Yang-Mills and Dirac Fields with Inhomogeneous Boundary Conditions

Abstract

Finite time existence and uniqueness of solutiolllS of the evolution equations of minimally coupled Yang-Mills and Dirac system are proJed for inhomogeneous boundary conditions. A characterization of the space of solutions lof minimally coupled Yang-Mills and Dirac equations is obtained in terms of the boundary dataand the Cauchy data satisfying. the constraint equation. The proof is based on~special gauge fixing and a singular perturbation result for the existence of continuous sJmigroups.

Geometry of the Constraint Sets for Yang–Mills–Dirac Equations with Inhomogeneous Boundary Conditions

“…Corollary 3. The space C β /GS(P ) 1 of GS(P 1 ) orbits in C β is a quotient manifold of C β , and C β has the structure of a principal fibre bundle over C β /GS(P ) 1 with structure group GS(P ) 1 .…”

“…Elements of the Lie algebra gs(P ) 1 of GS(P ) 1 are given by maps ξ ∈ H 3 (M, g) such that ξ | ∂M = 0, and n grad ξ = 0. Their action in P is given by the vector field…”

“…(8). Hence it preserves ω = dθ, and it is Hamiltonian with an equivariant momentum map J 1 such that, for every ξ ∈ gs(P ) 1 ,…”

“…Moreover, since the action of GS(P ) 1 in C β is continuous, proper and free, C β has the structure of the principal fibre bundle with base space C β /GS(P ) 1 and structure group GS(P ) 1 . A proof that a continuous, smooth, proper and free action of a Lie group in a finite dimensional manifold gives rise to the structure of a principal fibre bundle is given in Ref.…”

“…The Cauchy data (A, E, ) for the equations under investigation form the extended phase space (1) Here the Cauchy data (A, E) for the Yang-Mills fields are considered to be g-valued vector fields on M. The matter fields are Dirac spinor fields on M with values in the vector space V . In the preceding paper, [1], we have established a local in time existence and the uniqueness of solutions of the Yang-Mills-Dirac evolution equations subject to the gauge condition …”