Clifford
Algebra| Clifford
Algebra |
In General > s.a. Gamma
Matrices.
$ Def: Given an n-dimensional vector space V with metric
, define the matrices {
a}
satisfying {a,
b}
= 2 ab I;
Then the Clifford algebra is the 2n-dimensional
vector space generated by
A:= {I, a
,
ab ,
...,
a_1,...a_n}
, a...c:= [a ...
c]
.
* 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 R,
which includes a notion of direction.
* Properties: For all A, A2
= I, and A B = CABC,
with |CAB| = 1.
* Representations: It
has 1! equivalence class of IRRs (Pauli's fundamental theorem).
* Relationships: From
a representation of the Clifford algebra we get one of the Lie
algebra of O()
by 
ab:= 
[a b] .
@ General references: Hestenes 66, in(86), & Sobczyk
84; Chisholm & Common
ed-86; Crumeyrolle 90; Porteous 95; Snygg 97 [III]; Pavsic 01-gq/06;
Lundholm & Svensson a0907-ln [emphasis on applications].
@ Representations: Harnett JPA(92)
[on 4D bivectors]; Schray et al ht/94 [octonionic];
West ht/98;
Ulrych AACA(08)-a0707 [with
hyperbolic numbers]; Toppan & Verbeek a0903 ["alphabetic"].
@ Related topics: {S Rajeev, RL 21.01.1983}; 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].
In Physics > s.a. bell inequality; dirac
field theory;
kaluza-klein models; particle
statistics; unified theories; {& clifford
manifold below}.
* Applications: The Maxwell equations may be written in a very compact
form with the help of Clifford numbers.
@ And quantum mechanics: Finkelstein IJTP(82)
[and quantum sets]; Baugh et al JMP(01)ht/00;
Ferrante mp/02 [and
fiber bundles]; Beil & Ketner
IJTP(03).
@ And electromagnetic theory: Gull et al FP(93); > s.a. electromagnetism.
@ And special relativity: Baylis & Sobczyk mp/04.
@ And general relativity: Cohen AACA-gq/02 [quantum
gravity];
Capelas de Oliveira & Rodrigues IJMPD(04)mp/03 [spinor
fields];
Francis & Kosowsky AP(04)gq/03 [techniques];
Hestenes a0807-in
[gauge
gravity and electroweak theory].
@ Spinning particles: Pezzaglia gq/99-in
[in curved spacetime]; Rodrigues JMP(04)mp/02;
da Rocha & Vaz IJGMP(07)mp/04, mp/04, mp/04;
Coquereaux mp/05-in
[and fundamental interactions].
@ 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];
Castro & Pavsic
IJTP(03)ht/02 [and
conformal group]; Rausch de Traubenberg ht/05-in
[rev].
Related Topics
* Clifford analysis:
The theory of functions from Rn 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: Pavsic FP(03)gq/02-in
[intro], ht/04-in, FP(05)ht [and
generalized quantum field theory and strings].
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31 jul 2009