Boundary Operators in Quantum Field Theory

When quantum fields are studied on manifolds with boundary, the corresponding one-loop quantum theory for bosonic gauge fields with linear covariant gauges needs the assignment of suitable boundary operators for elliptic differential operators of Laplace type. It is here shown that, on requiring that the full boundary operator should be a projector, at least two sets of such operators are found to arise. The former leads to the Gilkey–Smith boundary-value problem, i.e. mixed boundary conditions involving both normal and tangential derivatives of the field. The latter leads instead to boundary conditions involving a nilpotent operator, two complementary projectors, a first-order differential operator on the boundary and the identity map. When the boundary operator is allowed to be pseudo-differential while remaining a projector, the conditions on its kernel leading to strong ellipticity of the boundary-value problem are studied in detail. This makes it possible to develop a theory of one-loop quantum gravity from first principles only, i.e. the physical principle of invariance under infinitesimal diffeomorphisms and the mathematical requirement of a good elliptic theory.