Nonlinear quantum fields in $\geq 4$ dimensions and cohomology of the infinite Heisenberg group
HTML articles powered by AMS MathViewer
- by J. Pedersen, I. E. Segal and Z. Zhou
- Trans. Amer. Math. Soc. 345 (1994), 73-95
- DOI: https://doi.org/10.1090/S0002-9947-1994-1204416-7
- PDF | 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.References
- John C. Baez, Irving E. Segal, and Zheng-Fang Zhou, The global Goursat problem and scattering for nonlinear wave equations, J. Funct. Anal. 93 (1990), no. 2, 239–269. MR 1073286, DOI 10.1016/0022-1236(90)90128-8
- John C. Baez, Irving E. Segal, and Zheng-Fang Zhou, Introduction to algebraic and constructive quantum field theory, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1992. MR 1178936, DOI 10.1515/9781400862504 M. A. B. Beg, Higgs particle(s) (A. Ali, ed.), Plenum Press, New York, 1990, pp. 7-38.
- Thomas P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), no. 2, 199–291. MR 904819, DOI 10.1016/0022-1236(87)90025-5
- Roe W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969), 55–76. MR 248285, DOI 10.1090/S0002-9947-1969-0248285-6
- O. W. Greenberg, Generalized free fields and models of local field theory, Ann. Physics 16 (1961), 158–176. MR 134228, DOI 10.1016/0003-4916(61)90032-X
- Stephen M. Paneitz and Irving E. Segal, Analysis in space-time bundles. I. General considerations and the scalar bundle, J. Functional Analysis 47 (1982), no. 1, 78–142. MR 663834, DOI 10.1016/0022-1236(82)90101-X
- S. M. Paneitz and I. E. Segal, Selfadjointness of the Fourier expansion of quantized interaction field Lagrangians, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 14, Phys. Sci., 4595–4598. MR 708444, DOI 10.1073/pnas.80.14.4595
- S. M. Paneitz, J. Pedersen, I. E. Segal, and Z. Zhou, Singular operators on boson fields as forms on spaces of entire functions on Hilbert space, J. Funct. Anal. 100 (1991), no. 1, 36–58. MR 1124292, DOI 10.1016/0022-1236(91)90101-A
- J. Pedersen, I. E. Segal, and Z. Zhou, Massless $\varphi ^q_d$ quantum field theories and the nontriviality of $\varphi ^4_4$, Nuclear Phys. B 376 (1992), no. 1, 129–142. MR 1164391, DOI 10.1016/0550-3213(92)90071-I
- Neils Skovhus Poulsen, On $C^{\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87–120. MR 0310137, DOI 10.1016/0022-1236(72)90016-x
- Irving Segal, Notes towards the construction of non-linear relativistic quantum fields. II. The basic nonlinear functions in general space-times, Bull. Amer. Math. Soc. 75 (1969), 1383–1389. MR 251991, DOI 10.1090/S0002-9904-1969-12428-6 —, Local non-commutative analysis, Problems in Analysis (R. C. Gunning, ed.), Princeton Univ. Press, 1970, pp. 111-130.
- I. E. Segal, The complex-wave representation of the free boson field, Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New York-London, 1978, pp. 321–343. MR 538026
- Irving Segal, Hilbert-Schmidt cohomology of Weyl systems, Aspects of mathematics and its applications, North-Holland Math. Library, vol. 34, North-Holland, Amsterdam, 1986, pp. 727–734. MR 849587, DOI 10.1016/S0924-6509(09)70290-9
Bibliographic 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