# 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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### Clifford algebra and exterior algebra

Let $E$ be a finite dimensional real vector space with $E^*$ its dual, and let $\langle \; , \; \rangle$ be an inner product on $E$. For any $e \in E$, denote by $e^* = \langle e, \; \rangle \in E^*$ ...
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### Deriving reduction formula in Geometric Algebra

I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes. I was reading the part about reduction formula (shown below) but couldn't ...
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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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### 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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### 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 quotientK^{-n}=M(...
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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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### 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 ...