Spectral Properties of Yang–mills Theory

Classical Yang–Mills theory in four dimensions is studied by using the Coulomb gauge, as a first step towards considering the mass gap problem in the quantum theory. The Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P which is studied both at perturbative and at non-perturbative level. Upon replacing space-time by a compact Riemannian 4-manifold without boundary, on which one can exploit the existence of discrete spectral resolutions of the Laplacian, the perturbative analysis of P at small coupling constant yields infinitely many inverse powers of the Laplacian. However, since P is an operator-valued function of the Laplacian, one can obtain a Cauchy integral representation for it which leads to the exact evaluation of its mean value on square-integrable fields. By virtue of the residues' theorem, such a mean value is found to vanish exactly. This property may prove useful in understanding whether a mass gap exists in the full quantum theory in Minkowski space-time.