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Questions tagged [clifford-algebras]

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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On the algebraic formulation of the Clifford algebra

I apologize in advance for a rather wordy question. As a physicist trying to learn new mathematics, I figured this was the place to ask. I am having trouble understanding the algebraic formulation of ...
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Clifford algebra from a bunch of commutation and anti-commutation relations

When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this: "Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =...
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Double cross product in 2D

Hello I have a question about a double cross product, appearing in centrifugal force \begin{align*} \mathbf{F}_{centrifugal} = -m \boldsymbol{\omega} \times [\boldsymbol{\omega} \times \mathbf{r}] \, ....
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Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here: Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\...
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wedging and contracting operators generate a clifford algebra

Im working on a paper and having trouble proving the following: The wedging and contracting operators $\phi(e_i), \phi^*(e_i)$, $1\leq i\leq n$ generate a Clifford Algebra with $W = V\oplus V^*$ and $...
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Definition of Clifford Algebra

Cliffard algebra defined by relation: $x*y+y*x=g(x,y)1$, where g(x,y) is bilinear symmetric form. What does mean $g(x,y)1$, why it's not just $g(x,y)$, without the identity?
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Trace of chiral matrix in higher odd dimension

Suppose we have $D$ gamma marices $$\gamma_{a}\gamma_{b}+\gamma_{b}\gamma_{a}=2\delta_{ab}I$$ with $a,b=1,2,...,D$. Chiral matrix is $$\gamma_{chir}=\gamma_{1}\gamma_{2}...\gamma_{D}$$ in even ...
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Rotation around a whole sphere by multiplying a single hypercomplex number forever?

In quaternion number system, any unit quaternion $\mathbf{q}\in\mathbb{H}$ can be written as $$ \mathbf{q} = \cos \theta + (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k})\sin \theta $$ for some $\...
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Combining the Laws of classical mechanics into a single equation using multi vectors

Maxwell's laws can be combined into a single equation using multi vectors. What about the Laws of classical mechanics? Can they be combined into a single equation? https://en.wikipedia.org/wiki/...
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Clifford algebra is isomorphic to exterior algebra, proof in Lawson's

This Proposition 1.2, pg 10 of Lawson's Spin Geometry. Context: We have a homomorphism of graded algebra $$ \wedge^*(V) \rightarrow G^*$$ induced on each component from the map $$\wedge^r(V) \...
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Clifford Multiplication and Spinor Bundles

I am trying to follow the discussion of Clifford multiplication on page 384 of The Wild World of 4-Manifolds, by Alexandru Scorpan (link, although I hope this will be totally self-contained), and I'm ...
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Geometric product of 2 vectors derived from Pauli matrices

The 3 Pauli matrices are: ${\color{blue}{\sigma_1}}$ = $ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ ${\color{blue}{\sigma_2}}$ = $ \begin{pmatrix} 0 & -i \\ i & 0 \...
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Identity matrix times i

Why is the psuedoscalar in 5 dimensions the identity matrix? In 3 dimensions its $\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}$ In 3 dimensions when you multiply all 3 vectors (Pauli ...
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Is the $\mathbb{Z}$-grading of a Clifford algebra basis independent?

Let $V$ be a finite dimensional vector space over a field $K$ of characteristic $\neq 2$, and let $q \colon V \to K$ be a quadratic form. One of the first things to show when learning the theory of ...
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How do the symmetries of a Lie manifold manifest in the metric tensor of that manifold?

Suppose we have some Lie manifold (ie a Lie group) that is also a Riemannian manifold endowed with a metric tensor How does the Lie group symmetry manifest in the properties of the metric tensor? I ...
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What is the motivation for the development of dual numbers?

I searched the backlog of this website for something along the lines of my question to no avail. Not to mention a myriad of PDFs online which just drop the definition and allege more details can be ...
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Clifford algebra Cliff(0) as reals, Cliff(1) as complex…

In this paper by John Baez: http://math.ucr.edu/home/baez/octonions/node6.html he says that assuming a Clifford algebra with $vw + wv = -2<v, w>$, you can see that Cliff(0) = $\mathbb{R}$, Cliff(...
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Wedge Product on $C\ell^+(0,3,0)$ Relationship to Quaternion Cross Product

The even Clifford sub-algebra $C\ell^+(0,3,0)$ is isomorphic to the quaternion algebra. The mapping between terms is $e_0 \mapsto 1$, $e_{23} \mapsto i$, $e_{31} \mapsto j$, $e_{12} \mapsto k$. In ...
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What does the notation $[,]$, $\{,\}$ in the Clifford algebra mean?

From Charbonneau, Harland: Deformations of nearly Kähler instantons: It is explained in the previous paragraph that the authors use the canonical identification $\operatorname{Cl}(V,g)=\Lambda^*V$. ...
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How to perform wedge product

I have heard all kinds of great things about Clifford/Geometric algebra, but I can't find any good resources. I have been looking EVERYWHERE for just one actual example of a wedge product being ...
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Proof of these equalities for Lorentz transformations

I'm working on Lorentz transformations for elements of Dirac algebra $\mathcal{D}$ (the Clifford algebra generated by $\mathbb{R}^{1,3}$). In this algebra we write an element $$ A=A_0+A_1+A_2+A_3+A_4 $...
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Transformation of metric tensor in clifford algebra for nonorthogonal transformations

I study physics so apologies for any nonstandard notation/terminology. In geometric (aka spacetime) algebra one speaks of basis $\gamma^{\mu}$ ( possibly represented by matrices) as transforming in a ...
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Transformation of metric tensor under clifford algebra?

I've had to rewrite the question as i made agrievious errors in the first go-round. Given a (pseudo)Riemannian metric g, we can identify its components with the symmetric product of gamma matrices: ...
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cross product not associative, outer product associative

The cross product is not associative. If $i=(1,0,0)$, $j=(0,1,0)$ and $k=(0,0,1)$, then \begin{eqnarray} i \times (i \times j) = i \times k = -j \\ (i \times i) \times j = 0 \end{eqnarray} However ...
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How to decompose a bivector into a sum of _orthogonal_ blades?

In Geometric Algebra, any bivector $B\in\Lambda^2\mathbb R^n$ is a sum of blades: $$B = B_1 + B_2 + \cdots$$ $$= \vec v_1\wedge\vec w_1 + \vec v_2\wedge\vec w_2 + \cdots$$ Each blade's component ...
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Primitive idempotents of Cl(1,3) over the complex numbers

Simply put, I need to find all primitive idempotents of the Clifford Algebra $Cl(1,3)$ over the complex numbers. I have found some general results but they're only applicable over the real numbers. I ...
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Clifford algebras of even rank, unique irrep.

I would like to prove that there is one and only one (non trivial) irreducible representation (up to equivalence) of the Clifford algebra $Cl(n)$ with $n=2p$ and $p\in\mathbb{Z}$. I have seen a proof ...
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K-theory and Clifford modules

I'm trying to wrap my head around the "Clifford modules" definition of K-theory. Let's just deal with K-theory of a point. One common definition of the $-n^\text{th}$ K-group is the quotient$$K^{-n}=M(...
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Orthogonality of Clifford algebra's Fueter polynomial in certain measure

In the article "Two integral operators in Clifford analysis" , https://www.sciencedirect.com/science/article/pii/S0022247X08012262, it said that $\langle V_{\alpha},V_{\alpha'}\rangle = \int_{\...
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“Square root” of a decomposition of a homogeneous polynomial to harmonic and $x^2 q$.

It is well known that any homogeneous polynomial $f \in \mathbb R[x_1, \ldots, x_n]$ can be uniquely split as $f = f_0 + x^2 f_1$, where $x^2 \equiv (x_1)^2 + \ldots + (x_n)^2$ and $f_0$ harmonic: $\...
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Geometric Calculus, Clifford Algebra, and Calculus of Variations

It has always bothered me that I was told in my Calculus of Variations class that it's only possible to minimize a function with respect to one objective. Obviously sometimes it is possible to ...
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Spinor chiral transformation by $\psi \to \gamma^5 \psi$

Let $\psi$ be a spinor. Let $\gamma^0,\gamma^1, \gamma^2, \gamma^3$ be the usual gamma matrices and the fifth $\gamma^5 : = i\gamma^0\gamma^1\gamma^2\gamma^3.$ Then if we define $\psi \to \psi' := \...
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Gamma matrices and special relativity

I understand how gamma matrices generate a Clifford algebra that corresponds to the Minkowski metric. So the next step for me is to understand how gamma matrices are used in the context of special ...
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For two unit non-oriented bivectors $A,B\in \mathbb{R}P^2\subset \Lambda^2\mathbb{R}^3$ is the mapping $\phi:(A,B)\rightarrow AB$ bijective?

For two non-oriented unit bivectors $A,B\in \mathbb{R}P^2\subset \Lambda^2\mathbb{R}^3$ is the mapping $\phi:\mathbb{R}P^2\times \mathbb{R}P^2/\mathbf{D} \rightarrow S^3$, where $\mathbf{D}$ is the ...
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Minkowski metric. Scalar or tensor?

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation $\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu =...
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Clifford algebra complex representation

A canonical basis for the geometric algebra $\mathcal{G}(3,0)$ has $2^3$ elements: $1, e_1, e_2, e_3, e_1e_2, e_1e_3, e_2e_3, e_1e_2e_3$ That's easy to understand. There are 3 dimensions and ...
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Clifford product of force and distance

$fd = force*distance$ $fd = (f_1\mathbf{e_1} + f_2\mathbf{e_2})(d_1\mathbf{e_1} + d_2\mathbf{e_2})$ $fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{...
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A confusing formula in Clifford algebra

I am reading a book named "An Introduction to Clifford Algebras and Spinors" by J. Vaz Jr. and R. da Rocha. In page 78, I met a confusing formula (3.89), written as: $$\gamma(\mathbf{v})\gamma(\...
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What is the Clifford/geometric product in terms of the inner and exterior product

The geometric product is defined generally as the algebra on the space of multivectors $\bigwedge \Lambda(V)$ $$ab = a \cdot b + a \wedge b$$ with $\cdot$ a product on the vector space $V$ and $\...
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Is the reverse really an anti-automorphism of a a Clifford algebra?

In the Wikipedia for the Spin group, the Clifford algebra is defined as the quotient of the tensor algebra by the ideal $v\otimes v + ||v||^2$. The reverse of an element $v$ is denoted $v^r$ and for a ...
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Geometric Algebra Rejection, Projection and reflection rotation, confused on how end result is actually calculated

I recently found a video that claimed to give intuition to Quaternions. And to my suprise it nearly did, but I have a few large hang ups on the lack of definitions of certain operations, it also ...
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adding bivector

I am reading the following notes on Clifford Algebra: http://www.av8n.com/physics/clifford-intro.htm#sec-addition And I have a confusion about bivector addition. The geometric interpretation of a ...
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Hodge duality in relating wedge and cross products?

I've been reading some quantum mechanics papers which involve Clifford Algebra. I am investigating it for an undergrad project but none of my professors seem to know anything about Clifford Algebras. ...
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Help with wedge product proof?

I am an undergraduate physics student trying to learn this on my own, so it's been a bear digging through either very general resources or resources that don't answer my questions. What I am trying ...
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Lie derivative in geometric algebra/Clifford algebra

What is the form of the Lie derivative in Clifford algebra? Context: Consider the Clifford algebra $\mathcal{C}l (p,q) $ with basis $\{e_i \}$. The geometric derivative following Hestenes is defined ...
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$spin(n)$ equivariance of Dirac operator

Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial x_j}$ where $E_j$ are $2^r \times 2^r$ matrices (where $n=2r$ or $n=2r+1$) satisfying $E_i^2=I, \ \ ...
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Real Clifford algebra for $SO(1,2)$

Is it possible to find 3 2x2 matrices $m_i$ with real coefficients such that their anticommutator $\{m_i,m_j\}=2 \eta_{ij}*1_2$, where $\eta = diag(1,-1,-1)$ and $1_2$ is the $2\times2$ identity ...
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What happens to Clifford algebra structure and periodicity when the field is weird?

The classification of the structure of real and complex Clifford algebras is well-known. Most sources concentrate only upon that case. Another resource. Also, I'm only interested in nondegenerate ...
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Clifford algebra vs 'universal' clifford algebra?

In (Lounesto, Ablamowicz; 2012; pg307) Hahn defines a Clifford algebra as follows (almost verbatim): Let $M=(M,q)$ with associated bilinear form $b$ be a quadratic module over $R$. A Clifford ...
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Generalizing the dot product to multivectors

I am studying the book Linear and Geometric Algebra (Macdonald), and I've been stuck on a couple related, seemingly-elementary problems for a couple of days. 5.3.4. Suppose that $\mathbf{a} \bot \...