Generalized Clifford algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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] who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882),[2] and organized by Cartan (1898)[3] and Schwinger.[4]

Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces.[5][6][7] The concept of a spinor can further be linked to these algebras.[6]

Definition and properties[edit]

Abstract definition[edit]

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by[8]

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} \,

j,k,l,m = 1,...,n.

Moreover, 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}}

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

More specific definition[edit]

In the more common cases of GCA,[6] the n-dimensional generalized Clifford algebra of order p has the property ωkj = ω, N_k=p   for all j,k, and \nu_{kj}=1. 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.[9]

Clifford algebra

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

Matrix representation[edit]

The Clock and Shift matrices can be represented[10] by n×n matrices in Schwinger's canonical notation as


V =
\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}
,    U =
\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}
,    
W =
\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}
.

Notably, Vn = 1, VU = ωUV (the Weyl braiding relations), and W−1VW = U (the Discrete Fourier transform). With e1 = V , e2 = VU, and e3 = U, one has three basis elements which, together with ω, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, V and U, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices V 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).

Specific examples[edit]

Case n = p = 2.

In this case, we have ω = −1, and


V =
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
,    
U =
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
,    
W =
\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}
,

which constitute the Pauli matrices.

Case n = p = 4,

In this case we have ω = i, and


V =
\begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
1&0&0&0
\end{pmatrix}
,    
U =
\begin{pmatrix}
1&0&0&0\\
0&i&0&0\\
0&0&-1&0\\
0&0&0&-i
\end{pmatrix}
,    
W =
\begin{pmatrix}
1&1&1&1\\
1&i&-1&-i\\
1&-1&1&-1\\
1&-i&-1&i
\end{pmatrix}

and e1, e2, e3 may be determined accordingly.

See also[edit]

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. ^ Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III . online and further.
  3. ^ Cartan, E. (1898). "Les groupes bilinéaires et les systèmes de nombres complexes." Annales de la faculté des sciences de Toulouse 12.1 B65-B99. online
  4. ^ 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
  5. ^ Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics 6 (5): 583. doi:10.1007/BF00715110. 
  6. ^ 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
  7. ^ A. K. Kwaśniewski: On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group
  8. ^ For a serviceable review, see Vourdas A. (2004), "Quantum systems with finite Hilbert space", Rep. Prog. Phys. 67 267. doi: 10.1088/0034-4885/67/3/R03.
  9. ^ See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
  10. ^ 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

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