Yang–Mills–Higgs equations
In mathematics, the Yang–Mills–Higgs equations are a set of
with a
where
- A is a connection on a vector bundle,
- DA is the exterior covariant derivative,
- FA is the curvature of that connection,
- Φ is a section of that vector bundle,
- ∗ is the Hodge star, and
- [·,·] is the natural, graded bracket.
These equations are named after
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
where is an invariant symmetric bilinear form on the adjoint bundle. This is sometimes written as due to the fact that such a form can arise from the trace on under some representation; in particular here we are concerned with the 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 is vanishing. Another common choice is , corresponding to a massive Higgs field.
This theory is a particular case of scalar chromodynamics where the Higgs field is valued in the adjoint representation as opposed to a general representation.
See also
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).