Related papers: On Clifford Algebras and binary integers
Using a binary representation for basis elements of an algebra combined with a framework of multiplier and index functions, a connection has been established between the structure of a large class of algebras and the XOR componentwise…
This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra…
Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2…
The representations of Clifford algebras and their involutions and anti-involutions are fully investigated since decades. However, these representations do sometimes not comply with usual conventions within physics. A few simple examples…
In this work we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially we present an general formulation through elements of left minimal ideals in tensor…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$…
In this paper we address the problem of constructing a class of representations of Clifford algebras that can be named "alphabetic (re)presentations". The Clifford algebras generators are expressed as m-letter words written with a…
Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical…
Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real…
A real representation theory of real Clifford algebra has been studied in further detail, especially in connection with Fierz identities. As its application, we have constructed real octonion algebras as well as related octonionic triple…
The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra…
We propose to represent both $n$--qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension $2n.$ In this framework, the Dirac formalism can be realized in…
We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a…
We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using…
General braided counterparts of classical Clifford algebras are introduced and investigated. Braided Clifford algebras are defined as Chevalley-Kahler deformations of the corresponding braided exterior algebras. Analogs of the spinor…
We present a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit…
We study briefly some properties of real Clifford algebras and identify them as matrix algebras. We then show that the representation space on which Clifford algebras act are spinors and we study in details matrix representations. The…
We present a short review of the action and coaction of Hopf algebras on Clifford algebras as an introduction to physically meaningful examples. Some q-deformed Clifford algebras are studied from this context and conclusions are derived.
In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.