Clifford Algebra| 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 + a ∧ b;
in 3 dimensions, a ∧ b = 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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