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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonlinear quantum fields in $ \geq 4$ 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
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 $ \phi $ denotes the free massless scalar field in d-dimensional Minkowski space ($ d \geq 4$, even) and if q is an even integer greater than or equal to 4, then $ {\smallint _{{{\mathbf{M}}_0}}}:\phi {(X)^q}:dX$ exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : $ \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$ and to corresponding scattering theory.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1204416-7
PII: S 0002-9947(1994)1204416-7
Article copyright: © Copyright 1994 American Mathematical Society