Clifford Algebra  

In General > s.a. Gamma Matrices; Geometric Algebra.
$ Def: Given an n-dimensional vector space V with metric η, define the matrices {γa} satisfying {γa, γb} = 2 ηab I; Then the Clifford algebra is the 2n-dimensional vector space generated by

ΓA:= {I, γa , γab , ..., γa1, ..., an} ,     γa ... c:= γ[a ... γc] .

* Outer product: Given by ab:= a · b + ab; in 3 dimensions, ab = i a × b; It is associative; Corresponds to an extension of \(\mathbb R\), which includes a notion of direction.
* Properties: For all A, γA2 = I, and γA γB = CAB γC, with |CAB| = 1.
* Representations: It has 1! equivalence class of irreducible representations (Pauli's fundamental theorem).
* Relationships: From a representation of the Clifford algebra we get one of the Lie algebra of O(η) by Σab:= \(1\over2\)Γ[a Γb] .
@ General references: Hestenes 66, in(86), & Sobczyk 84; Chisholm & Common ed-86; Crumeyrolle 90; Porteous 95; Snygg 97 [III]; Pavšič 01-gq/06; Lundholm & Svensson a0907-ln [emphasis on applications]; Lachièze-Rey a1007-conf [intro]; Garling 11; Todorov BulgJP(11)-a1106 [intro]; Boudet 11; Klawitter 15; de Gosson acad(16) [in symplectic geometry and quantum mechanics]; Shirokov a1709-ln.
@ Representations: Harnett JPA(92) [on 4D bivectors]; Schray & Manogue FP(96)ht/94 [octonionic]; West ht/98 [rev]; Ulrych AACA(08)-a0707 [with hyperbolic numbers]; Toppan & Verbeek JMP(09)-a0903 ["alphabetic"]; Budinich a1805 [complex representations].
@ Related topics: Ablamowicz et al CzJP(03)mp [classification of idempotents]; Beil & Ketner IJTP(03) [and Peirce logic]; da Rocha & Vaz AACA(06)mp [generalized, over Peano spaces]; Aragon-Camarasa et al a0810-proc [with Mathematica]; Dadbeh a1104 [inverses and determinants, up to dimension 5]; Hanson a1104 [real Clifford algebra as a Clifford module over itself]; Formiga a1209 [all possibles products between generators of the 4D Clifford algebra]; Kuusela a1905 [Mathematica package].
> Online resources: see Wikipedia page.

In Physics > s.a. bell inequality; kaluza-klein models; particle statistics; unified theories; {& clifford manifold below}.
* Electromagnetism: The Maxwell equations may be written in a compact form with the help of Clifford numbers.
* Quantum mechanics: There have been attempts at formulating quantum theory based on Clifford numbers instead of complex numbers, but it is more difficult than, for example, with quaternions, because this is not a division algebra.
@ Quantum theory: Finkelstein IJTP(82) [and quantum sets]; Baugh et al JMP(01)ht/00; Ferrante mp/02 [and fiber bundles]; Beil & Ketner IJTP(03); Hiley & Callaghan a1011; Binz et al FP(13)-a1112 [in symplectic geometry and quantum mechanics]; Hiley LNP-a1211 [starting from the notion of process].
@ Electromagnetic theory: Gull et al FP(93); Dressel et al PRP(15)-a1411 [comprehensive introduction]; > s.a. electromagnetism.
@ Special relativity: Baylis & Sobczyk IJTP(04)mp; Chappell et al a1101 [representation of 3D space]; Chappell et al PLoS(12)-a1106, a1205 [algebraic alternative to Minkowski spacetime, Clifford multivectors]; Castro FP(12) [extended relativity theories, superluminal particles]; > s.a. Rigid Body.
@ Classical gravity: Capelas de Oliveira & Rodrigues IJMPD(04)mp/03 [and spinor fields]; Francis & Kosowsky AP(04)gq/03 [techniques]; Hestenes a0807-MGXI [gauge gravity and electroweak theory]; Castro IJTP(13) [Lanczos-Lovelock and f(R) gravity].
@ Quantum gravity: Cohen AACA-gq/02; Pavšič a1104; Castro FP(14) [black-hole entropy, rainbow metrics, generalized dispersion & uncertainty relations].
@ Spinning particles: Pezzaglia gq/99-proc [in curved spacetime]; Rodrigues JMP(04)mp/02; da Rocha & Vaz IJGMP(07)mp/04, mp/04, mp/04; Coquereaux mp/05-ln [rev, and fundamental interactions]; > s.a. dirac field theory; fermions.
@ Quantum field theory, particle physics: Pavšič a1104, JPCS(13)-a1210 [quantum field theory]; Daviau & Bertrand JModP(14)-a1408, 15 [Standard Model]; Arnault a2105 [emergence from quantum automata].
@ Other applications: Dimakis & Müller-Hoissen CQG(91) [applications of clifforms in field theory]; Garrett PW(92)sep; Baylis 96; Chernitskii IJMMS(02)ht/00 [Born-Infeld theory]; Castro & Pavšič IJTP(03)ht/02 [and the conformal group]; Rausch de Traubenberg ht/05-ln [rev]; Berrondo et al AJP(12)oct [unifying the inertia and Riemann curvature tensors]; Trindade et al a2005 [quantum information]; > s.a. Kustaanheimo-Stiefel Transformation.

Related Topics > s.a. differential geometry; graph theory [operators].
* Clifford analysis: The theory of functions from \(\mathbb R\)n to the universal Clifford algebras, generalizing holomorphic functions.
* Clifford manifold: A "C-space" consisting not only of points, but also of 1-loops, 2-loops, etc.
@ Clifford analysis: Brackx, Delanghe & Sommen 82.
@ Clifford manifold: Pavšič FP(03)gq/02-conf [intro], ht/04-talk, FP(05)ht [and generalized quantum field theory and strings].

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