A new formulation of nonrelativistic scattering theory is developed which expresses the S matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the 8 matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving 8 function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed pathintegral expression for the T matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.
We study a power-series expansion for a conserved quantity K in the case of the two-dimensional Henon-Heiles potential. An alternative technique to that of Gustavson [Astron. J. 71, 670 (1966)] is applied to find the coefficients in the expansion for E. The technique is used to determine twelve orders for the conserved quantity E, more than twice as many as that calculated by Gustavson. We investigate the degree of constancy of our truncated E in regions where the motion is known to be chaotic and also where it is nonchaotic.
We generalize the path integral formalism of quantum mechanics to include the use of arbitrary infinitesimal generators, thus providing explicit expressions for solutions of a wide class of differential equations. In particular, we develop a method of calculating the eigenfunctions of a large class of operators.
I t is shown that unitarity allows fixed poles a t certain nonsense points of either right or wrong signature. The conditions for the existence of these poles are found. These conditions are then used to locate the poles allowed in hadronic reactions. Possible mechanisms for the poles are considered. I t is then argued that fixed poles provide the most natural explanation of the conspiracy phenomenon. 9 ILI. Jacob and G. C. Wick, Ann. Phps. (N. Y.) 7, 404 (1959). 10 Properties of the rotation functions of the first kind, dh,J, and second kind, ehPJ, may be found in W. Drechsler, Nuovo Cimento 53A, 115 (1968). 11 The unitarity condition also serves to Gx our normalization of +J(t).
I t is shown that the ordinary Born approximation for pn and pp charge-exchange scattering correctly accounts for (1) the shape of the forward peak for 0 5 (-t)Sw2/2 a t P L =~ GeV/c, and (2) the energy dependence of the cross sections a t t=O in the energy range PL=2-8 GeV/c. This result is analogous to the %yell-kn0~r.n success of the electric Born approximation in T+ photoproduction. I t is then shown that the simplest interpretation of this surprising result within the framework of Regge-pole theory is in terms of the fixed poles which are allowed by unitarity in hadronic amplitudes a t certain nonsense points of right signature. Finally, it is shown how such a fixed pole a t a nonsense point of one helicity amplitude affects another amplitude for which the corresponding point is sense.'
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