Show that in the nonrelativistic limit, the Gordon...

Show that in the nonrelativistic limit, the Gordon decomposition, (6.7), of the electron current, (6.6), separates the electron interaction with an electromagnetic field $A_{\mu}$ into a part arising from its charge, $-e$, and a part due to its magnetic moment, $-e / 2 m$. Assume that $A_{\mu}$ is independent of $t$, so that (6.4) becomes $$ T_{f i}=-i 2 \pi \delta\left(E_{f}-E_{i}\right) \int j_{\mu}^{f i} A^{\mu} d^{3} x $$ To identify the magnetic moment interaction $(-\mu \cdot \mathbf{B})$, it suffices to show that $$ \int\left[-\frac{e}{2 m} \bar{\psi}_{f} i \sigma_{\mu \nu}\left(p_{f}-p_{i}\right)^{\nu} \psi_{i}\right] A^{\mu} d^{3} x=\int \psi_{A}^{f \dagger}\left(\frac{e}{2 m} \mathbf{\sigma} \cdot \mathbf{B}\right) \psi_{A}^{i} d^{3} x $$ where $\psi_{A}$ denotes the upper two (or "large") components of $\psi$; compare with eqs. (5.31) and (5.32).

Show that in the nonrelativistic limit, the Gordon decomposition, (6.7), of the electron current, (6.6), separates the electron interaction with an electromagnetic field $A_{\mu}$ into a part arising from its charge, $-e$, and a part due to its magnetic moment, $-e / 2 m$. Assume that $A_{\mu}$ is independent of $t$, so that (6.4) becomes
$$
T_{f i}=-i 2 \pi \delta\left(E_{f}-E_{i}\right) \int j_{\mu}^{f i} A^{\mu} d^{3} x
$$
To identify the magnetic moment interaction $(-\mu \cdot \mathbf{B})$, it suffices to show that
$$
\int\left[-\frac{e}{2 m} \bar{\psi}_{f} i \sigma_{\mu \nu}\left(p_{f}-p_{i}\right)^{\nu} \psi_{i}\right] A^{\mu} d^{3} x=\int \psi_{A}^{f \dagger}\left(\frac{e}{2 m} \mathbf{\sigma} \cdot \mathbf{B}\right) \psi_{A}^{i} d^{3} x
$$
where $\psi_{A}$ denotes the upper two (or "large") components of $\psi$; compare with eqs. (5.31) and (5.32).

Key Concepts

Electromagnetic Coupling

Electromagnetic coupling in quantum mechanics is implemented via the minimal coupling prescription, where the momentum operator is replaced by p - eA. In the context of relativistic quantum field theory, the electron current couples to the electromagnetic four-potential A?. The analysis of this coupling using methods such as the Gordon decomposition helps separate the effects due to charge (leading to Coulomb interaction) from those due to the magnetic moment (leading to spin-magnetic field interactions).

Magnetic Moment Interaction

In the Dirac theory, the magnetic moment interaction stems from the intrinsic spin of the electron. When analyzed using the Gordon decomposition, one finds a term proportional to the spin operator coupled to the magnetic field. This term provides a natural explanation for the electron's magnetic dipole moment, with the appropriate gyromagnetic ratio emerging from the structure of the Dirac equation in the nonrelativistic limit. It explicitly shows how the intrinsic angular momentum of the electron couples to external magnetic fields.

Dirac Spinor Components

In the Dirac formalism, the four-component spinor can be divided into two parts: the large (upper) components and the small (lower) components. In the nonrelativistic limit, the large components dominate the physical behavior, effectively reducing the description to a two-component spinor. This reduction is central to deriving familiar nonrelativistic results, such as those appearing in the Pauli equation, and is essential for identifying the specific contributions of the electron's charge and magnetic moment to its interactions with electromagnetic fields.

Nonrelativistic Limit

This is the regime in which the velocities of particles are much smaller than the speed of light, so relativistic effects can be neglected. In the context of the Dirac equation, the nonrelativistic limit involves keeping only the large (upper) components of the Dirac spinor, which essentially reduces the four-component spinor to a two-component Pauli spinor. This limit simplifies the theory and allows one to clearly identify nonrelativistic quantum mechanical phenomena such as the magnetic moment and its interaction with external fields.

Gordon Decomposition

The Gordon decomposition is a mathematical technique used in quantum field theory to rewrite the Dirac current in a form that separates the contributions from different physical interactions. By rewriting the current, one can distinguish a part that involves the momentum (the convective or charge interaction) and a part involving the spin matrix (the magnetic moment interaction). This decomposition is particularly useful in clarifying how the electron’s charge and intrinsic magnetic moment couple to the electromagnetic field.

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