NONLINEAR QUANTUM-FIELDS IN GREATER-THAN-OR-EQUAL-TO-4 DIMENSIONS ANDCOHOMOLOGY OF THE INFINITE HEISENBERG-GROUP

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 l-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mappi ng from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, i s the quantization of a given classical interaction Lagrangian or Hami ltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if phi denotes the free massless scalar held in d-dimensional Minkowski space (d greater than or equal to 4, even) and if q is an even integer greater than or equal to 4, th en integral(M0) :phi(X)4 : dX exists as a nonvanishing, Poincare invar iant, hermitian, selfadjointly extendable operator, where : phi(X)4 : denotes the Wick power. Applications are also made to the rigorous est ablishment of basic symbolic operators in heuristic quantum held theor y, including certain massive field theories; to a class of pseudo-inte racting fields obtained by substituting the free held into desingulari zed expressions for the total Hamiltonian in the conformally invariant case d = q = 4 and to corresponding scattering theory.