View source for Generalized Clifford algebra

{{About|the algebra called generalized Clifford algebra (GCA)|(orthogonal) Clifford algebra|Clifford algebra|symplectic Clifford algebra|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]],<ref>[[Hermann Weyl|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)</ref> who  utilized and formalized  these [[Generalizations of Pauli matrices|'''clock-and-shift''']] operators introduced by [[James Joseph Sylvester|J. J. Sylvester]] (1882),<ref>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''' . 
[http://quod.lib.umich.edu/u/umhistmath/aas8085.0003.001/664?rgn=full+text;view=pdf;q1=nonions online]  and [http://quod.lib.umich.edu/u/umhistmath/AAS8085.0004.001/165?cite1=Sylvester;cite1restrict=author;rgn=full+text;view=pdf  further].
</ref> and organized by  [[Élie Cartan|Cartan]] (1898)<ref>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 online]</ref>  and  [[Julian Schwinger|Schwinger]].<ref>[[Julian Schwinger|Schwinger, J.]] (1960), "Unitary operator bases", ''Proc Natl Acad Sci U S A'', April; 46(4): 570–579, {{PMC|222876}}; ''ibid'', "Unitary transformations and the action principle", 46(6): 883–897, {{PMC|222951}}</ref>

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'''.<ref>{{Cite journal | last1 = Santhanam | first1 = T. S. | last2 = Tekumalla | first2 = A. R. | doi = 10.1007/BF00715110 | title = Quantum mechanics in finite dimensions | journal = Foundations of Physics | volume = 6 | issue = 5 | pages = 583 | year = 1976 | pmid =  | pmc = | bibcode = 1976FoPh....6..583S }}</ref><ref name="granik-et-al-1996"/><ref>A. K. Kwaśniewski: ''On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group''</ref> The concept of a [[spinor]] can further be linked to these algebras.<ref name="granik-et-al-1996"/>

The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.<ref>{{cite book|last1=Tesser|first1=Steven Barry|editor1-last=Micali|editor1-first=A.|editor2-last=Boudet|editor2-first=R.|editor3-last=Helmstetter|editor3-first=J.|title=Clifford algebras and their applications in mathematical physics|date=2011|publisher=Springer|location=Dordrecht|isbn=978-90-481-4130-2|pages=133–141|chapter=Generalized Clifford algebras and their representations}}</ref><ref>{{cite journal|last1=Childs|first1=Lindsay N.|title=Linearizing of n-ic forms and generalized Clifford algebras|journal=Linear and Multilinear Algebra|date=30 May 2007|volume=5|issue=4|pages=267–278|doi=10.1080/03081087808817206}}</ref><ref>{{cite journal|last1=Pappacena|first1=Christopher J.|title=Matrix pencils and a generalized Clifford algebra|journal=Linear Algebra and its Applications|date=July 2000|volume=313|issue=1-3|pages=1–20|doi=10.1016/S0024-3795(00)00025-2}}</ref><ref>{{cite journal|last1=Chapman|first1=Adam|last2=Kuo|first2=Jung-Miao|title=On the generalized Clifford algebra of a monic polynomial|journal=Linear Algebra and its Applications|date=April 2015|volume=471|pages=184–202|doi=10.1016/j.laa.2014.12.030|arxiv=1406.1981}}</ref>

== Definition and properties ==

===  Abstract definition ===
The {{mvar|n}}-dimensional generalized Clifford algebra is defined as an associative algebra over a field {{mvar|F}}, generated by<ref>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}}.</ref>
:<math>e_j e_k = \omega_{jk} e_k e_j \,</math>
:<math>\omega_{jk} e_l = e_l \omega_{jk} \,</math>
:<math>\omega_{jk} \omega_{lm} = \omega_{lm} \omega_{jk} \,</math>
and
:<math>e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \,</math>
{{math|∀ ''j'',''k'',''l'',''m'' {{=}} 1,...,''n''}}.

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that
:<math>\omega_{jk} = \omega_{kj}^{-1} = e^{2\pi i \nu_{kj}/N_{kj}}</math>
{{math|∀ ''j'',''k'' {{=}} 1,...,''n''}},  &nbsp; and <math>N_{kj} =</math>[[Greatest common divisor|gcd]]<math> (N_j,N_k)</math>.  The field {{mvar|F}} is usually taken to be the complex numbers '''C'''.

=== More specific definition ===
 {{Main article|Generalizations of Pauli matrices}}
In the more common cases of GCA,<ref name="granik-et-al-1996">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}}, [https://books.google.com/books?id=OpbY_abijtwC&pg=PA101 p. 101]–110</ref> the {{mvar|n}}-dimensional generalized Clifford algebra of order {{mvar|p}} has the property {{math| ''ω<sub>kj</sub>'' {{=}} ''ω''}}, <math>N_k=p</math> &nbsp;  for all ''j'',''k'', and <math>\nu_{kj}=1</math>.  It follows that 
:<math>e_j e_k = \omega \, e_k e_j \,</math>
:<math>\omega e_l = e_l \omega \,</math>
and
:<math>e_j^{p} = 1 = \omega^{p} \,</math>
for all ''j'',''k'',l = 1,...,''n'', and 
:<math>\omega = \omega^{-1} = e^{2\pi i /p}</math>
is the {{mvar|p}}th root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.<ref>See for example the review provided in: Tara L. Smith: [https://math.uc.edu/~tsmith/papers/CliffAlg.pdf ''Decomposition of Generalized Clifford Algebras'']</ref>

;Clifford algebra
In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with {{math|''ω'' {{=}} −1, and ''p'' {{=}} 2}}.

== Matrix representation ==
{{Main article|Generalizations of Pauli matrices#Construction: The clock and shift matrices}}
The Clock and Shift matrices can be represented<ref>[[Alladi Ramakrishnan]]: ''Generalized Clifford Algebra and its applications – A new approach to internal quantum numbers'', [http://www.imsc.res.in/xmlui/bitstream/handle/123456789/227/MR60.pdf Proceedings of the Conference on Clifford algebra, its Generalization and Applications], January 30 – February 1, 1971, [[Matscience]], Madras 20, pp.&nbsp;87–96</ref> by {{math|''n×n''}} matrices in Schwinger's canonical notation as
:<math>
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> , &nbsp;&nbsp;&nbsp;<math>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> , &nbsp;&nbsp;&nbsp;<math>
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, {{math|''V<sup>n</sup>'' {{=}} 1}},  {{math|''VU'' {{=}} ''ωUV''}} (the [[Stone–von Neumann theorem#Uniqueness of representation|Weyl braiding relations]]), and {{math| ''W<sup>−1</sup>VW'' {{=}} ''U''}} (the [[Discrete Fourier transform]]). 
With {{math|''e''<sub>1</sub> {{=}} ''V'' , ''e''<sub>2</sub> {{=}} ''VU'', and ''e''<sub>3</sub> {{=}} ''U''}}, one has  three basis elements which, together with {{mvar|ω}}, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, {{mvar|V}} and {{mvar|U}}, normally referred to as "[[Generalizations of Pauli matrices#Construction 2|shift and clock matrices]]",  were introduced by  [[James Joseph Sylvester|J. J. Sylvester]] in the 1880s. (Note that the matrices   {{mvar|V}} are cyclic [[permutation matrices]] that perform a [[circular shift]]; ''they are not to be confused'' with [[shift matrix|upper and lower shift matrices]] which have ones only either above or below the diagonal, respectively).

=== Specific examples ===
;Case {{math|''n'' {{=}} ''p'' {{=}} 2}}.
In this case,  we have  {{mvar|ω}} =  −1, and
:<math>
V =
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
</math> , &nbsp;&nbsp;&nbsp;<math>
U =
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
</math> , &nbsp;&nbsp;&nbsp;<math>
W =
\begin{pmatrix}
1&1\\
1&-1
\end{pmatrix}
</math>
thus
:<math>
e_1 =
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
</math> , &nbsp;&nbsp;&nbsp;<math>
e_2 =
\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix}
</math> , &nbsp;&nbsp;&nbsp;<math>
e_3 =
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
</math> ,
which constitute the [[Pauli matrices]].

;Case {{math|''n'' {{=}} ''p'' {{=}} 4}}, 
In this case we have  {{mvar|ω}} = {{mvar|i}},  and
:<math>
V =
\begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
1&0&0&0
\end{pmatrix}
</math>  , &nbsp;&nbsp;&nbsp;<math>
U =
\begin{pmatrix}
1&0&0&0\\
0&i&0&0\\
0&0&-1&0\\
0&0&0&-i
\end{pmatrix}
</math>  , &nbsp;&nbsp;&nbsp;<math>
W =
\begin{pmatrix}
1&1&1&1\\
1&i&-1&-i\\
1&-1&1&-1\\
1&-i&-1&i
\end{pmatrix} 
</math>
and {{math|''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>}} may be determined accordingly.

== See also ==
*[[Clifford algebra]]
*[[Generalizations of Pauli matrices]]
*[[DFT matrix]]
*[[Circulant matrix]]

== References ==
{{reflist}}

== Further reading ==
* R. Jagannathan, [https://arxiv.org/abs/1005.4300/ 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.&nbsp;13–41
* O. Morris (1967): ''On a Generalized Clifford Algebra'', Quart. J. Math (Oxford), '''18''', pp.&nbsp;7–12
* O. Morris (1968): ''On a Generalized Clifford Algebra II'', Quart. J. Math (Oxford), '''19''', pp.&nbsp;289–299

{{DEFAULTSORT:Generalized Clifford Algebra}}
[[Category:Algebras]]
[[Category:Clifford algebras]]
[[Category:Ring theory]]
[[Category:Quadratic forms]]
[[Category:Mathematical physics]]