# Causality groups of*S*-matrix

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## Keywords

Coral Gable Wightman Function Nonsingular Transformation Lorentz Trans Causality Group
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## References

- (1).
- (2).More rigorous procedures, but with equivalent results, can be obtained by assuming a topology for
*M*^{nℒ1,1}different than the usual one of*R*^{n}and implying some features of the Lorentz transformations (1).ADSMathSciNetCrossRefGoogle Scholar - (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 ℒ_{n}^{↑}−1,1 and ℛ_{nn}of*L*_{n−1,1/↑}and ℛ_{nn}, respectively, an inhomogeneization of ℒ_{n−1,1}is given by*I*(ℒ^{↑})=ℒ_{n}\(_{\Gamma a}^ \otimes \)ℒ_{n}^{↑}−1,1, where*Γ(a)*is a (real) representation of ℒ_{n}^{↑}−1,1 on*ℛ*_{n}. If we consider a simply connected component of*L*_{n−1,1}, then*Γ*(*a*) = ℒ(*a*_{L}) and conversely there is a representation*a*of*L*_{n−1,1}on*ℒ*_{n}such that*Γ(a)*is the representation of ℒ_{n−1,1}induced by*a*. Furthermore*Γ(a)*can be considered as ad*ℛn*where*Λ*is a representation of ℒ_{n}−1,1, on ℛ_{n}. It has been shown byBerzi andGorini that*ℒ*_{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 sum*I*(ℒ^{↑})=*U*⊕ ℱ_{n−1,1}⊕ ℒ_{n}^{↑}−1,1=*U*⊕ ℱ_{n}^{↑}−1,1 where*U*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.ADSMathSciNetCrossRefGoogle Scholar - (4).It must be noted that the above remarks are valid only for dim
*M*>2. Indeed, for dim*M*=2,*L*^{↑}becomes Abelian and*P*^{↑}is no longer normal in*C*.Google Scholar - (5).R. L. Ingraham: ICTP preprint (Trieste) IC/67/7.Google Scholar
- (6).G. Barucchi andG. C. Teppati:
*Nuovo Cimento*,**51**A, 529 (1967);**52**A, 50 (1967).ADSCrossRefGoogle Scholar - (7).L. Michel: Ecole d’été de Physique Théorique de Cargèse (1965).Google Scholar
- (9).R. Jost:
*Theoretical Physics in the 20th Century*(New York, 1960);E. H. Roffman:*Comm. Math. Phys.*,**4**, 237 (1967).Google Scholar - (10).A. Gamba andG. Luzzato:
*Nuovo Cimento*,**33**, 1732 (1964);J. Rosen: Brown preprint (Providence) NYO-2262TA-161.CrossRefGoogle Scholar - (12).H. Lehmann, K. Symanzik andW. Zimmermann:
*Nuovo Cimento*,**6**, 319 (1957).MathSciNetCrossRefGoogle Scholar - (13).
- (14).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 points*z*_{i}must be considered with their neighborhoods in account of possible zeros of Bessel functions.Google Scholar - (16).L. Sertorio andM. Toller:
*Nuovo Cimento*,**33**, 413 (1964);M. Toller:*Nuovo Cimento*,**37**, 631 (1965);**53**A, 671 (1968).MathSciNetCrossRefGoogle Scholar - (17).
- (18).
- (19).
- (20).As an example for illustrating the situation of causality for interacting fields we note that the causality group is valid also in a neighborhood
*N*of a point*x*ε*M*^{3.1}. Thus, any extended model of particles around*x*, different than the pointlike characterization of the axiomatic field theory, violates Zeeman causality in*N*. 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 the*Symposium on Relativistic Plasma at the CTS, Coral Gables, 1968*(to appear in the*Proceedings of the Symposium*).Google Scholar - (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 in*Nuovo Cimento*); CTS preprint (Coral Gables) M-67-2;R. M. Santilli:*Haag theorem and Lie-admissible algebras*, contributed paper to the*Symposium on Analytic Methods in Mathematical Physics at the Indiana University, 1968*.ADSMathSciNetCrossRefGoogle Scholar

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© Socictà Italiana di Fisica 1968