Gauge theories play a central role in modern physics, providing a unified framework for describing the fundamental interactions of nature. The introduction of local gauge invariance to the Dirac Lagrangian leads to the coupling of fermionic fields with gauge fields, paving the way for the development of quantum electrodynamics (QED) and other gauge theories. This document explores the formulation of the free Dirac Lagrangian, its invariance under global and local symmetries, and the derivation of the electromagnetic interaction through the introduction of a covariant derivative.
The free Dirac Lagrangian is
\[
L = i\bar{\psi}(x)\gamma^{\mu}\partial_{\mu}\psi(x) – m\bar{\psi}(x)\psi(x)
\]
Where:
\[
\bar{\psi}(x) = \psi(x)^{\dagger}\gamma^{0}
\]
\[
\gamma^{\mu} = \begin{pmatrix}
0 & \bar{\sigma}^{\mu} \
\sigma^{\mu} & 0
\end{pmatrix}
\]
The free Lagrangian above is invariant under a global phase symmetry \( \psi'(x) = e^{i\alpha}\psi(x) \):
\[
L’ = i\bar{\psi’}(x)\gamma^{\mu}\partial_{\mu}\psi'(x) – m\bar{\psi’}(x)\psi'(x)
\]
\[
= i\bar{\psi}(x)e^{-i\alpha}\gamma^{\mu}\partial_{\mu}\psi(x)e^{i\alpha} – m\bar{\psi}(x)e^{-i\alpha}\psi(x)e^{i\alpha}
\]
\[
= i\bar{\psi}(x)e^{-i\alpha}e^{i\alpha}\gamma^{\mu}\partial_{\mu}\psi(x) – m\bar{\psi}(x)e^{-i\alpha}e^{i\alpha}\psi(x)
\]
\[
L’ = L
\]
But now let’s look at what happens if the \( \alpha \) parameter of the phase symmetry becomes dependent on the position \( \alpha(x) \), and in this case, the new symmetry will be called a gauge symmetry:
\[
\psi'(x) = e^{i\alpha(x)}\psi(x)
\]
\[
\partial_{\mu}\psi'(x) = e^{i\alpha(x)}\partial_{\mu}\psi(x) + i\psi(x) e^{i\alpha(x)}\partial_{\mu}\alpha(x)
\]
In calculating the derivative \( \partial_{\mu}\psi'(x) \), we find an additional term \( i\psi(x)e^{i\alpha(x)}\partial_{\mu}\alpha(x) \) that cannot be undone.
To address this, we introduce a new type of derivative, the covariant derivative, defined as:
\[
\partial_{\mu}\psi \to D_{\mu}\psi \quad \text{and} \quad \partial_{\mu}\bar{\psi} \to \overline{D_{\mu}\psi}
\]
For the new term \( i\bar{\psi}(x)\gamma^{\mu}D_{\mu}\psi(x) \) to remain invariant, it is necessary that
\[
\bar{\psi’} = A\bar{\psi}, \quad (D_{\mu}\psi)’ = BD_{\mu}\psi, \quad A = B^{-1}
\]
That is,
\[
\bar{\psi}’ = e^{-i\alpha(x)}\bar{\psi}
\]
\[
(D_{\mu}\psi)’ = e^{i\alpha(x)}D_{\mu}\psi, \quad (\overline{D_{\mu}\psi})’ = e^{-i\alpha(x)}\overline{D_{\mu}\psi}
\]
Thus, the new Lagrangian invariant under local symmetry is:
\[
L = i\bar{\psi}(x)\gamma^{\mu}D_{\mu}\psi(x) + i\overline{D_{\mu}\psi(x)}\gamma^{\mu}\psi(x) – m\bar{\psi}(x)\psi(x)
\]
The simplest form of the covariant derivative is:
\[
D_{\mu}\psi = \partial_{\mu}\psi + ieA_{\mu}(x)\psi
\]
\[
\overline{D_{\mu}\psi} = (\partial_{\mu}\psi)^{\dagger}\gamma^{0} = \partial_{\mu}\bar{\psi} – ieA_{\mu}(x)\bar{\psi}
\]
The \( A_{\mu}(x) \) term transforms as:
\[
A_{\mu}'(x) = A_{\mu}(x) – \frac{1}{e}\partial_{\mu}\alpha(x)
\]
Finally, the dynamics of \( A_{\mu}(x) \) are introduced through the field tensor:
\[
[D_{\mu}, D_{\nu}] = ie(\partial_{\mu}A_{\nu} – \partial_{\nu}A_{\mu}) = ieF_{\mu\nu}
\]
Resulting in the Lagrangian:
\[
L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i\bar{\psi}(x)\gamma^{\mu}D_{\mu}\psi(x) – m\bar{\psi}(x)\psi(x)
\]
Or equivalently:
\[
L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i\bar{\psi}(x)\gamma^{\mu}\partial_{\mu}\psi(x) – m\bar{\psi}(x)\psi(x) – ie\bar{\psi}(x)\gamma^{\mu}A_{\mu}(x)\psi(x)
\]
The equation of motion for \( \psi(x) \) is:
\[
(i\hbar\gamma^{\mu}\partial_{\mu} – mc – ie\gamma^{\mu}A_{\mu}(x))\psi(x) = 0
\]
By requiring gauge symmetry in the Lagrangian, a new field \( A_{\mu}(x) \) is introduced, corresponding to the electromagnetic field and its quantum particle, the photon. The resulting interaction describes the electromagnetic coupling with fermions.