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 a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl,{{cite journal |author-link=Hermann Weyl |first=H. |last=Weyl |title=Quantenmechanik und Gruppentheorie |journal=Zeitschrift für Physik |volume=46 |issue= 1–2|pages=1–46 |year=1927 |doi=10.1007/BF02055756 |bibcode=1927ZPhy...46....1W |s2cid=121036548 }}
{{cite book |author-mask=1 |first=H. |last=Weyl |title=The Theory of Groups and Quantum Mechanics |publisher=Dover |orig-year=1931 |year=1950 |isbn=9780486602691 |url=https://archive.org/details/theoryofgroupsqu1950weyl|url-access=registration }} who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882),{{citation | last=Sylvester | first=J. J. |year=1882 |title=A word on Nonions |series=Johns Hopkins University Circulars |volume=I |pages=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 .

[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].

and organized by Cartan (1898){{cite journal |first=E. |last=Cartan |title=Les groupes bilinéaires et les systèmes de nombres complexes |journal=Annales de la Faculté des Sciences de Toulouse |volume=12 |issue=1 |pages=B65–B99 |year=1898 |url=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 }} and Schwinger.{{cite journal |author-link=Julian Schwinger |first=J. |last=Schwinger |title=Unitary operator bases |journal=Proc Natl Acad Sci U S A |volume=46 |issue=4 |pages=570–9 |date=April 1960 |doi=10.1073/pnas.46.4.570 |pmid=16590645 |pmc=222876|bibcode=1960PNAS...46..570S |doi-access=free }}
{{cite journal |author-mask=1 |first=J. |last=Schwinger |title=Unitary transformations and the action principle |journal=Proc Natl Acad Sci U S A |volume=46 |issue=6 |pages=883–897 |date=1960 |doi=10.1073/pnas.46.6.883 |pmid=16590686 |pmc=222951|bibcode=1960PNAS...46..883S |doi-access=free }}

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.{{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 | bibcode = 1976FoPh....6..583S | s2cid = 119936801 }}{{cite journal |first=A.K. |last=Kwaśniewski |title=On generalized Clifford algebra C⁽ⁿ⁾₄ and GL_q(2;C) quantum group |journal= Advances in Applied Clifford Algebras|volume=9 |issue= 2|pages=249–260 |year=1999 |doi=10.1007/BF03042380 |arxiv=math/0403061 |s2cid=117093671 }} The concept of a spinor can further be linked to these algebras.

The term generalized Clifford algebra can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.{{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|url=https://archive.org/details/cliffordalgebras1989hest|url-access=limited|date=2011|publisher=Springer |isbn=978-90-481-4130-2|pages=[https://archive.org/details/cliffordalgebras1989hest/page/n135 133]–141|chapter=Generalized Clifford algebras and their representations}}{{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}}{{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|doi-access=free}}{{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|s2cid=119280952}}

Definition and properties

= Abstract definition =

The {{mvar|n}}-dimensional generalized Clifford algebra is defined as an associative algebra over a field {{mvar|F}}, generated byFor a serviceable review, see {{cite journal |first=A. |last=Vourdas |title=Quantum systems with finite Hilbert space |journal=Reports on Progress in Physics |volume=67 |issue= 3|pages=267–320 |year=2004 |doi=10.1088/0034-4885/67/3/R03 |bibcode=2004RPPh...67..267V }}

: $\begin{align}
e_j e_k &= \omega_{jk} e_k e_j \\
\omega_{jk} e_\ell &= e_\ell \omega_{jk} \\
\omega_{jk} \omega_{\ell m} &= \omega_{\ell m} \omega_{jk}
\end{align}$

and

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

{{math|∀ j,k,ℓ,m {{=}} 1, . . . ,n}}.

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

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

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

= More specific definition =

In the more common cases of GCA,See for example: {{cite book |first1=A. |last1=Granik |first2=M. |last2=Ross |editor-first=R. |editor-last=Ablamowicz |editor2-first=J. |editor2-last=Parra |editor3-first=P. |editor3-last=Lounesto |chapter=On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics |chapter-url=https://books.google.com/books?id=OpbY_abijtwC&pg=PA101 |title=Clifford Algebras with Numeric and Symbolic Computation Applications |publisher=Birkhäuser |year=1996 |isbn=0-8176-3907-1 |pages=101–110 }} the {{mvar|n}}-dimensional generalized Clifford algebra of order {{mvar|p}} has the property {{math| ω_kj {{=}} ω}}, $N_k=p$ for all j,k, and $\nu_{kj}=1$ . It follows that

: $\begin{align}
e_j e_k &= \omega \, e_k e_j \,\\
\omega e_\ell &= e_\ell \omega \,
\end{align}$

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 {{mvar|p}}th root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.See for example the review provided in: {{cite web |first=Tara L. |last=Smith |title=Decomposition of Generalized Clifford Algebras |url=https://math.uc.edu/~tsmith/papers/CliffAlg.pdf |archive-url=https://web.archive.org/web/20100612050907/http://math.uc.edu/~tsmith/papers/CliffAlg.pdf |archive-date=2010-06-12 }}

; 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{{cite book |author-link=Alladi Ramakrishnan |first=Alladi |last=Ramakrishnan |chapter=Generalized Clifford Algebra and its applications – A new approach to internal quantum numbers |title=Proceedings of the Conference on Clifford algebra, its Generalization and Applications, January 30–February 1, 1971 |publisher=Matscience |location=Madras |year=1971 |pages=87–96 |url=http://www.imsc.res.in/xmlui/bitstream/handle/123456789/227/MR60.pdf}} by {{math|n×n}} matrices in Schwinger's canonical notation as

: $\begin{align}
V &= \begin{pmatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
0 & 0 & \ddots & 1 & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & 0 & 0 & \cdots & 0
\end{pmatrix}, &
U &= \begin{pmatrix}
1 & 0 & 0 & \cdots & 0\\
0 & \omega & 0 & \cdots & 0\\
0 & 0 & \omega^2 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & \omega^{(n-1)}
\end{pmatrix}, &
W &= \begin{pmatrix}
1 & 1 & 1 & \cdots & 1\\
1 & \omega & \omega^2 & \cdots & \omega^{n-1}\\
1 & \omega^2 & (\omega^2)^2 & \cdots & \omega^{2(n-1)}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & \omega^{n-1} & \omega^{2(n-1)} & \cdots & \omega^{(n-1)^2}
\end{pmatrix}
\end{align}$ .

Notably, {{math|Vⁿ {{=}} 1}}, {{math|VU {{=}} ωUV}} (the Weyl braiding relations), and {{math| W⁻¹VW {{=}} U}} (the discrete Fourier transform).

With {{math|e₁ {{=}} V , e₂ {{=}} VU, and e₃ {{=}} 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 "shift and clock matrices", were introduced by 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 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

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

thus

: $\begin{align}
e_1 &= \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, &
e_2 &= \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}, &
e_3 &= \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix},
\end{align}$

which constitute the Pauli matrices.

==Case {{math|''n'' {{=}} ''p'' {{=}} 4}}==

In this case we have {{mvar|ω}} = {{mvar|i}}, and

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

and {{math|e₁, e₂, e₃}} may be determined accordingly.