Nonlinear quantum fields in ≥4 dimensions and cohomology of the infinite Heisenberg group

Abstract

Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if ϕ \phi denotes the free massless scalar field in d-dimensional Minkowski space ( d ≥ 4 d \geq 4 , even) and if q is an even integer greater than or equal to 4, then ∫ M 0 : ϕ ( X ) q : d X {\smallint _{{{\mathbf {M}}_0}}}:\phi {(X)^q}:dX exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : ϕ ( X ) q \phi {(X)^q} : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case d = q = 4 d = q = 4 and to corresponding scattering theory.