Generalized Clifford algebra

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This article is about the algebra called generalized Clifford algebra (GCA). For (orthogonal) Clifford algebra, see Clifford algebra. For symplectic Clifford algebra, see Weyl algebra.

In mathematics, a Generalized Clifford algebra (GCA) is an associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl.[1] [2]


Definition and properties[edit]

General definition[edit]

This definition is brought forward by R. Jagannathan, in line with the work of his teacher Alladi Ramakrishnan. The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by

e_j e_k = \omega_{jk} e_k e_j \,
\omega_{jk} e_l = e_l \omega_{jk} \,
\omega_{jk} \omega_{lm} = \omega_{lm} \omega_{jk} \,

and

e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \,

for all j,k,l,m = 1,...,n. Additionally, in any irreducible matrix representation, relevant for physical applications, it is required that

\omega_{jk} = \omega_{kj}^{-1} = e^{2\pi i \nu_{kj}/N_{kj}}

for all j,k = 1,...,n, and N_{kj} =gcd (N_j,N_k).[3] The field F is usually taken to be the complex numbers C.

More specific definition[edit]

In the more common cases of GCA,[4][5] the n-dimensional generalized Clifford algebra of order p has the property \omega_{kj}=\omega, N_k=p for all j,k, and \nu_{kj}=1. From this it follows that

e_j e_k = \omega \, e_k e_j \,
\omega e_l = e_l \omega \,

and

e_j^{p} = 1 = \omega^{p} \,

for all j,k,l = 1,...,n, and \omega = \omega^{-1} = e^{2\pi i /p} is the pth root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.[6]

Clifford algebra[edit]

In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with \omega=-1 and p = 2.

Matrix representation[edit]

GCA matrices can be represented as:[7]


C =
\begin{pmatrix}
0&1&0&\cdots&0\\
0&0&1&\cdots&0\\
0&0&\cdots&1&0\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
1&0&0&\cdots&0
\end{pmatrix}
,    B =
\begin{pmatrix}
1&0&0&\cdots&0\\
0&\omega&0&\cdots&0\\
0&0&\omega^2&\cdots&0\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
0&0&0&\cdots&\omega^{(n-1)}
\end{pmatrix}
,    
U =
\begin{pmatrix}
1&1&1&\cdots&1\\
1&\omega&\omega^2&\cdots&\omega^{n-1}\\
1&\omega^2&(\omega^2)^2&\cdots&\omega^{2(n-1)}\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
1&\omega^{n-1}&\omega^{2(n-1)}&\cdots&\omega^{(n-1)^2}
\end{pmatrix}
,

Then C^n = 1 and C B = \omega B C and U^{-1} C U = B. Then with e_1=C and e_2=CB and e_3=B we have three basis elements which, together with \omega, fulfil the defining conditions of the Generalized Clifford Algebra (GCA).

These matrices C and B are often referred to as "shift and clock matrices", introduced by J. J. Sylvester in the 1880s. (Note that the matrices C are cyclic permutation matrices that perform a circular shift; they are not to be confused with upper and lower shift matrices which have ones only either above or below the diagonal, respectively).

Numerical examples[edit]

Case n = p = 2

In this case we have \omega=-1 and


C =
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
,    
B =
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
,    
U =
\begin{pmatrix}
1&1\\
1&-1
\end{pmatrix}

thus


e_1 =
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
,    
e_2 =
\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix}
,    
e_3 =
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
Case n = p = 4

In this case we have \omega=i and


C =
\begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
1&0&0&0
\end{pmatrix}
,    
B =
\begin{pmatrix}
1&0&0&0\\
0&i&0&0\\
0&0&-1&0\\
0&0&0&-i
\end{pmatrix}
,    
U =
\begin{pmatrix}
1&1&1&1\\
1&i&-1&-i\\
1&-1&1&-1\\
1&-i&-1&i
\end{pmatrix}

and e_1, e_2, e_3 can be determined accordingly.

Application to physics[edit]

Generalized Clifford algebras has been applied to quantum physics.[4][5][8] In particular, the concept of a spinor can be linked directly to these algebras.[4]


See also[edit]

Further reading[edit]

  • R. Jagannathan, On generalized Clifford algebras and their physical applications
  • K. Morinaga, T. Nono (1952): On the linearization of a form of higher degree and its representation, J. Sci. Hiroshima Univ. Ser. A, 16, pp. 13–41
  • O. Morris (1967): On a Generalized Clifford Algebra, Quart. J. Math (Oxford), 18, pp. 7–12
  • O. Morris (1968): On a Generalized Clifford Algebra II, Quart. J. Math (Oxford), 19, pp. 289–299

References[edit]

  1. ^ Weyl, H., "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, doi:10.1007/BF02055756. Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1931)
  2. ^ Schwinger, J. (1960), "Unitary operator bases", Proc Natl Acad Sci U S A, April; 46(4): 570–579, PMCID: PMC222876; ibid, "Unitary transformations and the action principle", 46(6): 883–897, PMCID: PMC222951
  3. ^ R. Jagannathan: On generalized Clifford algebras and their physical applications, arXiv:1005.4300v1 (submitted 24 May 2010)
  4. ^ a b c See for example: A. Granik, M. Ross: On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics, in: Rafal Ablamowicz, Joseph Parra, Pertti Lounesto (eds.): Clifford Algebras with Numeric and Symbolic Computation Applications, Birkhäuser, 1996, ISBN 0-8176-3907-1, p. 101–110
  5. ^ a b See for ex.: E.H.El Kinani: Between Quantum Virasoro Algebra Lc and Generalized Clifford Algebras arXiv:math-ph/0310044v1 (submitted 22 October 2003)
  6. ^ See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
  7. ^ Alladi Ramakrishnan: Generalized Clifford Algebra and its applications – A new approach to internal quantum numbers', Proceedings of the Conference on Clifford algebra, its Generalization and Applications, January 30 – February 1, 1971, Matscience, Madras 20, pp. 87–96
  8. ^ A. K. Kwaśniewski: On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group