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.
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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)
Albeverio, S., Høegh-Krohn, R.: The scattering matrix for some non-polynomial interactions, I and II. Helv. Phys. Acta46, 504–534, 536–545 (1973)
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)
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)
Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman path integrals. In: Lecture Notes in Mathematics, Vol. 535. Berlin, Heidelberg, New York: Springer 1976
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
Araki, H.: Hamiltonian formalism and the canonical commutation relations in quantum field theory. J. Math. Phys.1, 492–504 (1960)
Araki, H.: Expansional in Banach algebras. Ann. Sci. Ec. Norm. Sup.6, 67–84 (1973)
Bałaban, T., Raczka, R.: Second quantization of classical nonlinear relativistic field theory. I. Canonical formalism. J. Math. Phys.16, 1475–1481 (1975)
Bałaban, 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)
Bargman, V.: On a Hilbert space of analytic functions and an associated integral transform. Part I. Commun. Pure Appl. Math.14, 187–214 (1961)
Beaume, R., Manuceau, J., Pellet, A., Sirugue, M.: Translation invariant states in quantum mechanics. Commun. Math. Phys.38, 29–45 (1974)
Coester, F., Haag, R.: Representation of states in a field theory with canonical variables. Phys. Rev.117, 1137–1145 (1960)
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)
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)
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)
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)
Chebotarev, A.M., Maslov, V.P.: Jump processes and their applications in quantum mechanics, Viniti. Itogi Nauk Techn.15, 5–78 (1978)
Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Soc. A114, 243–265 (1927)
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 (φ4)3 quantum field theories. Ann. Phys.97, 80–135 (1976)
Friedrichs, K.O.: Mathematical aspects of the quantum theory of fields. New York: Interscience 1953
Friedrichs, K.O., Schapiro, L.: Integration over Hilbert space and outer extensions. Proc. Natl. Acad. Sci.43, 336–338 (1957)
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
Garding, L., Wightman, A.S.: Representations of the commutation relations. Proc. Natl. Acad. Sci.40, 622–626 (1954)
Gel'fand, I.M., Yaglom, A.M.: Integration in functional spaces and its applications in quantum physics. J. Math. Phys.1, 48–49 (1960)
Glimm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981
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
Haag, R.: On quantum field theories. Kgl. Danske Videnskab Selskab. Mat. Fys. Medd.29, No. 12 (1955)
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)
van Hove, L.: Les difficultés de divergence pour un modèle particulier de champ quantifié. Physica18, 145–159 (1957)
Høegh-Krohn, R.: On the spectrum of the space cut-offP(φ): Hamiltonian in two space-time dimensions. Commun. Math. Phys.21, 244–255 (1971)
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)
Magnen, J., Sénéor, R.: The infinite volume limit of the φ 43 model. Ann. Inst. H. Poincaré24, 95–159 (1976)
Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc.45, 99–124 (1949)
von Neumann, J.: In: Collected Works, Vol. 3, Taub, A. (ed.). New York: Pergamon Press 1963
Polley, L., Reents, G., Streater, R.F.: Some covariant representations of massless Boson field. Preprint Darmstadt (1980)
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
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)
Segal, I.E.: Explict formal construction of nonlinear quantum fields. J. Math. Phys.5, 269–282 (1964)
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
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
Simon, B.: TheP(φ2) Euclidean (quantum) field theory. Princeton, NJ: Princeton University Press 1974
Streater, R.F.: Canonical quantization. Commun. Math. Phys.2, 354–374 (1966)
Streit, L.: A generalization of Haag's theorem. Nuovo Cimento A10, 673–680 (1969)
Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys.7, 510–525 (1966)
Umemura, Y.: Carriers of continuous measures in a Hilbertian norm. Publ. R.I.M.S. (Kyoto) A1, 1–47 and 49–54 (1965)
Wentzel, G.: Quantum theory of fields. New York: Interscience 1949
Weyl, H.: The theory of groups and quantum mechanics. 2nd edn. London: Methuen 1931
Xia, Dao-Xing: Measure and integration on infinite dimensional spaces. New York: Academic Press 1972
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Albeverio, S., Blanchard, P., Combe, P. et al. Local relativistic invariant flows for quantum fields. Commun.Math. Phys. 90, 329–351 (1983). https://doi.org/10.1007/BF01206886
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DOI: https://doi.org/10.1007/BF01206886