Abstract
We generalize some notions of probability theory and theory of group representations to field theory and to states on the Borchers algebraS. It is shown that every field (relativistic and Euclidean, ...) can be decomposed into a countable number of prime fields and an infinitely divisible field. In terms of states this means that every state onS is a product of an infinitely divisible state and a countable number of prime states, and in this formulation it applies equally well to correlation functions of statistical mechanics and to moments of linear stochastic processes overS orD. Necessary and sufficient conditions for infinitely divisible states are given. It is shown that the fields of the ø 42 -theory are either prime or contain prime factors. Our results reduce the classification problem of Wightman and Euclidean fields to that of prime fields and infinitely divisible fields. It is pointed out that prime fields are relevant for a nontrivial scattering theory.
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Borchers, H. J.: On the structure of the algebra of field operators. Nuovo Cimento24, 214 (1962)
Borchers, H. J.: Algebraic aspects of Wightman field theory. In: Sen. R. N., Weil, C. (Eds.): Statistical mechanics and field theory. Haifa Lectures 1971. New York: Halsted Press 1972
Wyss, W.: On Wightman's theory of quantized fields. In: Lectures in theoretical physics. Boulder: University of Colorado 1968, New York: Gordon and Breach 1969
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
Loève, M.: Probability theory. Second edition. Princeton: van Nostrand 1960
Streater, R. F.: A continuum analogue of the lattice gas. Commun. math. Phys.12, 226 (1969)
Guichardet, A.: Symmetric Hilbert spaces and related topics. Lecture Notes in Mathematics. (Chapter 4.) Berlin-Heidelberg-New York: Springer 1972
Lukacs, E.: Characteristic functions, 2nd ed. London: Griffin 1970
Gelfand, I. M., Vilenkin, N. Ya.: Generalized functions, Vol. 4, Chapter III, § 4. New York: Academic Press 1964
Yngvason, J.: On the algebra of test functions for field operators. Commun. math. Phys.34, 315 (1973)
Trèves, F.: Topological vector spaces, distributions, and kernels, cf. Theorem 45.2. New York: Academic Press 1967
Velo, G., Wightman, A. (Editors): Constructive quantum field theory. Lecture Notes in Physics 25. Berlin-Heidelberg-New York: Springer 1973
Simon, B.: TheP(ø)2 Euclidean (quantum) field theory. Princeton: Princeton University Press 1974
Nelson, E.: Construction of quantum fields from Markoff fields. J. Funct. Anal.12, 97 (1973)
Hegerfeldt, G. C.: From Euclidean to relativistic fields and on the notion of Markoff fields. Commun. math. Phys.35, 155 (1974)
Osterwalder, K., Schrader, R.: Axioms for Euclidean Green's functions. II. Commun. math. Phys.42, 281 (1975)
Lebowitz, J.: GHS and other inequalities. Commun. math. Phys.35, 87 (1974)
Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2-model and other applications of high temperature expansions. In. Constructive quantum field theory. Lecture Notes in Physics 25, cf. p. 172. Berlin-Heidelberg-New York: Springer 1973
Newman, C. M.: Inequalities for Ising models and field theories which obey the Lee-Yang-theorem. Commun. math. Phys.41, 1 (1975)
Jost, R.: The general theory of quantized fields, p. 74. Providence: Am. Math. Soc. 1965
Rinke, M.: A remark on asymptotic completeness of local fields. Commun. math. Phys.12, 324 (1969)
Uhlmann, A.: Über die Definition der Quantenfelder nach Wightman und Haag. Wiss. Z. Karl-Marx-Univ. Leipzig, Math. Naturw. Reihe11, 213 (1962)
Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics. I. Commun. math. Phys.23, 199 (1971)
Streater, R. F.: Infinitely divisible representations of Lie algebras. Z. Wahrscheinlichkeitstheorie verw. Geb.19, 67–80 (1971)
Mathon, D., Streater, R. F.: Infinitely divisible representations of Clifford algebras. Z. Wahrscheinlichkeitstheorie verw. Geb.20, 308–316 (1971)
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Hegerfeldt, G.C. Prime field decompositions and infinitely divisible states on Borchers' tensor algebra. Commun.Math. Phys. 45, 137–151 (1975). https://doi.org/10.1007/BF01629244
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DOI: https://doi.org/10.1007/BF01629244