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From Euclidean to relativistic fields and on the notion of Markoff fields

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Abstract

Recently, Nelson [2] has constructed relativistic fields from Euclidean fields which satisfy the Markoff and reflection property as well as an additional domain assumption. In this paper we replace the Markoff and reflection property by a weaker condition, a very simple positivity condition (“T-positivity”) which can be very easily expressed in terms of the expectation functionalE(f)=〈ω, exp {i φ (f)} ω〉. We show that the special role of the Markoff property in Nelson's approach is entirely due to features also shared byT-positivity. The role of Nelson's domain assumption (A′) in by-passing the difficulties with the paper of Osterwalder and Schrader [4] are made transparent, and possible ways to weaken this assumption are pointed out. If the conditions of [4] should turn out to be sufficient after all, (A′) can be replaced by a simple differentiability condition onE(τf). Our approach can also be applied to Fermi fields. The notion of Markoff and reflection property is discussed and shown to implyT-positivity.

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  1. Institut für Theoretische Physik, Universität Göttingen, Göttingen, Germany

    Gerhard C. Hegerfeldt

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  1. Gerhard C. Hegerfeldt
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Communicated by A.S. Wightman

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Hegerfeldt, G.C. From Euclidean to relativistic fields and on the notion of Markoff fields. Commun.Math. Phys. 35, 155–171 (1974). https://doi.org/10.1007/BF01646614

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  • DOI: https://doi.org/10.1007/BF01646614

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