Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

general and the parameter of the transformation, θ, has been arbitrary. However, if we want the transformation to diagonalize the Hamiltonian, it is clear from (3.216) that we can choose the parameter of transformation to satisfy

In this case, we have

which, from (3.216), leads to the diagonalized Hamiltonian

We see from (3.219) that the Hamiltonian is now diagonalized in the positive and the negative energy spaces. As a result, the two components of the transformed spinor

would be decoupled in the energy eigenvalue equation and we can without any difficulty restrict ourselves to the positive energy sector where the energy eigenvalue equation takes the form

For |p| m, this leads to the non-relativistic equation in (3.185) to the lowest order and it can be expanded to any order in without any problem. We also note that with the parameter θ determined in (3.217), the unitary transformation in (3.213) takes the form

which has a natural non-relativistic expansion in powers of This analysis can be generalized even in the presence of interactions and the higher order terms in the interaction Hamiltonian are all well behaved without any problem of non-hermiticity.

There is a second limit of the Dirac equation, namely, the ultrarelativistic limit |p| m, for which the generalized Foldy-Wouthuysen transformation (3.213) is also quite useful. In this case, the transformation is known as the Cini-Touschek transformation and is obtained as follows. Let us note from (3.216) that if we choose the parameter of transformation to satisfy

this would lead to

As a result, in this case, the transformed Hamiltonian (3.216) will have the form

which has a natural expansion in powers of In fact, in this case, the unitary transformation (3.213) has the form

which clearly has a natural expansion in powers of (ultrarelativistic expansion). Therefore, we can think of the Foldy-Wouthuysen transformation (3.222) as transforming away the α · p term in the Hamiltonian (3.209) while the Cini-Touschek transformation rotates away the mass term βm from the Hamiltonian (3.209).

3.11Zitterbewegung

The presence of negative energy solutions for the Dirac equation leads to various interesting consequences. For example, let us consider the free Dirac Hamiltonian (1.100)

In the Heisenberg picture, where operators carry time dependence and states are time independent, the Heisenberg equations of motion take the forms (ħ = 1)

Here a dot denotes differentiation with respect to time.

The second equation in (3.228) shows that the momentum is a constant of motion as it should be for a free particle. The first equation, on the other hand, identifies α(t) with the velocity operator. Let us recall that, by definition,

where we have denoted the operator in the Schrödinger picture by

Furthermore, using (1.101) we conclude that

As a result, it follows that

In other words, even though the momentum of a free particle is a constant of motion, the velocity is not. Secondly, since the eigenvalues of α are ±1 (see, for example, (1.101)), it follows that the eigenvalues of α(t) are ±1 as well. This is easily understood from the fact that the eigenvalues of an operator do not change under a unitary transformation. More explicitly, we note that if

where λ denotes the eigenvalue of the velocity operator α, then, it follows that

where we have identified

Equation (3.234) shows that the eigenvalues of

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