mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Clifford module is a representation of a

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

. In general a Clifford algebra ''C'' is a central simple algebra over some

field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

''L'' of the field ''K'' over which the

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

''Q'' defining ''C'' is defined. The

abstract theory ''Abstract Theory'' is the debut solo album released by former Five (group), Five member Abz Love, Abs. The album was released on 1 September 2003, peaking at No. 29 on the UK Albums Chart. The album failed to find success elsewhere, and resulte ...

of Clifford modules was founded by a paper of

M. F. Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...

, R. Bott and

Arnold S. Shapiro Arnold Samuel Shapiro (1921, Boston, Massachusetts – 1962, Newton, Massachusetts) was an American mathematician known for his eversion of the sphere and Shapiro's lemma. He also was the author of an article on Clifford algebras and periodicit ...

. A fundamental result on Clifford modules is that the

Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...

class of a Clifford algebra (the equivalence class of the category of Clifford modules over it) depends only on the signature . This is an algebraic form of

Bott periodicity In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comple ...

Matrix representations of real Clifford algebras

We will need to study ''anticommuting''

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

() because in Clifford algebras orthogonal vectors anticommute :

A \cdot B = \frac( AB + BA ) = 0.

For the real Clifford algebra

\mathbb_

, we need mutually anticommuting matrices, of which ''p'' have +1 as square and ''q'' have −1 as square. :

\begin
\gamma_a^2 &=& +1 &\mbox &1 \le a \le p \\
\gamma_a^2 &=& -1 &\mbox &p+1 \le a \le p+q\\
\gamma_a \gamma_b &=& -\gamma_b \gamma_a &\mbox &a \ne b. \ \\
\end

Such a basis of gamma matrices is not unique. One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation. :

\gamma_ = S \gamma_ S^ ,

where ''S'' is a non-singular matrix. The sets ''γ''_''a''′ and ''γ''_''a'' belong to the same equivalence class.

Real Clifford algebra R_3,1

Developed by

Ettore Majorana Ettore Majorana (,, uploaded 19 April 2013, retrieved 14 December 2019 ; born on 5 August 1906 – possibly dying after 1959) was an Italian theoretical physicist who worked on neutrino masses. On 25 March 1938, he disappeared under mysteri ...

, this Clifford module enables the construction of a Dirac-like equation without complex numbers, and its elements are called Majorana

spinors In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...

. The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix. The

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

is (+++−). For the signatures (+−−−) and (−−−+) often used in physics, 4×4 complex matrices or 8×8 real matrices are needed.

References

* * . See als
the programme website
for a preliminary version. * . * {{citation, last1=Lawson, first1= H. Blaine, last2=Michelsohn, first2=Marie-Louise, author2-link=Marie-Louise Michelsohn, title=Spin Geometry, publisher= Princeton University Press, year=1989, isbn= 0-691-08542-0. Representation theory Clifford algebras

Matrix representations of real Clifford algebras

Real Clifford algebra R3,1

See also

References

Real Clifford algebra R_3,1