Reports on Mathematical Physics

Vol. 9(1976) No 3, pp. 377-385

LOGICS GENERATED BY CAUSALITY STRUCTURES.

COVARIANT REPRESENTATIONS OF THE GALILEAN LOGIC

W. CEGLA and A. Z. JADCZYK

Institute of Theoretical Physics, University of Wroctaw, 50-205, Wroclaw, Poland

*(Received June 13, 1975)*

*(Revised manuscript received November 12, ]975)*

We consider the causal structure of space-time in the logic approach. The general form of covariant representations of the Galilean logic, which correspond to a localizable Galilean system, is found.

**1. Introduction**

The goal of this work is to start a systematic investigation of the causal structure of space-time. This structure reflects itself in a structure of the algebra of observables of a quantum system via the correspondence O-->A(O) between space-time regions O and the algebras of observables related to O(see Haag and Kastler [5]). For an elementary particle we can take A(O) to be a yon Neumann algebra generated by projections localizing the particle in subregions of O. Unfortunately, by the theorem of Borchers [3], it is impossible to reconcile such a localization of a relativistic particle with the positivity of the energy. However, relativistic wave equations (finite or infinite component ones) are of great importance in spite of the fact that they admit negative-energy solutions. On the other hand, there is a problem (both relativistic and non-relativistic) of the satisfactory space-time description of unstable particles. Such an unstabilitv induces in space-time a causality structure which is stronger than in the stable case. This is why we try to formulate an abstract theory of the causality. The Galilean causality is then discussed in some details. A general form (not the most general) of a covariant representation of the persistent Galiean logic is found to correspond to a localization of a distinguished point of a free composed system (infinite-dimensional Galilean wave equations). A different approach to similar problems can be found in the work of Barut and Malin [2].

**2. Causality spaces**

**2a. Orthogonality spaces****3. Galilean causal logic****4. Covariant representations of the Galilean logic**

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Last modified on: May 21, 2000.

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