CLASSICAL YANG-MILLS THEORY WITH FERMIONS II: DIRAC’S OBSERVABLES

Abstract

For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.