J. Pedersen et al., NONLINEAR QUANTUM-FIELDS IN GREATER-THAN-OR-EQUAL-TO-4 DIMENSIONS ANDCOHOMOLOGY OF THE INFINITE HEISENBERG-GROUP, Transactions of the American Mathematical Society, 345(1), 1994, pp. 73-95
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.