arXiv subjects

George Sparling

Publications and source records attributed to George Sparling.

27 records · Page 2Linked to original sources

arXiv2006

The Xi-transform for conformally flat space-time

The Xi-transform is a new spinor transform arising naturally in Einstein's general relativity. Here the example of conformally flat space-time is discussed in detail. In particular it is shown that for this case, the transform coincides with two other naturally defined transforms: one a two-variable transform on the Lie group SU(2, C), the other a transform on the space of null split octaves. The key properties of the transform are developed.

gr-qc↗

arXiv2006

Observables of Angular Momentum as Observables on the Fedosov Quantized Sphere

In this paper we construct the commutators of the Fedosov * (a generalization of the Moyal star product) on the phase space of S2. It is shown that this product obeys the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is three-fold. One is to find an exact, non-perturbative solution to the commutators of the Fedosov *. The other is to verify that these commutation relations correspond to angular momentum commutation relations as in ordinary nonrelativistic quantum mechanics. The last is to show a more general computation of the Fedosov *; essentially undeforming Fedosov's algorithm.

quant-ph↗

arXiv2006

The structure of cosmic time

Following the approach of Julien Lesgourgues [astro-ph/0409426], we analyze the mathematical structure of the time co-ordinate of present day cosmological models, where these models include a cosmological constant term to account for the observed acceleration of the universe: we find that in all cases, except for a set of measure zero in the parameter space, the time is given by an (abelian) integral on a torus; the imaginary period of this integral then gives a natural periodicity in imaginary time for the universe; following Stephen Hawking, this periodicity may be interpreted either as giving a fundamental mass scale for the universe, or (using Planck's constant) a fundamental temperature, or both. The precise structure that emerges suggests that the structure of time can be regarded as an order parameter arising perhaps in a phase transition in the early universe; one might hope that this structure would be predictable in any fundamental theory.

gr-qc↗

arXiv2005

A dispersionless integrable system associated to Diff$(S^1)$ gauge theory

A dispersionless integrable system underlying 2+1-dimensional hyperCR Einstein--Weyl structures is obtained as a symmetry reduction of the anti--self--dual Yang--Mills equations with the gauge group Diff$(S^1)$. Two special classes of solutions are obtained from well known soliton equations by embedding SU(1,1) in ${Diff}(S^1)$.

nlin.SI↗

arXiv2004

Non-Commutative Time, the Quantum Hall Effect and Twistor Theory

It is proposed that Maxwell theory, with a topological term, in four non-commutative dimensions, where the co-ordinates obey the Heisenberg algebra, is an umbrella theory for the description of the two-dimensional Quantum Hall Effect (following the fluidic approach of Susskind and Polychronakos), the twistor analyses of self-dual gravity, solitons and instantons and the twistor theory of isolated horizons in space-time. Applied to the Quantum Hall case, the underlying metric is Lorentzian: two canonically conjugate co-ordinates are spatial, the other two co-ordinates naturally are null and represent time and a conjugate energy variable.

cond-mat.mes-hall↗

arXiv2004

A primordial theory

We review the twistor approach to the Zhang-Hu theory of the four-dimensional Quantum Hall effect. We point out the key role played by the group Spin(4,4), as the symmetry group of the boundary. It is argued that this group, which ignores the square root of minus one used in relativity and quantum mechanics, is the focal point of a primordial theory, one where the Cartan concept of triality is paramount, from which the standard theories emerge via a series of phase transitions of the Zhang-Hu fluid. An important role will be played by the Jordan cross-product algebras, particularly the exceptional Jordan algebra associated to the split octaves and by the associated Freudenthal phase space. The geometry and Hamiltonian theory of these spaces is examined in detail. A possible link to the theory of massive particles is outlined.

cond-mat.mes-hall↗

arXiv2002

Twistor theory and the four-dimensional Quantum Hall effect of Zhang and Hu

The construction by Zhang and Hu of a four-dimensional analogue of the Quantum Hall effect is generalized and recast as a purely geometrical theory, using the languages of Lie group theory and twistor theory. It emerges that the Zhang-Hu quantum liquid lies naturally in twistor space and is apparently more primitive than space-time itself, in accordance with twistor philosophy. The quantum liquid is then the glue that holds space-time together. It is argued that the theory is inherently chiral and time asymmetric, consonant with previous conjectures.

cond-mat.mes-hall↗

arXiv2002

On the Classical Solutions of the Perturbed, Massless Wave Equation with Singular Potential

This paper discusses the solutions to the perturbed wave equation containing a singular potential term in the Lorentzian metric. We present the classical solution to the problem using the separation of variables method for any dimension, n. Special solutions are obtained for even n's and properties of these solutions are discussed. Finally, we also consider the solution to the Cauchy problem for the case n=2.

math-ph↗

arXiv2001

Conformally invariant powers of the ambient Dirac operator

This paper constructs a family of conformally invariant differential operators acting on spinor densities with leading part a power of the Dirac operator. The construction applies for all powers in odd dimensions, and only for finitely many powers in even dimensions. These operators arise naturally as obstructions to formal solution of the Dirac equation on the Fefferman-Graham ambient space with prescribed boundary conditions.

math.DG↗