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), 7395
MSC:
Primary 81T05; Secondary 81T08, 81T70
MathSciNet review:
1204416
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Abstract: Aspects of the cohomology of the infinitedimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1cohomology 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 ddimensional 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 pseudointeracting 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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 , Introduction to algebraic and constructive quantum field theory, Princeton Univ. Press, 1992. MR 1178936 (93m:81002)
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 R. W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969), 5576. MR 0248285 (40:1537)
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 , Selfadjointness of the Fourier expansion of quantized interaction field Lagrangians, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), 45954598. MR 708444 (85f:81042)
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 , Local noncommutative analysis, Problems in Analysis (R. C. Gunning, ed.), Princeton Univ. Press, 1970, pp. 111130.
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 , The complexwave representation of the free boson field, Suppl. Studies, 3, Adv. in Math., Academic Press, 1978, pp. 321344. MR 538026 (82d:81069)
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 , HilbertSchmidt cohomology of Weyl systems, Aspects of Mathematics and its Applications (J. A. Barroso, ed.), Elsevier Science, Amsterdam, 1986, pp. 727734. MR 849587 (88h:47073)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412044167
PII:
S 00029947(1994)12044167
Article copyright:
© Copyright 1994
American Mathematical Society
