# Chris DoranUniversity of Cambridge | Cam · Sidney Sussex College

Chris Doran

PhD Theoretical Physics

## About

85

Publications

19,457

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

2,951

Citations

Introduction

## Publications

Publications (85)

In this paper we tackle the problem of correspondence and rotor estimation between models composed of geometric primitives of different types. We frame this problem as searching for the rotor that takes a query model to a reference model. The situations that we consider are those in which our query model: contains additional primitives not present...

In this talk, we present a novel technique to generate Signed Distance Fields (SDF) from vector paths. Unlike existing methods, instead of first rasterizing a path to a bitmap and then deriving the SDF, we can calculate the minimum distance for each pixel to the nearest segment directly from a path description comprised of line segments and Bezier...

We discuss a fully covariant Lagrangian-based description of 3D rigid body motion, employing spinors in 5D conformal space.
The use of this space enables the translational and rotational degrees of freedom of the body to be expressed via a unified
rotor structure, and the equations of motion in terms of a generalised ‘moment of inertia tensor’ are...

Twistors are re-interpreted in terms of geometric algebra as 4-d spinors with a position dependence. This allows us to construct
their properties as observables of a quantum system. The Robinson congruence is derived and extended to non-Euclidean spaces
where it is represented in terms of d-lines. Different conformal spaces are constructed through...

As leading experts in geometric algebra, Chris Doran and Anthony Lasenby have led many new developments in the field over the last ten years. This book provides an introduction to the subject, covering applications such as black hole physics and quantum computing. Suitable as a textbook for graduate courses on the physical applications of geometric...

Events in Minkowski space-time can be obtained from the intersection of two twistors with no helicity. These can be represented within the geometric (Clifford) algebra formalism, in a particular conformal space that is constructed from a quantum system of two particles. The realisation takes place in the multiparticle space-time algebra. This repre...

We study the scattering of massive spin-half waves by a Schwarzschild black hole using analytical and numerical methods. We begin by extending a recent perturbation theory calculation to next order to obtain Born series for the differential cross section and Mott polarization, valid at small couplings. We continue by deriving an approximation for g...

In previous work by two of the present authors, twistors were re-interpreted as 4-d spinors with a position dependence within the formalism of geometric (Clifford) algebra. Here we extend that approach and justify the nature of the position dependence. We deduce the spinor representation of the restricted conformal group in geometric algebra, and u...

Spacetime algebra (STA) is the name given to the geometric (Clifford) algebra where vectors are equipped with a product that is associative and distributive over addition. The essential feature of this product is that it mixes two different types of object such as scalars and bivectors. There is a growing realization that geometric algebra provides...

Motivated by conventional gauge theories, we consider a theory of gravity in which the Einstein-Hilbert action is replaced by a term that is quadratic in the Riemann tensor. We focus on cosmological solutions to the field equations in flat, open and closed universes. The gravitational action is scale invariant, so the only matter source considered...

We study the absorption of massive spin-half particles by a small Schwarzschild black hole by numerically solving the single-particle Dirac equation in Painleve-Gullstrand coordinates. We calculate the absorption cross section for a range of gravitational couplings Mm/m_P^2 and incident particle energies E. At high couplings, where the Schwarzschil...

We calculate the bound-state energy spectrum of the Dirac Equation in a Schwarzschild black hole background using a minimax variational method. Our method extends that of Talman to the case of non-Hermitian interactions, such as a black hole. The trial function is expressed in terms of a basis set that takes into account both the Hermitian limit of...

Conformal embedding of closed-universe models in a de Sitter background suggests a quantisation condition on the available conformal time. This condition implies that the universe is closed at no greater than the 10% level. When a massive scalar field is introduced to drive an inflationary phase this figure is reduced to closure at nearer the 1% le...

In the surface acoustic wave quantum computer, the spin state of an
electron trapped in a moving quantum dot comprises the physical qubit of
the scheme. Via detailed analytic and numerical modeling of the qubit
dynamics, we discuss the effect of excitations into higher-energy
orbital states of the quantum dot that occur when the qubits pass
through...

Kerr-Schild solutions to the vacuum Einstein equations are considered from the viewpoint of integral equations. We show that, for a class of Kerr-Schild fields, the stress-energy tensor can be regarded as a total divergence in Minkowski spacetime. If one assumes that Minkowski coordinates cover the entire manifold (no maximal extension), then Gauss...

A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what re...

Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this...

We review the applications of geometric algebra in electromag-netism, gravitation and multiparticle quantum systems. We discuss a gauge the-ory formulation of gravity and its implementation in geometric algebra, and apply this to the fermion bound state problem in a black hole background. We show that a discrete energy spectrum arises in an analogo...

Blending schemes based on circles provide smooth `fair' interpolations between series of points. Here we demonstrate a simple, robust set of algorithms for performing circle blends for a range of cases. An arbitrary level of G-continuity can be achieved by simple alterations to the underlying parameterisation. Our method exploits the computational...

We present a new approach to constructing inflationary models in closed universes. Conformal embedding of closed-universe models in a de Sitter background suggests a quantisation condition on the available conformal time. This condition implies that the universe is closed at no greater than the 10% level. When a massive scalar field is introduced t...

Two distinct representations for the density operators of multi-qubit systems are de-veloped and compared within the multiparticle spacetime algebra (MSTA). The two representations arise from two different methods for correlating imaginary struc-tures from different qubit spaces. The first is a direct transliteration of the usual representation by...

A new framework for analysing the gravitational fields in a stationary, axisymmetric configuration is introduced. The method is used to construct a complete set of field equations for the vacuum region outside a rotating source. These equations are under-determined. Restricting the Weyl tensor to type D produces a set of equations which can be solv...

We compute the spectrum of normalizable fermion bound states in a
Schwarzschild black hole background. The eigenstates have complex energies. The
real part of the energies, for small couplings, closely follow a hydrogen-like
spectrum. The imaginary parts give decay times for the various states, due to
the absorption properties of the hole, with sta...

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach - the Euclidea...

In this contribution we describe some applications of geometric algebra to the fleld of black hole physics. Our main focus is on the proper- ties of Dirac wavefunctions around black holes. We show the existence of normalised bound state solutions, with an associated decay rate controlled by an imaginary contribution to the energy eigenvalue. This i...

When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only two subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressi...

Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant...

Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems of PDEs, including schemes ba...

The multiparticle spacetime algebra (MSTA) is an extension of Dirac theory to a multiparticle setting, which was first studied by Doran, Gull and Lasenby. The geometric interpretation of this algebra, which it inherits from its one-particle factors, possesses a number of physically compelling features, including simple derivations of the Pauli excl...

When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only 2 subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing...

The differential cross section for scattering of a Dirac particle in a black hole background is found. The result is the gravitational analog of the Mott formula for scattering in a Coulomb background. The equivalence principle is neatly embodied in the cross section, which depends only on the incident velocity, and not the particle mass. The low a...

Using the spacetime algebra formulation of the Dirac equation, we demonstrate how to perform cross-section calculations following a method suggested by Hestenes (1982). Instead of an S-matrix, we use an operator that rotates the initial states into the scattered states. By allowing the scattering operator to become a function of the initial spin, w...

Course Notes to accompany Applications talk: SIGGRAPH 2001, Los Angeles – Inverse kinematics and dynamics 1 Scope This set of notes gives some background to the material presented in the applications lectures. More detail can be found in the references listed at the end of these notes. The notes cover some of the material presented in the lectures,...

Geometric algebra is an extremely powerful language for solving complex geometric problems in engineering [4, 8]. Its advantages are particularly clear in the treatment of rotations. Rotations of a vector are performed by the double-sided application of a rotor, which is formed from the geometric product of an even number of unit vectors. In three...

This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geome...

The late 18th and 19th centuries were times of great mathematical progress. Many new mathematical systems and languages wer introduced by some of the millennium'greatest mathematicians. Amongst these were the algebras of Clifford and Grassmann. Whil these algebras caused considerable interest at the time, they were largely abandoned with the introd...

The spacetime algebra provides an elegant language for studying the Dirac equation. Cross-section calculations can be performed in an intuitive way following a method suggested by D. Hestenes [Adv. Appl. Clifford Algebras 2, No. 2, 215–248 (1992; Zbl 0844.35099)]. The S-matrix is replaced with an operator which rotates the initial states into the s...

We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is...

A new form of the Kerr solution is presented. The solution involves a time coordinate which represents the local proper time for free-falling observers on a set of simple trajectories. Many physical phenomena are particularly clear when related to this time coordinate. The chosen coordinates also ensure that the solution is well behaved at the hori...

It is argued that geometric algebra, in the form of the multiparticle spacetime algebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometric algebra ap...

ABSTRACTA new method is proposed for modelling spherically symmetric inhomogeneities in the Universe. The inhomogeneities have finite size and are compensated, so they do not exert any measurable gravitational force beyond their boundary. The region exterior to the perturbation is represented by a Friedmann--Robertson--Walker (FRW) universe, which...

A new method for modelling spherically symmetric inhomogeneities is applied to the formation of clusters in an expanding Universe. We impose simple initial velocity and density perturbations of finite extent and we investigate the subsequent evolution of the density field. Photon paths are also calculated, allowing a detailed consideration of gravi...

It is shown that higher-weighted representations of rotation groups can be constructed using multilinear functions in geometric algebra. Methods for obtaining the irreducible representations are found, and applied to the spatial rotation group, SO(3), and the proper Lorentz group, SO+(1,3). It is also shown that the representations can be generaliz...

We discuss a coordinate-free approach to the geometry of computer vision problems. The technique we use to analyse the three-dimensional transformations involved will be that of geometric algebra: a framework based on the algebras of Clifford and Grassmann. This is not a system designed specifically for the task in hand, but rather a framework for...

We present new, massive, non-ghost solutions for the Dirac field coupled self-consistently to gravity. We employ a gauge-theoretic formulation of gravity which automatically identifies the spin of the Dirac field with the torsion of the gauge fields. Homogeneity of the field observables requires that the spatial sections be flat. Expanding and coll...

The spin-torsion sector of a new gauge-theoretic formulation of gravity is analysed and the relationship to the Einstein-Cartan-Kibble-Sciama theory of gravity is discussed. The symmetries of the Riemann tensor and the con-servation laws of the theory are derived. This formalism is applied to the problem of a Dirac field coupled self-consistently t...

We discuss three applications of a gauge theory of gravity to rotating astrophysical systems. The theory employs gauge fields in a flat Minkowski background spacetime to describe gravitational interactions. The iron fluorescence line observed in AGN is discussed, assuming that the line originates from matter in an accretion disk around a Kerr (rota...

We present a new model for the formation of spherically symmetric clusters in an expanding Universe. Both the Universe and the collapsing cluster are governed by the same pressure less fluid equations for which a uniform initial density profile is assumed. A simple perturbation imposed on the initial velocity field gives rise to an over-density whi...

We consider a fully relativistic method for the calculation of tunnelling times based on the streamlines of the conserved probability flux. This method is similar to that proposed by Leavens but is not based on the Bohmian interpretation. The method is applied to single- and two-particle tunnelling in Dirac theory.

A new gauge-theoretic approach to gravity is applied to the study of
rotating cylindrically symmetric strings. The interior and exterior
equations are reduced to a simple set of first-order differential
equations, and suitable matching conditions are obtained. The gauge
theory formulation affords a clear understanding of the physical
observables in...

An objective account of the action of a Stern-Gerlach apparatus on particles is given, using the Dirac equation. This generalizes earlier work on a causal interpretation of the Pauli equation to the relativistic domain, leading to a more natural choice for the current in the model.

Kerr-Schild solutions to the vacuum Einstein equations are considered from the viewpoint of a gauge-theoretic formulation of gravity. This for- mulation employs the spacetime algebra (the Clifiord algebra of space- time), which ofiers a number of novel insights into the nature of the solu- tions. Working with gauge flelds deflned over a background...

Cliiord's `geometric algebra' is presented as the natural language for expressing geometrical ideas in mathematical physics. Its spacetime version`spacetimeversion`spacetime algebra' i s i n troduced and is shown to provide a powerful, invariant description of relativistic physics. Applications to elec-tromagnetism and gravitation are discussed.

The STA is a mathematical system, rather than of itself containing new physics. However, when we employ it in the description of physical phenomena, we usually find that some fresh insight is obtained, often on old questions. The oldest question of 20th century physics is the interpretation of quantum mechanics, and in this lecture we aim to discus...

We begin by summarising the notations and conventions which we will employ throughout our series of lectures. Summation convention and natural units (ћ = c = ∈0 = G = l) are employed throughout, except where explicitly stated.

In this lecture we discuss applications of STA to problems in electromagnetism. In the STA, we write the electromagnetic field in terms of the 4-potential A $$F = \nabla \wedge A = \nabla A - \nabla \cdot A.$$ (8.1)

One of the simplest demonstrations of the insights provided by the STA formulation of both Maxwell and Dirac theories is in the treatment of characteristic surfaces. Suppose that we have a generic equation of the type $$\nabla \psi = f(\psi ,x),$$ (10.1) where ψ(x) is any multivector field (not necessarily a spinor field) and f(ψ, x) is some arbitr...

This series of five lectures introduces an approach to gravity which we have recently developed in Cambridge1. For many years physicists and mathematicians have attempted to express general relativity (GR) as a gauge theory, indexgravity!gauge theory of with the aim of placing GR on a similar conceptual footing to the modern theories of strong and...

After the development of the theory, it is now time to look at some applications. Historically the first solution to Einstein’s field equations was found by Schwarzschild, in 1916, who considered the fields of a mass-point and a spherically-symmetric star. Here we start with the vacuum part of this problem, and its extension to black holes. We will...

In Lecture I we introduced the \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} }\) and Ω(a) gauge fields. Their transformation properties are summarised in Table 13.1. Prom these, we need to construct ‘covariant’ quantities which transform in the same way that physical fields do. We start by defining the field strength R(a∧b) by $$...

In this Lecture we discuss a new approach to studying the gravitational field equations. In GR one works with the metric, usually encoded in the form of the line element, $$ds^2 = g_{\mu \nu } dx^\mu dx^\nu .$$ (15.1) The quantity gμν is derived from the \({\bar h}\)-function via $$g\mu \nu = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$...

We end this series of lectures by looking at three further applications of our approach to gravity — collapsing dust, cosmology, and cosmic strings.

This chapter is not intended to provide a balanced view of the STA approach to dynamics. We have instead chosen to concentrate on two topics normally covered in the Cambridge second-year undergraduate physics course: rigid body dynamics and elasticity. The level of presentation is variable, ranging from some elementary ideas through to more challen...

A new gauge-theory description of gravity is presented, employing gauge fields in a flat background spacetime. These fields ensure that all physical relations are independent of the position and orientation of the matter fields in this background. The language of "geometric algebra" best expresses the physical and mathematical content of the theory...

Clifford algebras have been studied for many years and their algebraic properties are well known. In particular, all Clifford algebras have been classified as matrix algebras over one of the three division algebras. But Clifford Algebras are far more interesting than this classification suggests; they provide the algebraic basis for a unified langu...

A new calculus, based upon the multivector derivative, is developed for Lagrangian mechanics and field theory, providing streamlined and rigorous derivations of the Euler-Lagrange equations. A more general form of Noether's theorem is found which is appropriate to both discrete and continuous symmetries. This is used to find the conjugate currents...

This paper employs the ideas of geometric algebra to investigate the physical content of Dirac's electron theory. The basis is Hestenes' discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation specifies a definite velocity, which might be interpreted as that of a r...

A reformulation of Grassmann calculus is presented in terms of geometric algebra—a unified language for physics based on Clifford algebra. In this reformulation, Grassmann generators are replaced by vectors, so that every product of generators has a natural geometric interpretation. The calculus introduced by Berezin [The Method of Second Quantizat...

It is shown that every Lie algebra can be represented as a bivector alge- bra; hence every Lie group can be represented as a spin group. Thus, the computa- tional power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thor- oughly analyzed, an...

The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum - and -matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar uni...

This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a geometric product of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are a...

In the preceding paper [the authors, ibid., 375-385 (1993; see the review above)] we described some aspects of a new theory of gravity, and found radially-symmetric static solutions to the free-field equations. Here we apply the theory to the universe on the largest scales by investigating the consequences of spatial homogeneity. A guiding principl...

We outline a theory of gravitational interactions utilizing the spacetime algebra – the geometric algebra of spacetime. The theory arises by demanding invariance active Poincaré transformations. Making this symmetry local results in a first-order theory with 40 degrees of freedom. The matter-free field equations are presented, and are solved for ra...

We present a new treatment of 2-spinors and twistors, using the spacetime algebra. The key rôle of bilinear covariants is emphasized. As a by-product, an explicit representation is found, composed entirely of real spacetime vectors, for the Grassmann entities of supersymmetric field theory.

A method of incorporating the results of Grassmann calculus within the framework of geometric algebra is presented, and shown to lead to a new concept, the multivector Lagrangian. A general theory for multivector Lagrangians is outlined, and the crucial role of the multivector derivative is emphasised. A generalisation of Noether’s theorem is deriv...

Stationary vacuum solutions of Kerr-Schild type are analysed within the framework of gauge-theory gravity. The complex structure at the heart of these flelds is shown to have a clear geometric origin, with the role of the unit imaginary fulfllled by the spacetime pseudoscalar. A number of general results for stationary Kerr-Schild flelds are obtain...

Preface This dissertation is the result of work carried out in the Department of Applied Mathe-matics and Theoretical Physics between October 1990 and October 1993. Sections of the dissertation have appeared in a series of collaborative papers | Except where explicit reference is made to the work of others, the work contained in this dissertation i...