A Short Introduction to Graphical Algebra

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The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory. The claim that Clifford algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry. Clifford algebra is used to develop a coordinate-free algebraic formulation of projective geometry. Major theorems of projective geometry are reduced to algebraic identities which apply as well to metrical geometry.

Improvements in the formulation of linear algebra are suggested to simplify its intimate relation to projective geometry.

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This paper is a comprehensive introduction to a CGA tool kit. Synthetic statements in classical geometry translate directly to coordinate-free algebraic forms. Invariant and covariant methods are coordinated by conformal splits, which are readily related to the literature using methods of matrix algebra, biquaternions, and screw theory. Designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented. 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, thus fulfilling Minkowski's original vision. In order to improve student reception we also focus on forces and the conservation of energy and momentum, which take a very simple form in GA, so that students can easily build on established intuitions in using these laws developed from studying Newtonian mechanics.

The potential formulation is also integrated into the presentation that allows an alternate solution path, as well as an introduction to the Lagrangian approach. In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate.


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We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is a wave equation. A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained.

In this work a comprehensive geometric model for paracatadioptric sensors has been presented.

The model is based on the equivalence between paracatadioptric projection and the inversion. The main reason for using the inversion is that it can be represented by a versor i. The advantage of this representation is that it can be applied not only to points but also to point-pairs, lines, circles, spheres and planes.

In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. Quaternions particularly have a catchy interpretation in the context of geometric algebra which can be used didactically.

The relation between quaternions and geometric algebra is presented with a view to analysing its didactical possibilities. In this presentation I explore the possible choices for the imaginary unit in the Dirac equation to show that SU 3 and SU 2 symmetries arise naturally from such choices. The quantum numbers derived from the imaginary unit are unusual but a simple conversion allows the derivation of electric charge and isospin, quantum numbers for two families of particles.

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This association to elementary particles is not final because further understanding of the role played by the imaginary unit is needed. In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, I give a quick review of classic results regarding geometric algebras, such as their classification in terms of matrix algebras, the connection to orthogonal and Spin groups, and their representation theory.

A number of lower-dimensional examples are worked out in a systematic way using so called norm functions, while general applications of representation theory include normed division algebras and vector fields on spheres. 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 exclusion principle and other nonlocal effects in quantum physics.

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 wavefunction and gives a measure of the degree of entanglement present in the system.

The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method to the geometric algebra representation. This latter form has the advantage of suggesting a generalisation to n subsystems.

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 geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed.

The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing. This paper surveys the application of geometric algebra to the physics of electrons.

It first appeared in and is reproduced here with only minor modifications. Subjects covered include non-relativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, propagators and scattering theory, spin precession, tunnelling times, spin measurement, multiparticle quantum mechanics, relativistic multiparticle wave equations, and semiclassical mechanics.

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 use it to show that the position dependence is the result of the action of the translation operator in the conformal space on the 4-d spinor.

We obtain the geometrical description of twistors through the conformal geometric algebra, and derive the Robinson congruence. 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.


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This representation allows us to identify an event with the wave-function for a non-charged Klein-Gordon particle. A more general point can also be obtained, if the space is complexified and the twistors have non-zero helicity. In this case, such a point is no longer an event, but it can be identified with the wave-function for a charged Klein-Gordon particle. Geometric Algebra GA is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Spacetime Algebra STA , and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics.

Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach.

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The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric Clifford Algebra is the most convenient formalism for that case.

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Generalization of formal complex plane to an an arbitrary plane in three dimensions and of usual Hopf fibration to the map generated by an arbitrary unit value element of even sub-algebra of the three-dimensional Geometric Algebra are resulting in more profound description of qubits compared to quantum mechanical Hilbert space formalism. Conventional quantum mechanical qubits can be lifted to states as even three dimensional geometric algebra operators that act on observables.

The operators may be implemented via the two types of Maxwell equation solution polarizations. Solution of Maxwell equation in geometric algebra formalism gives g-qubits which are exact lifts of conventional qubits. Therefore, it unambiguously reveals actual meaning of complex parameters of qubits of the commonly accepted Hilbert space quantum mechanics and, particularly, directly demonstrates the option of instant non-locality of states.

Recently suggested scheme of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.

Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the Clifford translation time factor and through the observable measurement results. The space so defined is really 4-dimensional because the null condition effectively reduces the dimensionality by one.

Besides generating GR and QM, the same space generates also 4-dimensional Euclidean space where dynamics can be formulated and is quite often equivalent to the relativistic counterpart. The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficient part of superanalysis one obtains quantum mechanics for systems with spin.

This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.

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The relationship between spinors and Clifford or geometric algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed.

Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U 1 , SU 2 , and SL 2,C spinors. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches.

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We introduce spinors, at a level appropriate for an undergraduate or first year graduate course on relativity, astrophysics or particle physics. The treatment assumes very little mathematical knowledge mainly just vector analysis and some idea of what a group is. The SU 2 -SO 3 homomorphism is presented in detail. Lorentz transformation, chirality, and the spinor Minkowski metric are introduced. Applications to electromagnetism, parity violation, and to Dirac spinors are presented. A new characterization of anticommutativity of unbounded self-adjoint operators is presented in connection with Clifford algebra.

Some consequences of the characterization and applications are discussed. An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface.

It is wrong to suppose that one need be a state-of-the-art differential geometer to make effective use of the exterior calculus. The formalism imposes demands certainly not greater than those of tensor analysis from which it arises as a powerful special case , and in practical applications does its work even more simply and swiftly than vector analysis. A large part of my effort here, therefore, will be directed to the the establishment of some degree of notational and conceptual commonality, and to construction of a clear statement of my own motivating interests in this area.

Of particular note are the matrix representation of geometric objects, effective algorithms for this geometry inversion, meet and join , and issues in efficiency and numerics.