Lessons from the 3d U(1) Gross-Neveu Model

The effectiveness of the Glasgow algorithm is explored via implementation in the 3d U(1) Gross-Neveu model and the realisation of the Goldstone mechanism in this model is compared and contrasted with its realisation in QCD.Comment: 6 pages, 5 eps figs, To appear in Proceedings of "QCD at Finite Baryon Density" workshop, Bielefeld, 27-30 April 199

The Definition of Mach's Principle

Two definitions of Mach's principle are proposed. Both are related to gauge theory, are universal in scope and amount to formulations of causality that take into account the relational nature of position, time, and size. One of them leads directly to general relativity and may have relevance to the problem of creating a quantum theory of gravity.Comment: To be published in Foundations of Physics as invited contribution to Peter Mittelstaedt's 80th Birthday Festschrift. 30 page

Mod-discrete expansions

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the $n$'th random variable $X_n$ is by a particular member $R_n$ of a given family of distributions, whose variance increases with $n$. The basic assumption is that the ratio of the characteristic function of $X_n$ and that of R_n$ converges to a limit in a prescribed fashion. Our results cover a number of classical examples in probability theory, combinatorics and number theory

Towards the Unification of Gravity and other Interactions: What has been Missed?

Faced with the persisting problem of the unification of gravity with other fundamental interactions we investigate the possibility of a new paradigm, according to which the basic space of physics is a multidimensional space ${\cal C}$ associated with matter configurations. We consider general relativity in ${\cal C}$. In spacetime, which is a 4-dimensional subspace of ${\cal C}$, we have not only the 4-dimensional gravity, but also other interactions, just as in Kaluza-Klein theories. We then consider a finite dimensional description of extended objects in terms of the center of mass, area, and volume degrees of freedom, which altogether form a 16-dimensional manifold whose tangent space at any point is Clifford algebra Cl(1,3). The latter algebra is very promising for the unification, and it provides description of fermions.Comment: 11 pages; Talk presented at "First Mediterranean Conference on Classical and Quantum Gravity", Kolymbari, Crete, Greece, 14-18 September 200

Interacting vector fields in Relativity without Relativity

Barbour, Foster and \'{O} Murchadha have recently developed a new framework, called here {\it{the 3-space approach}}, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric--matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -- and more -- without presupposing relativity. This paper completes the recovery of the presently known bosonic sector within the 3-space approach. We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang--Mills fields minimally coupled to gravity.Comment: Replaced with final version accepted by Classical and Quantum Gravity (14 pages, no figures

The geometry of the Barbour-Bertotti theories II. The three body problem

We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for $N\neq 3$, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there.Comment: 16 pages, 2 eps figures, to appear in Classical and Quantum Gravit

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

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

Can a flavour-conserving treatment improve things ?

In this work I would like to present some ideas on how to improve on the gauge sector in our lattice simulations at finite baryon density. The long standing problem, that we obtain an onset in thermodynamic quantities at a much smaller chemical potential than expected, could be related to an unphysical proliferation of flavours due to hard gluons close to the Brillouin edges. These hard gluons produce flavour non-conserving vertices to the fermion sector. They also produce excessive number of small instantons due to lattice dislocations. Both unphysical effects could increase the propagation in (di)-quarks to give the early onset in $\mu$. Thus we will present here a modified action that avoids large fields close to the lattice cutoff. Some of these ideas have been tested for SU(2) and are being implemented for SU(3).Comment: Talk presented at the Intl. Workshop on QCD at Finite Baryon Density in Bielefeld, April 98. 5 pp in Latex, uses espcrc1.st