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

Authors:
J. Pedersen, I. E. Segal and Z. Zhou

Journal:
Trans. Amer. Math. Soc. **345** (1994), 73-95

MSC:
Primary 81T05; Secondary 81T08, 81T70

DOI:
https://doi.org/10.1090/S0002-9947-1994-1204416-7

MathSciNet review:
1204416

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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 denotes the free massless scalar field in *d*-dimensional Minkowski space (, even) and if *q* is an even integer greater than or equal to 4, then exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : : 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 and to corresponding scattering theory.

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DOI:
https://doi.org/10.1090/S0002-9947-1994-1204416-7

Article copyright:
© Copyright 1994
American Mathematical Society