MacDowell symmetry and fermion regge trajectories

Phase shift analyses of pion-nucleon scattering have led to the discovery of a large number of excited baryonic states having positive and negative parity. A fascinating challenge is presented by the classification of these states, and the search for the fundamental laws of Nature which determine their spectrum.Previous study of this problem has taken place along two different lines. In one, the use of symmetry groups is made, and the pion-nucleon resonances are allocated to different representations of these groups. The other has been the study of the underlying forces involved, including dynamical models such as bootstrap theory. Both these approaches have been adequately discussed in the report of the Trieste Conference (19651, and are not treated further in this work.Recently, the importance of complex angular momentum and Regge theory in this problem was demonstrated by Barger and Cline, who showed that the known pion-nucleon resonances could be fitted on families of Regge trajectories.An important theoretical concept in baryonic systems is MacDowell symmetry which is a relationship between parity conserving partial wave amplitudes for one parity at positive energy, to the wave having opposite parity and negative energy. The application of MacDowell symmetry and Regge theory to the pion nucleon system shows that two Regge trajectories α(±w) may be defined. The physical Regge recurrences on α( +w) have one parity, and the trajectory α( -w) corresponds to a trajectory in which the Regge recurrences have opposite parity.The work of Barger and Cline is discussed in detail in Chapter II, and from the experimental fits it is shown that the two trajectories α(w), α( -w) are approximately the same, so parity degeneracy occurs. There are some notable exceptions to this result. Several states predicted by this parity degeneracy are missing, such as the lowest member of the highest ranking N₈ trajectory (the S₁₁), the lowest member of the Nδ (P₁₃), and the lowest members of the Δγ (D₃₃, G₃₇) . The usual spectroscopic notation L₂₁,2₂ⱼ is used in the classification of the baryons.This thesis is concerned with a study of the pion-nucleon resonances in the framework of Regge theory and MacDowell symmetry. Attempts .are made to explain the form of the Regge trajectories for the system, and special attention is paid to the missing mass states. The scope has been restricted to the nucleon Nα and Nß trajectories, but the theory may be generalised to other trajectories using SU symmetry.In Chapter I the concept of MacDowell symmetry is stated and proved for the parity conserving partial wave amplitudes of Gell-Mann, Goldberger, Low and Zachariasen. A discussion of generalised MacDowell symmetry which depends on field theoretic arguments has been given by Hara. The approach to MacDowell symmetry used in this thesis depends on crossing symmetry, and to the author's knowledge this has not been done before.Chapter II is an introduction to pion-nucleon scattering and the application of MacDowell symmetry and Regge poles. The original work in this thesis starts at section 2.5, in which a discussion of Riemann sheets and their application to missing mass states, is given.In Chapter III a potential scattering model is described, and its possible applications to the pion-nucleon system and missing mass states is discussed.Chapter IV is concerned with parametrisations of Regge trajectories, and a critical discussion is given of models which produce distortions of the Regge trajectory near the missing mass states.Finally, in Chapter V possible dynamical models for fernion Regge trajectories are discussed, and a review is given of their applications to the higher pion-nucleon resonances

The Use and Development of American Waterways

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Leibniz, Acosmism, and Incompossibility

Leibniz claims that God acts in the best possible way, and that this includes creating exactly one world. But worlds are aggregates, and aggregates have a low degree of reality or metaphysical perfection, perhaps none at all. This is Leibniz’s tendency toward acosmism, or the view that there this no such thing as creation-as-a-whole. Many interpreters reconcile Leibniz’s acosmist tendency with the high value of worlds by proposing that God sums the value of each substance created, so that the best world is just the world with the most substances. I call this way of determining the value of a world the Additive Theory of Value (ATV), and argue that it leads to the current and insoluble form of the problem of incompossibility. To avoid the problem, I read “possible worlds” in “God chooses the best of all possible worlds” as referring to God’s ideas of worlds. These ideas, though built up from essences, are themselves unities and so well suited to be the value bearers that Leibniz’s theodicy requires. They have their own value, thanks to their unity, and that unity is not preserved when more essences are added