moment of a particle is defined in general as (c = 1)
Since S =
where the nuclear magneton is defined to be
with mP 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
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
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 has been quite general and the parameter of the transformation, θ, has been arbitrary. However,
