In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
that takes the place of the
Dirac fermion
In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category.
Description
In particle physics, all fermions in the standard model ...
s in "ordinary"
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the
Abelian Higgs mechanism.
Matter content and Lagrangian
Matter content
The model consists of a complex scalar field
minimally coupled to a gauge field
.
This article discusses the theory on flat spacetime
(
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
) so these fields can be treated (naïvely) as functions
, and
. 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
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-conne ...
, specifically a principal
connection.
Lagrangian
The dynamics is given by the Lagrangian density
where
*
is the electromagnetic field strength, or
curvature of the connection.
*
is the covariant derivative of the field
*
is the electric charge
*
is the potential for the complex scalar field.
Gauge-invariance
This model is invariant under gauge transformations parameterized by
. This is a real-valued function
Differential-geometric view
From the geometric viewpoint,
is an infinitesimal change of trivialization, which generates the finite change of trivialization
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
, this model exhibits the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
. 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
times the minimum value of
. As shown in 1973 by Nielsen and Olesen, this model, in
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
) and appears as a topological charge associated with the topological current
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
:
The potential is minimized at
, which is chosen to be greater than zero. This produces a circle of minima, with values
, for
a real number.
Scalar chromodynamics
This theory can be generalized from a theory with
gauge symmetry containing a scalar field
valued in
coupled to a gauge field
to a theory with gauge symmetry under the gauge group
, a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
.
The scalar field
is valued in a representation space of the gauge group
, making it a vector; the label of scalar field refers only to the transformation of
under the action of the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, so it is still referred to as a scalar field. The gauge-field is a
-valued 1-form, where
is the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of G.
References
*
* Peskin, M and Schroeder, D. ;''An Introduction to Quantum Field Theory'' (Westview Press, 1995)
{{Quantum field theories
Quantum electrodynamics