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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 Nâ
trajectory (the Sââ),
the lowest member of the NÎŽ (Pââ), and
the lowest members of the ÎÎł (Dââ, Gââ) . The usual spectroscopic
notation
Lââ,2âⱌ
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 NĂ
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
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 Leibnizâs tendency toward acosmism, or the view that there this no such thing as creation-as-a-whole. Many interpreters reconcile Leibnizâs 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 âpossible worldsâ in âGod chooses the best of all possible worldsâ as referring to Godâs ideas of worlds. These ideas, though built up from essences, are themselves unities and so well suited to be the value bearers that Leibnizâs theodicy requires. They have their own value, thanks to their unity, and that unity is not preserved when more essences are added
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