Generalized Clifford algebra

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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]

The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.[8][9][10][11]

Definition and properties

Abstract definition

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by[12]

[math]\displaystyle{ e_j e_k = \omega_{jk} e_k e_j \, }[/math]
[math]\displaystyle{ \omega_{jk} e_l = e_l \omega_{jk} \, }[/math]
[math]\displaystyle{ \omega_{jk} \omega_{lm} = \omega_{lm} \omega_{jk} \, }[/math]

and

[math]\displaystyle{ e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \, }[/math]

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

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that

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

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

More specific definition

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

[math]\displaystyle{ e_j e_k = \omega \, e_k e_j \, }[/math]
[math]\displaystyle{ \omega e_l = e_l \omega \, }[/math]

and

[math]\displaystyle{ e_j^{p} = 1 = \omega^{p} \, }[/math]

for all j,k,l = 1,...,n, and

[math]\displaystyle{ \omega = \omega^{-1} = e^{2\pi i /p} }[/math]

is the pth root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.[13]

Clifford algebra

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

Matrix representation

The Clock and Shift matrices can be represented[14] by n×n matrices in Schwinger's canonical notation as

[math]\displaystyle{ 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} }[/math] ,    [math]\displaystyle{ 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} }[/math] ,    [math]\displaystyle{ 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} }[/math] .

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

Case n = p = 2.

In this case, we have ω = −1, and

[math]\displaystyle{ V = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} }[/math] ,    [math]\displaystyle{ U = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} }[/math] ,    [math]\displaystyle{ W = \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} }[/math]

thus

[math]\displaystyle{ e_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} }[/math] ,    [math]\displaystyle{ e_2 = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} }[/math] ,    [math]\displaystyle{ e_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} }[/math] ,

which constitute the Pauli matrices.

Case n = p = 4,

In this case we have ω = i, and

[math]\displaystyle{ V = \begin{pmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0 \end{pmatrix} }[/math] ,    [math]\displaystyle{ U = \begin{pmatrix} 1&0&0&0\\ 0&i&0&0\\ 0&0&-1&0\\ 0&0&0&-i \end{pmatrix} }[/math] ,    [math]\displaystyle{ W = \begin{pmatrix} 1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i \end{pmatrix} }[/math]

and e1, e2, e3 may be determined accordingly.

See also

References

  1. Weyl, H. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift für Physik 46 (1–2): 1–46. doi:10.1007/BF02055756. Bibcode1927ZPhy...46....1W. 
    (1950) [1931]. The Theory of Groups and Quantum Mechanics. Dover. ISBN 9780486602691. https://archive.org/details/theoryofgroupsqu1950weyl. 
  2. Sylvester, J. J. (1882), A word on Nonions, Johns Hopkins University Circulars, I, pp. 241–2 ; 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. http://archive.numdam.org/ARCHIVE/AFST/AFST_1898_1_12_2/AFST_1898_1_12_2_B65_0/AFST_1898_1_12_2_B65_0.pdf. 
  4. Schwinger, J. (April 1960). "Unitary operator bases". Proc Natl Acad Sci U S A 46 (4): 570–9. doi:10.1073/pnas.46.4.570. PMID 16590645. Bibcode1960PNAS...46..570S. 
    (1960). "Unitary transformations and the action principle". Proc Natl Acad Sci U S A 46 (6): 883–897. doi:10.1073/pnas.46.6.883. PMID 16590686. Bibcode1960PNAS...46..883S. 
  5. Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics 6 (5): 583. doi:10.1007/BF00715110. Bibcode1976FoPh....6..583S. 
  6. 6.0 6.1 6.2 See for example: Granik, A.; Ross, M. (1996). "On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics". in Ablamowicz, R.; Parra, J.; Lounesto, P.. Clifford Algebras with Numeric and Symbolic Computation Applications. Birkhäuser. pp. 101–110. ISBN 0-8176-3907-1. https://books.google.com/books?id=OpbY_abijtwC&pg=PA101. 
  7. Kwaśniewski, A.K. (1999). "On generalized Clifford algebraC(n)4 andGLq(2;C) quantum group". AACA 9 (2): 249–260. doi:10.1007/BF03042380. 
  8. Tesser, Steven Barry (2011). "Generalized Clifford algebras and their representations". in Micali, A.; Boudet, R.; Helmstetter, J.. Clifford algebras and their applications in mathematical physics. Springer. pp. 133–141. ISBN 978-90-481-4130-2. https://archive.org/details/cliffordalgebras1989hest. 
  9. Childs, Lindsay N. (30 May 2007). "Linearizing of n-ic forms and generalized Clifford algebras". Linear and Multilinear Algebra 5 (4): 267–278. doi:10.1080/03081087808817206. 
  10. Pappacena, Christopher J. (July 2000). "Matrix pencils and a generalized Clifford algebra". Linear Algebra and Its Applications 313 (1–3): 1–20. doi:10.1016/S0024-3795(00)00025-2. 
  11. Chapman, Adam; Kuo, Jung-Miao (April 2015). "On the generalized Clifford algebra of a monic polynomial". Linear Algebra and Its Applications 471: 184–202. doi:10.1016/j.laa.2014.12.030. 
  12. For a serviceable review, see Vourdas, A. (2004). "Quantum systems with finite Hilbert space". Rep. Prog. Phys 67 (3): 267–320. doi:10.1088/0034-4885/67/3/R03. Bibcode2004RPPh...67..267V. 
  13. See for example the review provided in: Smith, Tara L.. "Decomposition of Generalized Clifford Algebras". Archived from the original on 2010-06-12. https://web.archive.org/web/20100612050907/http://math.uc.edu/~tsmith/papers/CliffAlg.pdf. 
  14. Ramakrishnan, Alladi (1971). "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. Madras: Matscience. pp. 87–96. http://www.imsc.res.in/xmlui/bitstream/handle/123456789/227/MR60.pdf. 

Further reading

  • Jagannathan, R. (2010). "On generalized Clifford algebras and their physical applications". arXiv:1005.4300 [math-ph].
  • Morinaga, K.; Nono, T. (1952). "On the linearization of a form of higher degree and its representation". J. Sci. Hiroshima Univ. Ser. A 16: 13–41. doi:10.32917/hmj/1557367250. 
  • Morris, A.O. (1967). "On a Generalized Clifford Algebra". Quart. J. Math (Oxford 18 (1): 7–12. doi:10.1093/qmath/18.1.7. Bibcode1967QJMat..18....7M. 
  • Morris, A.O. (1968). "On a Generalized Clifford Algebra II". Quart. J. Math (Oxford 19 (1): 289–299. doi:10.1093/qmath/19.1.289. Bibcode1968QJMat..19..289M.