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References
E. C. Zeeman:Journ. Math. Phys.,5, 490 (1964).
More rigorous procedures, but with equivalent results, can be obtained by assuming a topology forM nℒ1,1 different than the usual one ofR n and implying some features of the Lorentz transformations (1).
It is interesting to note that the Zeeman causality group can be considered in the framework of the inhomogeneization of semisimple Lie algebras recently investigated by the following authors:Y. Ne’eman:Comm. Math. Phys.,3, 181 (1966);V. Berzi andV. Gorini: Milano preprint, IFUM044/SP 1967;J. Rosen:Nuovo Cimento,45 A, 234 (1966);46 B, 1 (1966). Indeed, by introducing the Lie algebras ℒ ↑ n −1,1 and ℛ nn ofL n−1,1/↑ and ℛ nn , respectively, an inhomogeneization of ℒ n−1,1 is given byI(ℒ↑)=ℒ n \(_{\Gamma a}^ \otimes \)ℒ ↑ n −1,1, whereΓ(a) is a (real) representation of ℒ ↑ n −1,1 onℛ n . If we consider a simply connected component ofL n−1,1, thenΓ(a) = ℒ(a L ) and conversely there is a representationa ofL n−1,1 onℒ n such thatΓ(a) is the representation of ℒ n−1,1 induced bya. FurthermoreΓ(a) can be considered as adℛn whereΛ is a representation of ℒ n −1,1, on ℛ n . It has been shown byBerzi andGorini thatℒ n is given by a direct sum of two invariant subspaces (not necessarity irreducible)U and ℱ n−1,1. Then the most general inhomogeneization of ℒ n−1, is given by the vector space direct sumI(ℒ↑)=U ⊕ ℱ n−1,1 ⊕ ℒ ↑ n −1,1=U ⊕ ℱ ↑ n −1,1 whereU is the center,i.e. Ad U Λ ↑=0 or[ℒ, U]=0, and ℱ n−1,1, is an Abelian ideal. Since dilatations appear in the center of Aut (℘ ↑ n −1,1) we can say that the causality group is isomorphic to the inhomogenization of the orthochronous Lorentz groups with normal center. We also recall that, as has been shown byRosen, a classification of all inhomogeneizations of a semisimple Lie algebra can be given in terms of their irreducible representations.
It must be noted that the above remarks are valid only for dimM>2. Indeed, for dimM=2,L ↑ becomes Abelian andP ↑ is no longer normal inC.
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As an example for illustrating the situation of causality for interacting fields we note that the causality group is valid also in a neighborhoodN of a pointx εM 3.1. Thus, any extended model of particles aroundx, different than the pointlike characterization of the axiomatic field theory, violates Zeeman causality inN. This statement seems to be in contrast with some experiments of scattering of high-energy electrons on protons which show that nucleons have an extended structure of about 10−14 cm. The situation can also be investigated, perhaps more directly, in classical relativistic systems (we recall that Zeeman causality is a pure classical formulation) in account of the so-called no-interaction theorem for any theory which is invariant under the Poincaré group. See in this connectionR. M. Santilli:Causality and relativistic plasma, contributed paper to theSymposium on Relativistic Plasma at the CTS, Coral Gables, 1968 (to appear in theProceedings of the Symposium).
Let us recall in this connection that interesting possibilities for avoiding the limitations of the Haag theorem are offered by the enlargment of the analytical dynamics and algebraic formulations in terms of the Lie-admissible structure elsewhere proposed. These procedures have been proved to be justified at a classical level by the assumption of the dissipativity of the interpolating region (which corresponds for particle physics to consider one or more Feynman lines as external), and conceivably they seem to be extendable to quantum theories too. SeeR. M. Santilli:Nuovo Cimento,51 A, 570 (1967); CTS preprint (Coral Gables) M-67-1 (to appear inNuovo Cimento); CTS preprint (Coral Gables) M-67-2;R. M. Santilli:Haag theorem and Lie-admissible algebras, contributed paper to theSymposium on Analytic Methods in Mathematical Physics at the Indiana University, 1968.
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Notes on a lecture given at the University of South Carolina in Columbia on March 7, 1968. Research supported by U. S. Air Force Office of Scientific Research, Washington, D. C., U. S. Air Force Contract no. 49(638)-1260 and NASA Contract no. NASA 10-007-010.
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Santilli, R.M. Causality groups ofS-matrix. Nuovo Cimento B (1965-1970) 55, 578–586 (1968). https://doi.org/10.1007/BF02711667
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DOI: https://doi.org/10.1007/BF02711667