Towards a Spectral Proof of the Mass Gap in Qcd?

The positive-definiteness of the Hamiltonian operator for Yang–Mills theory in four dimensions is studied by using the Coulomb gauge. It was indeed well known that the Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P built from infinitely many inverse powers of the Laplacian. If space-time is replaced by a compact Riemannian four-manifold without boundary, on which one can exploit the existence of discrete spectral resolutions of the Laplacian, such an infinite sum is found to contribute a positive-definite term to the energy. This is found by direct evaluation of the action of P on square-integrable fields. The decompactification limit is then the key element in trying to prove the existence of the mass gap from the point of view of spectral theory. Talk prepared for the Fifth Leipzig Workshop on Quantum Field Theory under the Influence of External Conditions. Leipzig, September 10–14, 2001.