Self-adjointness of the Fourier expansion of quantized interaction field Lagrangians

Abstract

Regularity properties significantly stronger than were previously known are developed for four-dimensional non-linear conformally invariant quantized fields. The Fourier coefficients of the interaction Lagrangian in the interaction representation-i.e., evaluated after substitution of the associated quantized free field-is a densely defined operator on the associated free field Hilbert space K. These Fourier coefficients are with respect to a natural basis in the universal cosmos M, to which such fields canonically and maximally extend from Minkowski space-time M(0), which is covariantly a submanifold of M. However, conformally invariant free fields over M(0) and M are canonically identifiable. The kth Fourier coefficient of the interaction Lagrangian has domain inclusive of all vectors in K to which arbitrary powers of the free hamiltonian in M are applicable. Its adjoint in the rigorous Hilbert space sense is a(-k) in the case of a hermitian Lagrangian. In particular (k = 0) the leading term in the perturbative expansion of the S-matrix for a conformally invariant quantized field in M(0) is a self-adjoint operator. Thus, e.g., if varphi(x) denotes the free massless neutral scalar field in M(0), then integralM(0):varphi(x)(4):d(4)x is a self-adjoint operator. No coupling constant renormalization is involved here.