Quaternions, Clifford Algebras and Relativistic Physics

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Clifford algebra

Be the first to like this. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Quaternions clifford algebras and relativistic physics 1. Girard 2. The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have privileged a geometric approach, this book uses an algebraic approach that can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics.

It proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism, and general relativity. You just clipped your first slide! Clipping is a handy way to collect important slides you want to go back to later. Relativistic kinematics via H C. Group of conformal transformations.

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Contents xi. The three-dimensional vector calculus is used in nonrelativistic physics and also in classical electromagnetism which is a relativistic theory. A satisfactory treatment of classical electromagnetism, special relativity and general relativity is given by the tensor calculus.

Yet, the tensor calculus does not allow a double representation of the Lorentz group and thus seems incompatible with relativistic quantum mechanics. A third calculus is then introduced, the spinor calculus, to formulate relativistic quantum mechanics. The set of mathematical tools used in physics thus appears as a succession of more or less coherent formalisms.

Is it possible to introduce more coherence and unity in this set? Computations within these algebras have become straightforward with software like Mathematica which allows us to perform extended algebraic computations and to simplify them. This page intentionally left blank Chapter 1 Quaternions The abstract quaternion group, discovered by William Rowan Hamilton in , is an illustration of group structure. Then the quaternion algebra and the classical vector calculus are treated as an application. Let F and G be two groups. Cyclic group Cn of order n the elements of which are b, b2 , b3 ,.

Quaternions 2. The symmetry group of the equilateral triangle see Fig. The symmetry group of the square see Fig. This 1. Historically, these two products were obtained by W. Quaternion algebra H 11 1.

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The equations 1. With a 14 Chapter 1. Quaternions Example.

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Give the polar form of a and b. Compute a, b, a. Find x. The expression of a plane symmetry is obtained as follows. Rotation groups SO 4 and SO 3 2. The set of proper and improper rotations constitute the group O 3. The same property applies to the cube dual of the octaeder the 8 vertices of which are the centers of the faces of the above octaeder.

Express the basis vectors ei in the basis at rest. Find the velocity and the acceleration in the mobile basis. This page intentionally left blank Chapter 3 Complex quaternions From the very beginning of special relativity, complex quaternions have been used to formulate that theory [45]. This chapter establishes the expression of the Lorentz group using complex quaternions and gives a few applications.

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Complex quaternions 3. The hyperplanes are supposed to go through the origin. Theorem 3. Complex quaternions Proper orthochronous Lorentz transformation Consider the transformation obtained by combining an even number of time symmetries and an even number of space symmetries.

Let us take for example two time symmetries minquats f and g followed by two space symmetries minquats m and n. Other Lorentz transformations Call n the number of time symmetries and p the number of space symmetries. In combining these symmetries, one obtains the following Lorentz transformations L: 1. Group O 1, 3 : summarizing table The whole set of Lorentz transformations is given in the table below where n is the number of time symmetries and p the number of space symmetries [49]. A general transformation proper, orthochronous is 42 Chapter 3.

Complex quaternions obtained by combining a rotation and a pure Lorentz transformation. Reciprocally, a general Lorentz transformation can be decomposed into a pure Lorentz transformation and a rotation. The problem of the decomposition of a Lorentz transformation into a pure Lorentz transformation and a rotation is thus solved in the most general case.

As an immediate application, consider the combination of two pure Lorentz transformations b1 , b2. The resulting Lorentz transformation will thus contain a rotation; this is the principle of the Thomas precession [52], [51]. Hence, one sees that H C already allows us to develop a few notions of a multivector calculus.

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Relativistic kinematics via H C 47 3. Explicitly, equation 3. Explicitly, one obtains [26, p. The transformations 3. Equation 3. Complex quaternions Equation 3. Exercises 55 using complex quaternions. Theorem 4. A complete basis of the algebra is given in the following table. Conway and D. On quaternions and octonions, their geometry, arithmetic, and symmetry. Quantum energetic causal sets, See note added in proof, Phys. D version, and updated abstract, arXiv version 2. D, 90 ,. The universe as a process of unique events, See updated abstract, arXiv version 3.

D, 90 Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics Publishers. Conference Proceedings, in press. World Scientific. Causal sets and the deep structure of spacetime. Article contributed to ed Abhay Ashtekar. Non-embeddable relational configurations - Edwards, William arXiv The physics of what happens. Annals Phys. An algebraic causal set constructed at the fundamental level out of particles not space-time points.

Hand-written notes emailed L. Smolin on Sept 2,, per his request. Please see amendment to abstracts of [24] and [25], arXiv versions. A unified theory of ideals. D 86 Generations: three prints, in colour.

JHEP, 10 Charge quantization from a number operator - Furey, C. Quaternions, Clifford algebras and relativistic physics. Birkhauser Velag,. Greiter and D.

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Imaginary in all directions. J Phys. Quark structure and the octonions. Quark statistics and octonions.