Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

to correspond to

Of course, this shows that a point Dirac particle has a magnetic moment corresponding to a gyro-magnetic ratio

Let us recall that the magnetic moment of a particle is defined in general as (c = 1)

Since S = σ for a two component electron, comparing with (3.199) we obtain g = 2. Quantum mechanical corrections (higher order corrections) in an interacting theory such as quantum electrodynamics, however, change this value slightly and the experimental deviation of g from the value of 2 (g − 2 experiment) for the electron agrees exceptionally well with the theoretical predictions of quantum electrodynamics. Particles with a nontrivial structure (that is particles which are not point like and have extended structures), however, can have g-factors quite different from 2. In this case, one says that there is an anomalous contribution to the magnetic moment. Thus, for example, for the proton and the neutron, we know that the magnetic moments are given by

where the nuclear magneton is defined to be

with m_P denoting the mass of the proton.

Anomalous magnetic moments can be accommodated through an additional interaction Hamiltonian (in the Dirac system) of the form (this is known as a non-minimal coupling)

where

denotes the electromagnetic field strength tensor and κ represents the anomalous magnetic moment of the particle. This is commonly known as the Pauli coupling or the Pauli interaction.

3.10Foldy-Wouthuysen transformation

In the last two sections, we have described how the non-relativistic limit of a Dirac theory can be taken in a simple manner. In the non-relativistic limit, the relevant expansion parameter is and the method works quite well in the lowest order of expansion, as we have seen explicitly. However, at higher orders, this method runs into difficulty. For example, if we were to calculate the electric dipole interaction of an electron in a background electromagnetic field using the method described in the earlier sections, the electric dipole moment becomes imaginary at order (namely, the Hamiltonian becomes non-Hermitian). This puzzling feature can be understood in a simple manner as follows. The process of eliminating the “small” components from the Dirac equation described in the earlier sections can be understood mathematically as

where the matrix A, in the case of the free Dirac equation, for example, has the form (see (3.182))

for the positive energy spinors. The matrix T that takes us to the two component “large” spinors (from the original spinor) in (3.206) has the form

It is clear from the form of the matrix in (3.208) that it is not unitary and this is the reason that the Hamiltonian becomes non-Hermitian at higher orders in the inverse mass expansion (non-relativistic expansion). This difficulty in taking a consistent non-relativistic limit to any order in the expansion in was successfully solved by Foldy and Wouthuysen and also independently by Tani which we describe below.

Since the lack of unitarity in (3.208) is the source of the problem in taking the non-relativistic limit consistently, the main idea in the works of Foldy-Wouthuysen as well as Tani is to ensure that the relevant transformation used in going to the non-relativistic limit is manifestly unitary. Thus, for example, let us look at the free Dirac theory where we know that the Hamiltonian has the form (see (1.100) as well as (1.101))

Let us next look for a unitary transformation that will diagonalize the Hamiltonian in (3.209). In this case, such a transformation would also transform the spinor into two 2-component spinors that will be decoupled and we do not have to eliminate one in favor of the other (namely, avoid the problem with “large” and “small” spinors). Let us consider a transformation of the type

where the real scalar parameter of the transformation is a function of p and m,

From the properties of the gamma matrices in (1.83) or (1.91), we note that

and using this we can simplify and write

It follows now that

which leads to

Namely, the transformation (3.210) is indeed unitary.

Under the unitary transformation (3.210), the free Dirac Hamiltonian (3.209) would transform as

So far, our discussion

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