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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Nonlinear quantum fields in $\geq 4$ dimensions and cohomology of the infinite Heisenberg group
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by J. Pedersen, I. E. Segal and Z. Zhou PDF
Trans. Amer. Math. Soc. 345 (1994), 73-95 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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