By S. de Filippo

Specialists commonly relativity, particle physics and mathematical physics talk about facets in their fresh learn. the most emphasis is at the geometrical and algebraic tools utilized in fixing quite a lot of problems.

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15. Salamin, Phys. Rev. Dowling, Phys. Rev. A, 36, 649; 2550 (1987), and (1988 - in press). 16. Ünal, J. Math. Physics, 27, 3055 (1986). 17. Ünal, Physica, 142 A , 467 and 488 (1987). 18. Barut, Ann. der Physik, 4 3 , 83 (1985). 19. Barut, Found, of Physics, 18, 95 (1988). 20. Barut, Ann. der Physik, 4 5 , 31 (1988): Pound, of Phys. Lett, 1, 47 (1988). Geometrical and Algebraic Aspects of Nonlinear Field Theory S. De Filippo, M. Marinaro, G. Marmo and G. V. , Sezione di Trieste, Italy In the last few years a considerable amount of work work has been spent in trying to reconcile the operator formalism in conformai field theory and string theory,which was originally formulated in the complex plane, with the fact that in Polyakov string theory Riemann surfaces of arbitrary genus must be taken into account.

Dorda,M. Pepper: Phys. Rev. Lett. Rev. Lett. Simon:Phys. Rev. Cambridge Philos. Soc. Semenoff:Proc. of Positano Symposium on Quantum Field Theory, F. Singer:Advances In Math. Aoki, Rep. Prog. Phys. 50, 655 (1987) Geometrical and Algebraic Aspects of Nonlinear Field Theory S. De Filippo, M. Marinaro, G. Marmo and G. V. Barut Department of Physics University of Colorado Boulder, CO 80309 Abstract An essay on some features of Schrödinger and Dirac equations in particular their nonlinear generalizations to include self-potentials for radiative processes and relativistic two body interactions.

3) follows, as we have indicated, from an action principle, from the coupled Schrödinger-Maxwell equation. We now give the relativistic version of it, which is in some respect much simpler. The action of the Dirac field coupled to electromagnetism is given by W = J dx | φ ( - Λ Λ - m)Φ - eVYVA» - ^„F""} (7) leading to the field equations μ {η ίθμ - τη)Ψ = μ βη ^{χ)Αμ{χ) ΠΑμ = -βΨ-7μ* μ in the gauge Α = 0. Eliminating Α μ (8) between these two equations, we obtain μ μ (η ίθμ - m ) # = eYV(x) j dyD{x - y)V{y)^V(y) (9) where D(x — y) is the Green's function of • with the boundary conditions to be properly specified.