James M. Chappell

University of Adelaide · School of Electrical and Electronic Engineering

The principle of equivalence is used to argue that the known law of decreasing acceleration for high speed motion, in a low acceleration regime, produces the same result as found for a weak gravitational field, with subsequent implications for stronger fields. This result coincides with Hilbert's little explored equation of 1917, regarding the velo...

The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which " flows equably without relation to anything external. " In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that he could provide a unif...

This paper presents an approach to classification of substances based on their terahertz spectra. We use geometric algebra to provide a concise mathematical means for attacking the classification problem in a coordinate-free form. For the first time, this allows us to perform classification independently of dispersion and, hence, independently of t...

In this paper we develop a framework allowing a natural extension of the Lorentz transformations. To begin, we show that by expanding conventional four-dimensional spacetime to eight-dimensions that a natural generalization is indeed obtained. We then find with these generalized coordinate transformations acting on Maxwell's equations that the elec...

There are a wide variety of different vector formalisms currently utilized in science. For example, Gibbs three-vectors, spacetime four-vectors, complex spinors for quantum mechanics, quaternions used for rigid body rotations and Clifford multivectors. With such a range of vector formalisms in use, it thus appears that there is as yet no general ag...

A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question is to understand the conditions under which a classical game-theoretic setting can be transf...

A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question, is to understand the conditions under which a classical game-theoretic setting can be trans...

Understanding the nature of time has become one of the key unsolved problems in science. Newton in his Principia proposed an absolute universal time that `flows equably without relation to anything external'. Hamilton also proposed a representation of time within the mathematical framework of the quaternions before the Minkowski space-time unificat...

Are there two-player games in which a strategy pair can exist as a Nash equilibrium only when the game is played quantum mechanically? To find an answer to this question, we study two-player games that are played in generalized Einstein-Podolsky-Rosen setting. Considering particular strategy pairs, we identify sets of games for which the pair can e...

Are there two-player games in which a strategy pair can exist as a Nash equilibrium only when the game is played quantum mechanically? To find an answer to this question, we study two-player games that are played in generalized Einstein-Podolsky-Rosen setting. Considering particular strategy pairs, we identify sets of games for which the pair can e...

The Minkowski formulation of special relativity reveals the essential four-dimensional nature of spacetime, consisting of three space and one time dimension. Recognizing its fundamental importance, a variety of arguments have been proposed over the years attempting to derive the Minkowski spacetime structure from fundamental physical principles. In...

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and thre...

A quantum game-theoretic setting enables the analysis of strategic interaction among agents with access to quantum resources. One of the original motivations for re-expressing quantum algorithms, and quantum communication protocols, in terms of competing quantum agents was to provide new insights into their workings. Looking closely, one finds that...

A Bayesian game is a game of incomplete information in which the rules of the game are not fully known to all players. We consider the Bayesian game of Battle of Sexes that has several Bayesian Nash equilibria and investigate its outcome when the underlying prob-ability set is obtained from generalized Einstein-Podolsky-Rosen experiments. We find t...

Eisert et al.'s quantization scheme for a bimatrix game involves players' local unitary transformations on a pair of qubits and the subsequent quantum measurement generating four quantum probabilities. When quantum probabilities obey factorizability relations, the quantum game reduces itself to the corresponding mixed-strategy classical game. In th...

In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate a...

The idealized Kish-Sethuraman (KS) cipher is theoretically known to offer perfect security through a classical information channel. However, realization of the protocol is hitherto an open problem, as the required mathematical operators have not been identified in the previous literature. A mechanical analogy of this protocol can be seen as sending...

While information-theoretic security is often associated with the one-time pad and quantum key distribution, noisy transport media leave room for classical techniques and even covert operation. Transit times across the public internet exhibit a degree of randomness, and cannot be determined noiselessly by an eavesdropper. We demonstrate the use of...

While information-theoretic security is often associated with the one-time pad and quantum key distribution, noisy transport media leave room for classical techniques and even covert operation. Transit times across the public internet exhibit a degree of randomness, and cannot be determined noiselessly by an eavesdropper. We demonstrate the use of...

Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension ict, with the unit imaginary producing the correct spacetime distance x^2 -c^2 t^2 , and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for...

Wave Mechanics. The Dirac equation in two dimensions, wave mechanics and Maxwell’s equations. (PDF)

We explore the consequences of space and time described within the Clifford multivector of three dimensions $ Cl_{3,0}$, where space consists of three-vectors and time is described with the three bivectors of this space. When describing the curvature around massive bodies, we show that this model of spacetime when including the Hubble expansion nat...

A Bayesian game is a game of incomplete information in which the rules of the game are not fully known by all players. We consider a Bayesian game of Battle of Sexes that has several Bayesian Nash equilibria and investigate its outcome when the underlying probability set is obtained from the generalized Einstein-Podolsky-Rosen experiments. We find...

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford's geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, tha...

Large astronomical objects such as stars or planets, produce approximately spherical shapes due to the large gravitational forces, and if the object is rotating rapidly, it becomes an oblate spheroid. In juxtaposition to this, we conduct a thought experiment regarding the properties of a planet being in the form of a perfect cube. We firstly calcul...

Following the development of the special theory of relativity in 1905, Minkowski proposed a unified space and time structure consisting of three space dimensions and one time dimension, with relativistic effects then being natural consequences of this spacetime geometry. In this paper, we illustrate how Clifford's geometric algebra that utilizes mu...

The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made....

The nonvolatile memory property of a memristor enables the realization of new methods for a variety of computational engines ranging from innovative memristive-based neuromorphic circuitry through to advanced memory applica-tions. The nanometer-scale feature of the device creates a new opportunity for realization of innovative circuits that in some...

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford's geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, tha...

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game beco...

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is of significant interest to describe it in the most efficient mathematical formalism. We show firstly, that Clifford's formalism of geometric algebra, provides a significantly more efficient representation than the conventional Bra-ket not...

We calculate the Newtonian gravitational potential and field of a cubic, homogeneous asteroid and we apply it to the orbit of possible satellites. Large astronomical objects such as stars or planets, naturally tend to form spherical shapes due to the dominance of the gravitational forces, but as a thought experiment, we consider the properties of a...

Early researchers attempting to simulate complex quantum mechanical interactions on digital computers discovered that they very quickly consumed the computers’ available memory resources, because the state space of a quantum system typically grows exponentially with problem size. Consequently, Richard Feynman proposed in 1982 that perhaps the only...

We use the formalism of Clifford Geometric Algebra (GA) to develop an analysis of quantum versions of three-player non-cooperative games. The quantum games we explore are played in an Einstein-Podolsky-Rosen (EPR) type setting. In this setting, the players' strategy sets remain identical to the ones in the mixed-strategy version of the classical ga...

Geometric Algebra. Boost-rotation form of a multivector, the exponential of a general multivector and useful results from geometric calculus. (PDF)

Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring symmetry in the error definitions, an exact formula can be found. This new approach may also have value in sol...

Historically, there have been many attempts to produce the appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions, Gibbs' vector system using the dot and cross products. We illustrate however, that Clifford's geomet...

A new simplified approach for teaching electromagnetism is presented using the formalism of geometric algebra (GA) which does not require vector calculus or tensor index notation, thus producing a much more accessible presentation for students. The four-dimensional spacetime proposed is completely symmetrical between the space and time dimensions,...

We construct quantum games from a table of non-factorizable joint probabilities, coupled with a symmetry constraint, requiring symmetrical payoffs between the players. We give the general result for a Nash equilibrium and payoff relations for a game based on non-factorizable joint probabilities, which embeds the classical game. We study a quantum v...

We analyze the quantum penny flip game using geometric algebra and so determine all possible unitary transformations which enable the player Q to implement a winning strategy. Geometric algebra provides a clear visual picture of the quantum game and its strategies, as well as providing a simple and direct derivation of the winning transformation, w...