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

MathSciNet review:
1204416

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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**, 10.1016/0022-1236(90)90128-8**[2]**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****[3]**M. A. B. Beg,*Higgs particle*(*s*) (A. Ali, ed.), Plenum Press, New York, 1990, pp. 7-38.**[4]**Thomas P. Branson,*Group representations arising from Lorentz conformal geometry*, J. Funct. Anal.**74**(1987), no. 2, 199–291. MR**904819**, 10.1016/0022-1236(87)90025-5**[5]**Roe W. Goodman,*Analytic and entire vectors for representations of Lie groups*, Trans. Amer. Math. Soc.**143**(1969), 55–76. MR**0248285**, 10.1090/S0002-9947-1969-0248285-6**[6]**O. W. Greenberg,*Generalized free fields and models of local field theory*, Ann. Physics**16**(1961), 158–176. MR**0134228****[7]**Stephen M. Paneitz and Irving E. Segal,*Analysis in space-time bundles. I. General considerations and the scalar bundle*, J. Funct. Anal.**47**(1982), no. 1, 78–142. MR**663834**, 10.1016/0022-1236(82)90101-X**[8]**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**, 10.1073/pnas.80.14.4595**[9]**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**, 10.1016/0022-1236(91)90101-A**[10]**J. Pedersen, I. E. Segal, and Z. Zhou,*Massless 𝜙^{𝑞}_{𝑑} quantum field theories and the nontriviality of 𝜙⁴₄*, Nuclear Phys. B**376**(1992), no. 1, 129–142. MR**1164391**, 10.1016/0550-3213(92)90071-I**[11]**Neils Skovhus Poulsen,*On 𝐶^{∞}-vectors and intertwining bilinear forms for representations of Lie groups*, J. Functional Analysis**9**(1972), 87–120. MR**0310137****[12]**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**0251991**, 10.1090/S0002-9904-1969-12428-6**[13]**-,*Local non-commutative analysis*, Problems in Analysis (R. C. Gunning, ed.), Princeton Univ. Press, 1970, pp. 111-130.**[14]**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****[15]**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**, 10.1016/S0924-6509(09)70290-9

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
81T05,
81T08,
81T70

Retrieve articles in all journals with MSC: 81T05, 81T08, 81T70

Additional Information

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

Article copyright:
© Copyright 1994
American Mathematical Society