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

摘要

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 M0, which is covariantly a submanifold of ˜M. However, conformally invariant free fields over M0 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 M0 is a self-adjoint operator. Thus, e.g., if ϕ(x) denotes the free massless neutral scalar field in M0, then ∫M0:ϕ(x)⁴:d4x is a self-adjoint operator. No coupling constant renormalization is involved here.