The History of Contract in Early English Equity

“Mr. Barbour’s contribution to the Studies is an attempt to characterize with some precision and detail the functions of the Chancery in the fifteenth century. The court was gradually differentiated from the King’s Council, and the writs of Edward III’s time calling on persons to appear under penalty of a fine or imprisonment (subpoena), and other special injunctions, was generally framed in terms which leave it undecided whether proceedings were to be taken by the King’s Council, or by the Council under the chairmanship of the Chancellor himself with or without the aid of assessors. By the time of Richard II, however, the personal jurisdiction of the Chancellor had acquired a fairly definite range, and was assuming the aspect of a standing institution.” --Prefacehttps://repository.law.umich.edu/books/1069/thumbnail.jp

Scale-Invariant Gravity: Geometrodynamics

We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scale-invariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t scaling developed in the parallel particle dynamics paper by one of the authors. In spatially-compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented. Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.Comment: 33 pages. Published version (has very minor style changes due to changes in companion paper

Entropic Dynamics, Time and Quantum Theory

Quantum mechanics is derived as an application of the method of maximum entropy. No appeal is made to any underlying classical action principle whether deterministic or stochastic. Instead, the basic assumption is that in addition to the particles of interest x there exist extra variables y whose entropy S(x) depends on x. The Schr\"odinger equation follows from their coupled dynamics: the entropy S(x) drives the dynamics of the particles x while they in their turn determine the evolution of S(x). In this "entropic dynamics" time is introduced as a device to keep track of change. A welcome feature of such an entropic time is that it naturally incorporates an arrow of time. Both the magnitude and the phase of the wave function are given statistical interpretations: the magnitude gives the distribution of x in agreement with the usual Born rule and the phase carries information about the entropy S(x) of the extra variables. Extending the model to include external electromagnetic fields yields further insight into the nature of the quantum phase.Comment: 29 page

New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split

I show how there is an ambiguity in how one treats auxiliary variables in gauge theories including general relativity cast as 3 + 1 geometrodynamics. Auxiliary variables may be treated pre-variationally as multiplier coordinates or as the velocities corresponding to cyclic coordinates. The latter treatment works through the physical meaninglessness of auxiliary variables' values applying also to the end points (or end spatial hypersurfaces) of the variation, so that these are free rather than fixed. [This is also known as variation with natural boundary conditions.] Further principles of dynamics workings such as Routhian reduction and the Dirac procedure are shown to have parallel counterparts for this new formalism. One advantage of the new scheme is that the corresponding actions are more manifestly relational. While the electric potential is usually regarded as a multiplier coordinate and Arnowitt, Deser and Misner have regarded the lapse and shift likewise, this paper's scheme considers new {\it flux}, {\it instant} and {\it grid} variables whose corresponding velocities are, respectively, the abovementioned previously used variables. This paper's way of thinking about gauge theory furthermore admits interesting generalizations, which shall be provided in a second paper.Comment: 11 page

Relational Particle Models. II. Use as toy models for quantum geometrodynamics

Relational particle models are employed as toy models for the study of the Problem of Time in quantum geometrodynamics. These models' analogue of the thin sandwich is resolved. It is argued that the relative configuration space and shape space of these models are close analogues from various perspectives of superspace and conformal superspace respectively. The geometry of these spaces and quantization thereupon is presented. A quantity that is frozen in the scale invariant relational particle model is demonstrated to be an internal time in a certain portion of the relational particle reformulation of Newtonian mechanics. The semiclassical approach for these models is studied as an emergent time resolution for these models, as are consistent records approaches.Comment: Replaced with published version. Minor changes only; 1 reference correcte

Pointlike structure for super p-branes

We present an efficient method to understand the p-brane dynamics in a unified framework. For this purpose, we reformulate the action for super p-branes in the form appropriate to incorporate the pointlike (parton) structure of higher dimensional p-branes and intend to interpret the p-brane dynamics as the collective dynamics of superparticles. In order to examine such a parton picture of super p-branes, we consider various superparticle configurations that can be reduced from super p-branes, especially, a supermembrane, and study the partonic structure of classical p-brane solutions.Comment: 22 pages, corrected typos, to appear in Phys. Rev. D58, 085018 (1998

Triangleland. I. Classical dynamics with exchange of relative angular momentum

In Euclidean relational particle mechanics, only relative times, relative angles and relative separations are meaningful. Barbour--Bertotti (1982) theory is of this form and can be viewed as a recovery of (a portion of) Newtonian mechanics from relational premises. This is of interest in the absolute versus relative motion debate and also shares a number of features with the geometrodynamical formulation of general relativity, making it suitable for some modelling of the problem of time in quantum gravity. I also study similarity relational particle mechanics (`dynamics of pure shape'), in which only relative times, relative angles and {\sl ratios of} relative separations are meaningful. This I consider firstly as it is simpler, particularly in 1 and 2 d, for which the configuration space geometry turns out to be well-known, e.g. S^2 for the `triangleland' (3-particle) case that I consider in detail. Secondly, the similarity model occurs as a sub-model within the Euclidean model: that admits a shape--scale split. For harmonic oscillator like potentials, similarity triangleland model turns out to have the same mathematics as a family of rigid rotor problems, while the Euclidean case turns out to have parallels with the Kepler--Coulomb problem in spherical and parabolic coordinates. Previous work on relational mechanics covered cases where the constituent subsystems do not exchange relative angular momentum, which is a simplifying (but in some ways undesirable) feature paralleling centrality in ordinary mechanics. In this paper I lift this restriction. In each case I reduce the relational problem to a standard one, thus obtain various exact, asymptotic and numerical solutions, and then recast these into the original mechanical variables for physical interpretation.Comment: Journal Reference added, minor updates to References and Figure

Note and Comment

The Law School - In common with all other law schools requiring college work for admission, this school has suffered a very heavy loss in attendance because of war conditions. This, however, is a matter for pride and not for discouragement for it means that our students have gone into the army or navy or other branches of the national service in very high ratio to their total number. And this is by no means due only to the effect of the Selective Service Act for from the very beginning our men have volunteered in great spirit and promptness. In 1917 fewer than two-thirds of the then senior class were present at the Commencement exercises to receive their degrees. Most of them had gone by the middle of May. All this indicates that the profession is living up to one of its high traditions of patriotic service

Radiative capture of protons by deuterons

The differential cross section for radiative capture of protons by deuterons is calculated using different realistic NN interactions. We compare our results with the available experimental data below $E_x = 20 MeV$. Excellent agreement is found when taking into account meson exchange currents, dipole and quadrupole contributions, and the full initial state interaction. There is only a small difference between the magnitudes of the cross sections for the different potentials considered. The angular distributions, however, are practically potential independent.Comment: 4 pages (twocolumn), 4 postscript figures included, submitted for publication, revised versio

Quantum Cosmological Relational Model of Shape and Scale in 1-d

Relational particle models are useful toy models for quantum cosmology and the problem of time in quantum general relativity. This paper shows how to extend existing work on concrete examples of relational particle models in 1-d to include a notion of scale. This is useful as regards forming a tight analogy with quantum cosmology and the emergent semiclassical time and hidden time approaches to the problem of time. This paper shows furthermore that the correspondence between relational particle models and classical and quantum cosmology can be strengthened using judicious choices of the mechanical potential. This gives relational particle mechanics models with analogues of spatial curvature, cosmological constant, dust and radiation terms. A number of these models are then tractable at the quantum level. These models can be used to study important issues 1) in canonical quantum gravity: the problem of time, the semiclassical approach to it and timeless approaches to it (such as the naive Schrodinger interpretation and records theory). 2) In quantum cosmology, such as in the investigation of uniform states, robustness, and the qualitative understanding of the origin of structure formation.Comment: References and some more motivation adde