×

Local relativistic invariant flows for quantum fields. (English) Zbl 0538.46053

Author’s abstract. For quantum fields with trigonometric interaction in arbitrary space dimension we construct a representation of the Lorentz group by automorphisms on a Banach space generated by the Weyl algebra.
Reviewer: P.Hillion

MSC:

46L60 Applications of selfadjoint operator algebras to physics
81T05 Axiomatic quantum field theory; operator algebras
22E70 Applications of Lie groups to the sciences; explicit representations
46N99 Miscellaneous applications of functional analysis
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81T20 Quantum field theory on curved space or space-time backgrounds
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Albeverio, S., Høegh-Krohn, R.: Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions. Commun. Math. Phys.30, 171-200 (1973)
[2] Albeverio, S., Høegh-Krohn, R.: The scattering matrix for some non-polynomial interactions, I and II. Helv. Phys. Acta46, 504-534, 536-545 (1973)
[3] Albeverio, S., Høegh-Krohn, R.: Uniqueness and the global Markov property for Euclidean fields. The case of trigonometric interactions. Commun. Math. Phys.68, 95-128 (1979) · Zbl 0443.60099
[4] Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space time. J. Funct. Anal.16, 39-82 (1974) · Zbl 0279.60095
[5] Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman path integrals. In: Lecture Notes in Mathematics, Vol. 535. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0337.28009
[6] Albeverio, S., Høegh-Krohn, R.: Feynman path integrals and the corresponding method of stationary phase. In: Proceedings of the Conference on Feynman Path Integral, Marseille (1978). Lecture Notes in Physics, Vol. 106. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0424.28014
[7] Araki, H.: Hamiltonian formalism and the canonical commutation relations in quantum field theory. J. Math. Phys.1, 492-504 (1960) · Zbl 0099.22906
[8] Araki, H.: Expansional in Banach algebras. Ann. Sci. Ec. Norm. Sup.6, 67-84 (1973) · Zbl 0257.46054
[9] Ba?aban, T., Raczka, R.: Second quantization of classical nonlinear relativistic field theory. I. Canonical formalism. J. Math. Phys.16, 1475-1481 (1975)
[10] Ba?aban, T., Jezuita, K., Raczka, R.: Second quantization of classical nonlinear relativistic field theory. Part II. Construction of relativistic interacting local quantum field. Commun. Math. Phys.48, 291-311 (1976)
[11] Bargman, V.: On a Hilbert space of analytic functions and an associated integral transform. Part I. Commun. Pure Appl. Math.14, 187-214 (1961) · Zbl 0107.09102
[12] Beaume, R., Manuceau, J., Pellet, A., Sirugue, M.: Translation invariant states in quantum mechanics. Commun. Math. Phys.38, 29-45 (1974) · Zbl 0288.46056
[13] Coester, F., Haag, R.: Representation of states in a field theory with canonical variables. Phys. Rev.117, 1137-1145 (1960) · Zbl 0093.21605
[14] Combe, Ph., Rodriguez, R., Sirugue-Collin, M.: A uniqueness theorem for anticommutation relations and commutation relations of quantum spin systems. Commun. Math. Phys.63, 219-235 (1978)
[15] Combe, Ph., Høegh-Krohn, R., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Poisson processes on groups and Feynman path integrals. Commun. Math. Phys.77, 269-288 (1980) · Zbl 0526.22003
[16] Combe, Ph., Høegh-Krohn, R., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Feynman path integrals with piecewise classical paths. J. Math. Phys.23, 405-411 (1982), and Generalized Poisson processes in quantum mechanics and field theory. Phys. Rep.77, 221-233 (1981) · Zbl 0526.22003
[17] Combe, Ph., Høegh-Krohn, R., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Zero-mass, 2-dimensional real sine-Gordon model without ultraviolet cut-off. Ann. Inst. H. Poincaré37, 115-127 (1982)
[18] Chebotarev, A.M., Maslov, V.P.: Jump processes and their applications in quantum mechanics, Viniti. Itogi Nauk Techn.15, 5-78 (1978) · Zbl 0516.28016
[19] Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Soc. A114, 243-265 (1927) · JFM 53.0847.01
[20] Feldman, J.S., Osterwalder, K.: In: International symposium on mathematical problems in theoretical physics, Araki, H. (ed.). Berlin, Heidelberg, New York: Springer 1975, and The Wightman axioms and the mass gap for weakly coupled (?4)3 quantum field theories. Ann. Phys.97, 80-135 (1976)
[21] Friedrichs, K.O.: Mathematical aspects of the quantum theory of fields. New York: Interscience 1953 · Zbl 0052.44504
[22] Friedrichs, K.O., Schapiro, L.: Integration over Hilbert space and outer extensions. Proc. Natl. Acad. Sci.43, 336-338 (1957) · Zbl 0077.31303
[23] Fröhlich, J.: Classical and quantum statistical mechanics in one and two dimensions: two-component Yukawa- and Coulomb systems. Commun. Math. Phys.47, 233-268 (1976), and In: Constructive field theory, Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1975 · Zbl 1092.82505
[24] Garding, L., Wightman, A.S.: Representations of the commutation relations. Proc. Natl. Acad. Sci.40, 622-626 (1954) · Zbl 0057.09604
[25] Gel’fand, I.M., Yaglom, A.M.: Integration in functional spaces and its applications in quantum physics. J. Math. Phys.1, 48-49 (1960) · Zbl 0092.45105
[26] Glimm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0461.46051
[27] Gross, L.: In: Proceedings of conference on theory and application of analysis in function spaces, Martin, W.Ted., Segal, I.E. (eds.). Cambridge, MA: MIT Press 1972, and in Proceedings of the Vth Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley 1968
[28] Haag, R.: On quantum field theories. Kgl. Danske Videnskab Selskab. Mat. Fys. Medd.29, No. 12 (1955) · Zbl 0067.21102
[29] Heisenberg, W., Pauli, W.: Zur Quantendynamik der Wellenfelder. Z. Phys.56, 1-61 (1929); Zur Quantentheorie der Wellenfelder. II. Z. Phys.59, 168-190 (1930) · JFM 55.0519.07
[30] van Hove, L.: Les difficultés de divergence pour un modèle particulier de champ quantifié. Physica18, 145-159 (1957)
[31] Høegh-Krohn, R.: On the spectrum of the space cut-offP(?): Hamiltonian in two space-time dimensions. Commun. Math. Phys.21, 244-255 (1971)
[32] Klauder, J.R.: Continuous-representation theory. I. Postulates of continuous-representation theory, and II. Generalized relation between quantum and classical dynamics. J. Math. Phys.4, 1055-1057 and 1058-1073 (1963) · Zbl 0127.18701
[33] Magnen, J., Sénéor, R.: The infinite volume limit of the ? 3 4 model. Ann. Inst. H. Poincaré24, 95-159 (1976)
[34] Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc.45, 99-124 (1949) · Zbl 0031.33601
[35] von Neumann, J.: In: Collected Works, Vol. 3, Taub, A. (ed.). New York: Pergamon Press 1963 · Zbl 0188.00105
[36] Polley, L., Reents, G., Streater, R.F.: Some covariant representations of massless Boson field. Preprint Darmstadt (1980) · Zbl 0484.46056
[37] Ruelle, D.: Statistical mechanics. New York: Benjamin 1969 · Zbl 0177.57301
[38] Segal, I.: Distributions in Hilbert space and canonical systems of operators. Trans. Am. Math. Soc.88, 12-41 (1958); and Foundations of the theory of dynamical systems of infinitely many degrees of freedom. I. Kgl. Danske Videnskab Selskab. Mat. Phys. Medd.31, No. 12 (1959) · Zbl 0099.12104
[39] Segal, I.E.: Explict formal construction of nonlinear quantum fields. J. Math. Phys.5, 269-282 (1964) · Zbl 0136.47103
[40] Segal, I.E.: In: Proceedings of the conference on theory and applications of analysis on function space, Martin, W.Ted., Segal, I.E. (eds.). Cambridge, MA: MIT Press 1972
[41] Segal, I.E.: In: Differential geometric methods in mathematics and physics, Marsden, J. (ed.). In: Lecture Notes in Mathematics Berlin, Heidelberg, New York: Springer 1979
[42] Simon, B.: TheP(?2) Euclidean (quantum) field theory. Princeton, NJ: Princeton University Press 1974
[43] Streater, R.F.: Canonical quantization. Commun. Math. Phys.2, 354-374 (1966) · Zbl 0178.28402
[44] Streit, L.: A generalization of Haag’s theorem. Nuovo Cimento A10, 673-680 (1969) · Zbl 0192.61304
[45] Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys.7, 510-525 (1966)
[46] Umemura, Y.: Carriers of continuous measures in a Hilbertian norm. Publ. R.I.M.S. (Kyoto) A1, 1-47 and 49-54 (1965) · Zbl 0181.41502
[47] Wentzel, G.: Quantum theory of fields. New York: Interscience 1949 · Zbl 0033.23701
[48] Weyl, H.: The theory of groups and quantum mechanics. 2nd edn. London: Methuen 1931 · JFM 58.1374.01
[49] Xia, Dao-Xing: Measure and integration on infinite dimensional spaces. New York: Academic Press 1972 · Zbl 0275.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.