Yang–Mills–Higgs equations

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 Higgs field, given by a section of a vector bundle. These equations are

$D_{A}*F_{A}+[\Phi ,D_{A}\Phi ]=0,$

$D_{A}*D_{A}\Phi =0$

with a boundary condition

$\lim _{|x|\rightarrow \infty }|\Phi |(x)=1.$

These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the 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.

See also

• Yang–Mills equations

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).