A Hamiltonian analysis of Yang-Mills equations
The paper under review is to state some results for Yang-Mills equation on Minkowski space-time and bounded region in \(\mathbb{R}^3\). Basic theme is that the solution set of Yang-Mills equation has some symplectic structure with gauge group symmetry (the action is Hamiltonian). I. Let \(P\) be the space of Cauchy data, \((A, E)\) a pair of time-dependent vector fields on \(\mathbb{R}^3\) with Lie algebra value for some compact Lie group \(G\). The Yang-Mills equations on the Minkowski space-time \(\mathbb{R}^3 \times \mathbb{R}\) can be expressed as evolution equations for \((A, E)\) over \(P\). The key point in this paper is to introduce a weakly symplectic structure on \(P\) (an exact form) as well on the solution set of the Yang-Mills equations with \(w = d \theta\) and \(\langle\theta (A, E)|(a, e)\rangle = \int_{\mathbb{R}^3}E \cdot a d_3 x\). The author claims that (i) the gauge group action is Hamiltonian, (ii) the solution set of Yang-Mills equations is precisely the zero set of the induced moment map from the gauge group action, (iii) the zero value is regular for the moment map, hence the symplectic quotient of the solution set under the gauge symmetry is also a symplectic manifold, (iv) the energy function restricted to the symplectic quotient is Hamiltonian with respect to the induced symplectic structure. II. Then the author looks for a contractible bounded domain \(M\) in \(\mathbb{R}^3\) and considers the Yang-Mills equations on \(M \times\mathbb{R}\) with proper boundary condition (a bag boundary condition, mixed type). In this case the zero value is no longer regular for the induced moment map. But there is still a symplectic stratified structure on the symplectic reduction by \textit{R. Sjamaar} and \textit{E. Lerman}'s work [Ann. Math., II. Ser. 134, 375-422 (1991; Zbl 0759.58019)]. The explicit stratified symplectic spaces are described by the conjugate classes of the compact Lie group \(G\). Similar results for inhomogeneous boundary conditions are stated. Some physical interpretation and the historical background are given in Sections 3 and 4.
scientific article; zbMATH DE number 1486460