
<A>_I = (1/2) (A - A^†)

When using basis elements that are derived from the Minkowski metric, we see that we can change the sign on the i's merely be reversing the order of all products of basis elements. For this reason, the Hermitian conjugate is also sometimes referred to as the "Reversal operator". In other words, if we can represent any element as the product of a set of real elements, the Reversal operator merely reverses the order of all of the factors - which ends up having the same effect as the Hermitian conjugate.