Gauge symmetries

Abstract

The simplest example of a phenomenological application of the theory of spinor fields is electrodynamics, originally formulated by P.A.M. Dirac to provide a relativistic description of electrons and photons. The theory is based on a pair of spinor fields of opposite chiralities, ξ_R and ξ_L; invariance under phase multiplication of the spinor fields is assumed, and the corresponding conserved current is identified with the electromagnetic current. No scalar field is present. Under these assumptions, the free-field Lagrangian is given by

$$ \mathcal{L}_0 = i\xi _R^\dag \bar \sigma _\mu \partial ^\mu \xi _R + i\xi _L^\dag \sigma _\mu \partial ^\mu \xi _L - m\left( {\xi _R^\dag \xi _L + \xi _L^\dag \xi _R } \right). $$

(6.1)

$\mathcal{L}_0$ is manifestly invariant under the phase transformations

$$ {\xi _R}\left( x \right) \to {e^{ie\Lambda }}{\xi _R}\left( x \right)s $$

(6.2)

$$ {\xi _L}\left( x \right) \to {e^{ie}}^\Lambda {\xi _L}\left( x \right), $$

(6.3)

where e, Λ are real constants. Notice that we have inserted a Dirac mass term, which is allowed by the assumed invariance properties, while Majorana mass terms are not, as mentioned in Section 5.2. The conserved current is given by

$$ J_\mu = - e\left( {\xi _R^\dag \bar \sigma _\mu \xi _R + \xi _L^\dag \sigma _\mu \xi _L } \right);{\mathbf{ }}\partial ^\mu J_\mu = 0, $$

(6.4)

and the constant e plays the role of elementary charge.