Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

This is identical to (2.11) and the energy eigenvalues would then correspond to the roots of this equation given in (2.12). (Note that this method of evaluating a determinant is not valid, in general, for matrices involving submatrices that do not commute. In the present case, however, the submatrices are both diagonal and, therefore, commute which is why this simpler method works.)

We can obtain the solutions (wave functions) of the Dirac equation, in this case, by directly solving the set of four coupled equations in (2.9). Alternatively, we can introduce two component wave functions and and write

where

We note that for the positive energy solutions

the set of coupled equations takes the form

Writing out explicitly, (2.17) leads to

The two component function can be solved in terms of and we obtain from the second relation in (2.18)

Let us note here parenthetically that the first relation in (2.18) also leads to the same relation (as it should), namely,

where we have used the property of the Pauli matrices, namely, (in the first line). Note also that if the relation (2.19) obtained from the second equation in (2.18) is substituted into the first relation, it will hold trivially (because of the Einstein relation). Therefore, the positive energy solution is completely determined by the relation (2.19) in terms of

Choosing the two independent solutions for û as

we obtain respectively

and

This determines the two positive energy solutions of the Dirac equation (remember that the energy eigenvalues are doubly degenerate). (The question of which components can be chosen independently follows from an examination of the dynamical equations. Thus, for example, from the second of the two two-component Dirac equations in (2.18), we note that must vanish in the rest frame while remains arbitrary. Thus, can be thought of as the independent solution.)

Similarly, for the negative energy solutions we write

and the set of equations (2.9) becomes

We can solve these as

Choosing the independent solutions as

we obtain respectively

and

and these determine the two negative energy solutions of the Dirac equation.

The independent two component wave functions in (2.21) and (2.27) are reminiscent of the spin up and spin down states of a two component spinor. Thus, from the fact that we can write

the positive and the negative energy solutions have the explicit forms

The notation is suggestive and implies that the wave function corresponds to that of a spin particle. (We will determine the spin of the Dirac particle shortly.) It is because of the presence of negative energy solutions that the wave function becomes a four component column matrix as opposed to the two component spinor we expect in non-relativistic systems. (The correct counting for the number of components of the wave function for a massive, relativistic particle of spin s in the presence of both positive and negative energies follows to be 2(2s + 1), unlike the nonrelativistic counting (2s + 1).)

From the structure of the wave function, it is also clear that, for the case of general motion, where

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