Clifford algebras are associative algebras constructed from quadratic forms on vector spaces. They can be viewed as generalizations of the real numbers, complex numbers, and quaternions. These algebras have applications in geometry and theoretical physics.
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Composition of two axis-angle rotations
Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
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The product of a simple k-vector with its reverse in Clifford algebra
Is there a simple proof that the product of a simple k-vector with its reverse always yields a scalar? Assume real Clifford algebra of arbitrary but finite dimension with unspecified metric. By simple ...
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Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form?
I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question.
Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
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Mathematical significance of the “Dirac conjugate”
Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
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Matrices defining Clifford Algebra
Given the two matrices:
$\sigma_i$ and $\sigma_j$
we can construct a Clifford algebra based on the anti commutator rule:
$$\{\sigma_i,\sigma_j\}=\delta_{ij}1$$
where $\delta_{ij}$ is the Kronecker ...
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product $abababab…$ in clifford algebra
Let $a,b$ are vectors in vector space $V \leq \mathcal{Cl}_n(V)$.
I would like to know if product $ababab...ab=(ab)^r$ can be written in form $\sum_{\alpha \in A} F_\alpha(a) G_\alpha(b)$. For some ...
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Choosing a spinor space
I still have a hard time on trying to fully understand spinors (on real spaces). Let $V$ be a $n$-dimensional real quadratic space $V$ with a non degenerated signature $(p, q)$.
From what I had ...
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Hermitian conjugation and representations of the Lorentzian Clifford algebras
The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
