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Il Nuovo Cimento B (1965-1970)

, Volume 55, Issue 2, pp 578–586 | Cite as

Causality groups ofS-matrix

  • R. M. Santilli
Lettere Alla Redazione

Keywords

Coral Gable Wightman Function Nonsingular Transformation Lorentz Trans Causality Group 
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References

  1. (1).
    E. C. Zeeman:Journ. Math. Phys.,5, 490 (1964).ADSMathSciNetCrossRefGoogle Scholar
  2. (2).
    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).ADSMathSciNetCrossRefGoogle Scholar
  3. (3).
    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 ℒn−1,1 and ℛnn ofL n−1,1/↑ and ℛnn, respectively, an inhomogeneization of ℒn−1,1 is given byI(ℒ)=ℒn \(_{\Gamma a}^ \otimes \)n−1,1, whereΓ(a) is a (real) representation of ℒn−1,1 on 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 on n such thatΓ(a) is the representation of ℒn−1,1 induced bya. FurthermoreΓ(a) can be considered as adℛn whereΛ is a representation of ℒn−1,1, on ℛn. It has been shown byBerzi andGorini that nis given by a direct sum of two invariant subspaces (not necessarity irreducible)U and ℱn−1,1. Then the most general inhomogeneization of ℒn−1, is given by the vector space direct sumI(ℒ)=U ⊕ ℱn−1,1 ⊕ ℒn−1,1=U ⊕ ℱn−1,1 whereU is the center,i.e. AdU Λ =0 or[ℒ, U]=0, and ℱn−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.ADSMathSciNetCrossRefGoogle Scholar
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    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.Google Scholar
  5. (5).
    R. L. Ingraham: ICTP preprint (Trieste) IC/67/7.Google Scholar
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    D. Hall andA. S. Wightman:Kgl. Danske Videnskab. Selskab., Math.-Fys. Medd.,31, no. 5 (1957). Let us note that in order to construct a causal automorphism from the invariance of a Wight-man function the pointsz i must be considered with their neighborhoods in account of possible zeros of Bessel functions.Google Scholar
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  17. (20).
    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).Google Scholar
  18. (21).
    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.ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Socictà Italiana di Fisica 1968

Authors and Affiliations

  • R. M. Santilli
    • 1
  1. 1.Center for Theoretical StudiesUniversity of MiamiCoral Gables

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