Scalar electrodynamics

In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism.

The model consists of a complex scalar field ${\displaystyle \phi (x)}$ minimally coupled to a gauge field ${\displaystyle A_{\mu }(x)}$.

This article discusses the theory on flat spacetime ${\displaystyle \mathbb {R} ^{1,3}}$ (Minkowski space) so these fields can be treated (naïvely) as functions ${\displaystyle \phi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} }$, and ${\displaystyle A_{\mu }:\mathbb {R} ^{1,3}\rightarrow (\mathbb {R} ^{1,3})^{*}}$. The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one. The gauge field is also known as a principal connection, specifically a principal ${\displaystyle {\text{U}}(1)}$ connection.

The dynamics is given by the Lagrangian density

${\displaystyle {\begin{array}{lcl}{\mathcal {L}}&=&(D_{\mu }\phi )^{*}D^{\mu }\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\\&=&(\partial _{\mu }\phi )^{*}(\partial ^{\mu }\phi )-ie((\partial _{\mu }\phi )^{*}\phi -\phi ^{*}(\partial _{\mu }\phi ))A^{\mu }+e^{2}A_{\mu }A^{\mu }\phi ^{*}\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\end{array}}}$

where

This model is invariant under gauge transformations parameterized by ${\displaystyle \lambda (x)}$. This is a real-valued function ${\displaystyle \lambda :\mathbb {R} ^{1,3}\rightarrow \mathbb {R} .}$

${\displaystyle \phi '(x)=e^{ie\lambda (x)}\phi (x)\quad {\textrm {and}}\quad A_{\mu }'(x)=A_{\mu }(x)+\partial _{\mu }\lambda (x).}$