Lane P. Hughston

Geometric Quantum Mechanics

Abstract The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Fubini–Study metric. According to the principles of geometric quantum mechanics, the physical characteristics of a given quantum system can be represented by geometrical features that are preferentially identified in this complex manifold. Here we construct a number of examples of such features as they arise in the state spaces for spin 1 2 , spin 1, spin 3 2 and spin 2 systems, and for pairs of spin 1 2 systems. A study is then undertaken on the geometry of entangled states. A locally invariant measure is assigned to the degree of entanglement of a given state for a general multi-particle system, and the properties of this measure are analysed for the entangled states of a pair of spin 1 2 particles. With the specification of a quantum Hamiltonian, the resulting Schrodinger trajectories induce an isometry of the Fubini–Study manifold, and hence also an isometry of each of the energy surfaces generated by level values of the expectation of the Hamiltonian. For a generic quantum evolution, the corresponding Killing trajectory is quasiergodic on a toroidal subspace of the energy surface through the initial state. When a dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of a general state is in its applicability in various attempts to go beyond the standard quantum theory, some of which admit a natural phase-space characterisation.

On a Quadratic First Integral for the Charged Particle Orbits in the Charged Kerr Solution

Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.

Martingale models for quantum state reduction

Stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory. Here we examine the correspondence of the two theories in detail, making a systematic use of the methods of martingale theory. An analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred. We consider the general projection postulate of L¨ uders applicable in the case of a possibly degenerate eigenvalue spectrum, and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state. We also analyse the associated Lindblad equation for the evolution of the density matrix, and obtain an exact time-dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the L¨ uders density matrix. Finally, we apply Girsanov’s theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric Brownian motions, thereby constructing an explicit unravelling of the Lindblad equation.

Beyond Hazard Rates: A New Framework for Credit-Risk Modelling

A new approach to credit risk modelling is introduced that avoids the use of inaccessible stopping times. Default events are associated directly with the failure of obligors to make contractually agreed payments. Noisy information about impending cash flows is available to market participants. In this framework, the market filtration is modelled explicitly, and is assumed to be generated by one or more independent market information processes. Each such information process carries partial information about the values of the market factors that determine future cash flows. For each market factor, the rate at which true information is provided to market participants concerning the eventual value of the factor is a parameter of the model. Analytical expressions that can be readily used for simulation are presented for the price processes of defaultable bonds with stochastic recovery. Similar expressions can be formulated for other debt instruments, including multi-name products. An explicit formula is derived for the value of an option on a defaultable discount bond. It is shown that the value of such an option is an increasing function of the rate at which true information is provided about the terminal payoff of the bond. One notable feature of the framework is that it satisfies an overall dynamic consistency condition that makes it suitable as a basis for practical modelling situations where frequent recalibration may be necessary.

A generalised Kerr-Robinson theorem

The Kerr and Robinson theorems in four-dimensional spacetime together provide the general null solution of Maxwells equations. Robinsons theorem reduces the problem to that of obtaining certain null foliations. The Kerr theorem shows how to represent such foliations in terms of analytic varieties in complex projective 3-space. The authors generalise these results to spinor fields of higher valence in spacetimes of arbitrary even dimension. They first review the theory of spinors and twistors for these higher dimensions. They define the appropriate generalisations of Maxwells equations, and null solutions thereof. It is then proved that the Kerr and Robinson theorems generalise to all even dimensions. The authors discuss various applications, examples and further generalisations. The generalised Robinson theorem can be seen to extend to curved spaces which admit such null foliations. In the case of Euclidean reality conditions, the generalised Kerr theorem determines all complex structures compatible with the flat metric in terms of freely specified complex analytic varieties in twistor space. Interpretations of the generalised Kerr theorem are also provided for Lorentzian and ultrahyperbolic signatures.

Interest Rates and Information Geometry

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square–integrable functions, and is characterized by the Fisher–Rao metric. In the non–parametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term–structure modelling, we show that the derivative of the discount function with respect to the time left until maturity gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitragefree yield–curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.

Spacetimes with Killing tensors

For Einstein-Maxwell fields for which the Weyl spinor is of type {2, 2}, and the electromagnetic field spinor is of type {1, 1} with its principal null directions coaligned with those of the Weyl spinor, the integrability conditions for the existence of a certain valence two Killing tensor are shown to reduce to a simple criterion involving the ratio of the amplitude of the Weyl spinor to the amplitude of a certain test solution of the spin two zero restmass field equations. The charged Kerr solution provides an example of a spacetime for which the criterion is satisfied; the chargedC-metric provides an example for which it is not.

Homogeneous Electromagnetic and Massive Vector Meson Fields in Bianchi Cosmologies

Maxwell equations and equations of massive-vector- meson fields in spatially homogeneous Bianchi cosmologies, obtaining formula for homogeneous EM field

Chaos and coherence: a new framework for interest–rate modelling

A set of elementary axioms for stochastic finance is presented wherein a prominent role is played by the state–price density, which in turn determines the stochastic dynamics of the interest–rate term structure. The fact that the state–price density is a potential implies the existence of an asymptotic random variable X∞ with the property that its conditional variance is the state–price density. The Wiener chaos expansion technique can then be applied to X∞, thus enabling us to ‘parametrize’ the dynamics of the discount–bond system in terms of the deterministic coefficients of the chaos expansion. Using this method, we find that there is a natural map from the space of all admissible term–structure trajectories to the symmetric Fock space F naturally associated with the space of square-integrable random variables on the underlying probability space. An element of F is either coherent or incoherent, and a stochastic bond–price system is necessarily represented by an incoherent element of F. Making use of the linearity of F we derive simple analytic formulae for the bond–price system, the volatility structure, the short rate, and the risk premium associated with an arbitrary admissible term–structure model. Extensions to foreign–exchange markets and general asset systems are also developed.

A chaotic approach to interest rate modelling

Abstract.This paper presents a new approach to interest rate dynamics. We consider the general family of arbitrage-free positive interest rate models, valid on all time horizons, in the case of a discount bond system driven by a Brownian motion of one or more dimensions. We show that the space of such models admits a canonical mapping to the space of square-integrable Wiener functionals. This is achieved by means of a conditional variance representation for the state price density. The Wiener chaos expansion technique is then used to formulate a systematic analysis of the structure and classification of interest rate models. We show that the specification of a first-chaos model is equivalent to the specification of an admissible initial yield curve. A comprehensive development of the second-chaos interest rate theory is presented in the case of a single Brownian factor, and we show that there is a natural methodology for calibrating the model to at-the-money-forward caplet prices. The factorisable second-chaos models are particularly tractable, and lead to closed-form expressions for options on bonds and for swaptions. In conclusion we outline a general “international” model for interest rates and foreign exchange, for which each currency admits an associated family of discount bonds, and show that the entire system can be generated by a vector of Wiener functionals.