Scalar electrodynamics

Matter content and Lagrangian

Matter content

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

This article discusses the theory on flat spacetime \mathbb{R}^{1,3} (Minkowski space) so these fields can be treated (naïvely) as functions \phi:\mathbb{R}^{1,3}\rightarrow \mathbb{C}, and 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 \text{U}(1) connection.

Lagrangian

The dynamics is given by the Lagrangian density

\begin{array}{lcl} \mathcal{L} & = & (D_\mu \phi)^* D^\mu \phi - V(\phi^\phi) -\frac14 F_{\mu\nu}F^{\mu\nu} \ & = & (\partial_\mu \phi)^(\partial^\mu \phi)-ie((\partial_\mu \phi)^\phi-\phi^(\partial_\mu \phi))A^\mu +e^2A_\mu A^\mu\phi^\phi - V(\phi^\phi) -\frac14 F_{\mu\nu}F^{\mu\nu} \end{array}

where

• F_{\mu\nu}=(\partial_\mu A_\nu - \partial_\nu A_\mu) is the electromagnetic field strength, or curvature of the connection.
• D_\mu\phi=(\partial_\mu \phi - i e A_\mu \phi) is the covariant derivative of the field \phi
• e = -|e| is the electric charge
• V(\phi^*\phi) is the potential for the complex scalar field.

Gauge-invariance

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

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

Differential-geometric view

From the geometric viewpoint, \lambda is an infinitesimal change of trivialization, which generates the finite change of trivialization e^{ie\lambda}:\mathbb{R}^{1,3}\rightarrow \text{U}(1). In physics, it is customary to work under an implicit choice of trivialization, hence a gauge transformation really can be viewed as a change of trivialization.

Higgs mechanism

If the potential is such that its minimum occurs at non-zero value of |\phi|, this model exhibits the Higgs mechanism. This can be seen by studying fluctuations about the lowest energy configuration: one sees that the gauge field behaves as a massive field with its mass proportional to e times the minimum value of |\phi|. As shown in 1973 by Nielsen and Olesen, this model, in 2+1 dimensions, admits time-independent finite energy configurations corresponding to vortices carrying magnetic flux. The magnetic flux carried by these vortices are quantized (in units of \tfrac{2\pi}{e}) and appears as a topological charge associated with the topological current

J_{top}^\mu =\epsilon^{\mu\nu\rho} F_{\nu\rho}\ .

These vortices are similar to the vortices appearing in type-II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions.

Example

A simple choice of potential for demonstrating the Higgs mechanism is

:V(|\phi|^2) = \lambda(|\phi|^2 - \Phi^2)^2.

The potential is minimized at |\phi| = \Phi, which is chosen to be greater than zero. This produces a circle of minima, with values \Phi e^{i\theta}, for \theta a real number.

Scalar chromodynamics

This theory can be generalized from a theory with U(1) gauge symmetry containing a scalar field \phi valued in \mathbb{C} coupled to a gauge field A_\mu to a theory with gauge symmetry under the gauge group G, a Lie group.

The scalar field \phi is valued in a representation space of the gauge group G, making it a vector; the label of "scalar" field refers only to the transformation of \phi under the action of the Lorentz group, so it is still referred to as a scalar field, in the sense of a Lorentz scalar. The gauge-field is a \mathfrak{g}-valued 1-form, where \mathfrak{g} is the Lie algebra of G.

References

• {{cite journal

• Peskin, M and Schroeder, D.; An Introduction to Quantum Field Theory (Westview Press, 1995)