Rafal Ablamowicz

Lectures on Clifford (geometric) algebras and applications

Preface (Rafal Ablamowicz and Garret Sobczyk) * Lecture 1: Introduction to Clifford Algebras (Pertti Lounesto) * 1.1 Introduction * 1.2 Clifford algebra of the Euclidean plane * 1.3 Quaternions * 1.4 Clifford algebra of the Euclidean space R3 * 1.5 The electron spin in a magnetic field * 1.6 From column spinors to spinor operators * 1.7 In 4D: Clifford algebra Cl4 of R4 * 1.8 Clifford algebra of Minkowski spacetime * 1.9 The exterior algebra and contractions * 1.10 The Grassmann-Cayley algebra and shuffle products * 1.11 Alternative definitions of the Clifford algebra * 1.12 References * Lecture 2: Mathematical Structure of Clifford Algebras (Ian Porteous) * 2.1 Clifford algebras * 2.2 Conjugation * 2.3 References * Lecture 3: Clifford Analysis (John Ryan) * 3.1 Introduction * 3.2 Foundations of Clifford analysis * 3.3 Other types of Clifford holomorphic functions * 3.4 The equation Dkf = 0 * 3.5 Conformal groups and Clifford analysis * 3.6 Conformally flat spin manifolds * 3.7 Boundary behavior and Hardy spaces * 3.8 More on Clifford analysis on the sphere * 3.9 The Fourier transform and Clifford analysis * 3.10 Complex Clifford analysis * 3.11 References * Lecture 4: Applications of Clifford Algebras in Physics (William E. Baylis) * 4.1 Introduction * 4.2 Three Clifford algebras * 4.3 Paravectors and relativity * 4.4 Eigenspinors * 4.5 Maxwells equation * 4.6 Quantum theory * 4.7 Conclusions * 4.8 References * Lecture 5: Clifford Algebras in Engineering (J.M. Selig) * 5.1 Introduction * 5.2 Quaternions * 5.3 Biquaternions * 5.4 Points, lines, and planes * 5.5 Computer vision example * 5.6 Robot kinematics * 5.7 Concluding remarks * 5.8 References * Lecture 6: Clifford Bundles and Clifford Algebras (Thomas Branson) * 6.1 Spin Geometry * 6.2 Conformal Structure * 6.3 Tractor constructions * 6.4 References * Appendix (Rafal Ablamowicz and Garret Sobczyk) * 7.1 Software forClifford algebras * 7.2 References * Index

Clifford Algebras and Spinor Structures

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Clifford Algebra Computations with Maple

Maple Computer Algebra System provides a convenient environment for computations in the real Clifford algebra Cl(B) of an arbitrary symbolic bilinear form B on a vector space V. It is well known that the symmetric part of B determines a unique Clifford structure on Cl(B) while it is less known that the antisymmetric part of B, if present, changes the multilinear structure of Cl(B). In our computations with Maple we assume that the bilinear form B is symbolic, real, possibly degenerate, and that it may have a non-trivial antisymmetric part. Any element (multivector) of Cl(B) is represented in Maple as a multivariate Clifford polynomial in basis monomials which form a standard basis for the algebra. Multiplication of these polynomials is based on recursive application of Cartan’s decomposition of Clifford product xu = x ∧ u + x ⌟ u into the wedge product part x ∧ u and the left contraction part x ⌟ u for any x ∈ V and u ∈ ∧ V. A package ‘Clifford’ of Maple procedures described below implements, among other features, Clifford and wedge/exterior multiplications, left contraction, grade involution, reversion, conjugation, Clifford and exterior exponentiation, computation of a symbolic inverse, scalar and vector parts, matrix representations. It affords computations in division rings of quaternions and octonions including conjugation, norm, and inverse, and it allows for implementation of rotations in a 3-dimensional Euclidean space in terms of quaternions.

Square Roots of –1 in Real Clifford Algebras

It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [33] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cl 3,0 of ℝ3. Further research on general algebras Cl p,q has explicitly derived the geometric roots of –1for p + q≤4 [20]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of –1f ound in the different types of Clifford algebras, depending on the type of associated ring (ℝ,ℍ,ℝ2,ℍ2, or ℂ). At the end of the chapter explicit computer generated tables of representative square roots of –1 are given for all Clifford algebras with n = 5,7, and s = 3 (mod 4) with the associated ring ℂ. This includes, e.g., Cl 0,5 important in Clifford analysis, and Cl 4,1 which in applications is at the foundation of conformal geometric algebra. All these roots of –1 are immediately useful in the construction of new types of geometric Clifford–Fourier transformations.

Mathematics of CLIFFORD -A Maple Package for Clifford and Graßmann Algebras

Abstract.CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C ℓ (B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM - based on Chevalley’s recursive formula, and cmulRS - based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

On Clifford algebras of a bilinear form with an antisymmetric part

We explicitly demonstrate with a help of a computer that Clifford algebra Cl(B) of a bilinear form B with a non-trivial antisymmetric part A is isomorphic as an associative algebra to the Clifford algebra Cl(Q) of the quadratic form Q induced by the symmetric part of B [in characteristic ≠ 2], However, the multivector structure of Cl(B) depends on A and is therefore different than the one of Cl(Q). Operation of reversion is still an anti-automorphism of Cl(B). It preserves a new kind of gradation in ⋀ V determined by A but it does not preserve the gradation in ⋀ V. The demonstration is given for Clifford algebras in real and complex vector spaces of dimension ≤ 9 with a help of a Maple package ‘Clifford’. The package has been developed by one of the authors to facilitate computations in Clifford algebras of an arbitrary bilinear form B.

STRUCTURE OF SPIN GROUPS ASSOCIATED WITH DEGENERATE CLIFFORD ALGEBRAS

Clifford algebras over finite‐dimensional vector spaces endowed with degenerate quadratic form contain a nontrivial two‐sided nilpotent ideal (the Jacobson radical) generated by the orthogonal complement of such spaces. Thus, they cannot be faithfully represented by matrix algebras. Following the theory of spin representations of classical Clifford algebras, the left regular (spin) representations of these degenerate algebras can be studied in suitably constructed left ideals. First, structure of the group of units of such algebras is examined for a quadratic form of arbitrary rank. It is shown to be a semidirect product of a group generated by the radical and the group of units of a maximal nondegenerate Clifford subalgebra. Next, in the special case of corank 1, Clifford, pin, and spin groups are defined an their structures are described. As an example, a Galilei–Clifford algebra over the Galilei space‐time is considered. A covering theorem is then proved analogous to the one well known in the theory of...

Bilinear covariants and spinor fields duality in quantum Clifford algebras

Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounestos spinor field classification. A physical interpretation of the deformed parts and the underlying Z-grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity [S. W. Hawking, “The unpredictability of quantum gravity,” Commun. ...

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Clifford algebras are naturally associated with quadratic forms. These algebras are ℤ2 -graded by construction. However, only a ℤn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) ↔ Λ V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the ℤn-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a ℤn -grading which we now call Clifford algebras of multi-vectors or quantum Clifford algebras. These quantum Clifford algebras are in fact Clifford algebras of a bilinear form in a functorial way. It turns out that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonaliz-ability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Clp,q which can be decomposed in the symmetric case into a tensor product Clp-1,q-1 ⊕ Cl1,1. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q -deformed structures through nontrivial vacuum states in quantum theories is outlined.

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V, Q) with a non-degenerate quadratic form Q of any signature (p, q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra Cℓ(V*, Q) of linear functionals (multiforms) acting on the universal Clifford algebra Cℓ(V, Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of Cℓ(V, Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of Cℓ(V, Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].