View source for Yang–Mills–Higgs equations

{{Short description|Yang–Mills coupled to a Higgs field}}
In mathematics, the '''Yang–Mills–Higgs equations''' are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a [[Physics:Higgs field|Higgs field]], given by a section of a [[Vector bundle|vector bundle]] (specifically, the [[Adjoint bundle|adjoint bundle]]). These equations are 
:<math>\begin{align}
  D_A*F_A + [\Phi, D_A\Phi] &= 0, \\
  D_A*D_A\Phi &= 0
\end{align}</math>

with a boundary condition
:<math>\lim_{|x|\rightarrow \infty}|\Phi|(x) = 1</math>

where
: ''A'' is a connection on a vector bundle,
: ''D''{{sub|''A''}} is the exterior covariant derivative,
: ''F''{{sub|''A''}} is the curvature of that connection,
: Φ is a section of that vector bundle,
: &lowast; is the Hodge star, and
: [·,·] is the natural, graded bracket.

These equations are named after Chen Ning Yang, [[Biography:Robert Mills (physicist)|Robert Mills]], and [[Biography:Peter Higgs|Peter Higgs]].  They are very closely related to the [[Physics:Ginzburg–Landau theory|Ginzburg–Landau equations]], when these are expressed in a general geometric setting.

M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimensional Minkowski space have a unique  global solution with no restrictions at the spatial infinity. Furthermore,  the solution has the finite propagation speed property.

== Lagrangian ==
The equations arise as the equations of motion of the Lagrangian density
{{Equation box 1
|title='''Yang–Mills–Higgs Lagrangian density'''
|indent=:
|equation = <math>\mathcal{L} = -\frac{1}{4} \left\langle F_{\mu\nu},F^{\mu\nu}\right\rangle + \frac{1}{2}\left\langle D_\mu \phi ,D^\mu \phi\right\rangle - V(\phi) </math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

where <math>\langle\cdot,\cdot\rangle</math> is an invariant symmetric bilinear form on the adjoint bundle. This is sometimes written as <math>\text{tr}</math> due to the fact that such a form can arise from the trace on <math>\mathfrak{g}</math> under some representation; in particular here we are concerned with the [[Adjoint representation|adjoint representation]], and the trace on this representation is the [[Killing form]]. 

For the particular form of the Yang–Mills–Higgs equations given above, the potential <math>V(\phi)</math> is vanishing. Another common choice is <math>V(\phi) = \frac{1}{2}m^2\langle \phi, \phi\rangle</math>, corresponding to a massive Higgs field.

This theory is a particular case of [[Physics:Scalar chromodynamics|scalar chromodynamics]] where the Higgs field <math>\phi</math> is valued in the adjoint representation as opposed to a general representation.

== See also ==

* [[Yang–Mills equations]]
* [[Physics:Scalar chromodynamics|Scalar chromodynamics]]

==References==
*M.V. Goganov and L.V. Kapitansii, "Global solvability of the initial problem for Yang-Mills-Higgs equations", Zapiski LOMI 147,18–48, (1985); J. Sov. Math, 37, 802–822 (1987).

{{Quantum field theories}}

{{DEFAULTSORT:Yang-Mills-Higgs equations}}
[[Category:Partial differential equations]]



{{Sourceattribution|Yang–Mills–Higgs equations}}