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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Geometric product of a bivector and a vector anticommute

I want to prove that $\left(e_1\wedge e_2\right)e_1 = -e_1 \left(e_1\wedge e_2\right)$. I did this in two ways, but I didn't understand why the second way is wrong. First way: $\left(e_1\wedge ...
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Isomorphism between those two Algebras

Why is there isomorphism between Clifford Algebra and Exterior Algebra? Maybe, better said, what does it mean to have isomorphism between those two Algebras?
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Is there a relationship between Rotors and the Rodrigues' rotation formula

I am trying to understand quaternion in general, and it seems like the path to making sense of how they actually work is to first understand rotors and other techniques related to rotations. By ...
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Lift of inclusion $U(k)$ into $SO(2k)$ to $Spin^c$ group

In "Clifford Modules" by Atiyah, Bott and Shapiro (p.10) or "Dirac Operators in Riemannian Geometry" by Friedrich (p.28) one finds some sort of a lift of the natural inclusion $\operatorname{U}(k)\to ...
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The relation between $\nabla\nabla f$ and $H_f$

I've been reading Lounesto's text on Clifford Algebras and Spinors, and I came across the following in section 1.11: Using the multiplication table of ${C\mathcal{l}}_2$ we notice that the ...
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Random matrix on Clifford algebra within a specific grade

There has been some discussions about random matrices on generic Clifford algebra arXiv:1312.6291. However I would like to consider a more specific case by restricting the random matrix within a ...
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37 views

Trace of six gamma matrices

I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$ I know that I can express this as: $$ ...
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Exponents with bivectors

According to the Wikipedia page on bivectors: ...if $B$ is a bivector, then the rotor $R$ is $e^{B/2}$ and rotations are generated [by] $v'=RvR^{-1}$. But how do you take an exponent between a ...
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Showing $ \epsilon_{\pm} =(-1)^{\frac{1}{8}(s-t)(s-t\pm 2)} = \zeta^t (-1)^{\frac{1}{2} t(t-1) + \frac{1}{8}d(d-\zeta 2)}$

I want to show the following equation holds \begin{align} \epsilon_{\pm} =(-1)^{\frac{1}{8}(s-t)(s-t\pm 2)} = \zeta^t (-1)^{\frac{1}{2} t(t-1) + \frac{1}{8}d(d-\zeta 2)} \end{align} where $\zeta = ...
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Why are “innermorphisms” not useful?

I commonly studied type of linear function in geometric algebra is the outermorphism. For reference, here's Wikipedia's definition: Let $f$ be an $\Bbb R$-linear map from $V$ to $W$. The ...
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Clifford Algebra: Expression for $\sum_b(\gamma_a^{\;b})_{\alpha\beta}(\gamma_b^{\;c})_{\rho\delta}$

Is there any simple expression for this in general $D$-dimensions? $$\sum_b(\gamma_a^{\;b})_{\alpha\beta}(\gamma_b^{\;c})_{\rho\delta}$$ where $\gamma^a$ are the usual $D$-dimensional gamma matrices ...
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mapping of local Pauli operators

Let $A, B \subseteq P_n$, 2 finite sets of k-local commuting Pauli operators from the Pauli group $P_n$. Can we always a finite depth unitary $U$ such that $U^ \dagger AU=B$?
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50 views

Norm of clifford operator on forms

I am a beginner in differential geometry and would appreciate some pointers on how to answer the following question. Let $M$ be a closed orientable Riemannian manifold with $\{e^1,...,e^n\}$ an ...
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38 views

The deconposition of $\mathfrak{so}(V \oplus V^*)$

Let $V$ be an n dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ in $V\oplus V^*$ such that $(v+\xi , ...
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Isomorphism of the Clifford bundle of a Riemannain manifold

Let $(M,g)$ be an oriented Riemannian manifold and $Cl(M):=\bigcup_{x\in M}Cl(T_xM,g_x)$ be the clifford bundle of $(M,g)$. (Here $Cl(T_xM,g_x)$ denotes the clifford algebra of the vector space ...
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Representation of complex Clifford algebra on exterior algebras when quadratic form has odd index

Overview This problem entails the explicit construction of representation of Clifford algebra upon the exterior algebra, using orthogonal complex structure or polarization, namely, given a ...
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What operation is “$\oplus$” in Lounesto's introduction to Clifford Algebras

I'm reading Lounesto's CLifford Algebras and Spinors and on page 26 (also below) he states the following: \begin{align} C\mathcal{l}_2=\mathbb{R}\oplus\mathbb{R}^2\oplus\bigwedge^2\mathbb{R}^2. ...
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bibliography reference to clifford algebras

I am starting to study Clifford algebras and geometric calculus. I am not finding references for these topics. Someone could point me a reference for these topics? Thanks in advance!
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Difference between $P(V,q)$ and $\tilde P(V,q)$ (about Lawson, Michelson “Spin Geometry”, page 19)

I'm confused by discussion on page 19 of the "Spin Geometry" book by Blaine Lawson and Marie-Louise Michelson. It claims, that when field $k$ is a spin field, $\widetilde P(V,q)/P(V,q)$ is either $0$ ...
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troubles showing existence of Clifford-algebra

We had the following definition in class: Let $V$ be a vector space, $K$ a field and $Q$ be a quadratic form. We call $(C(V,Q),j)=C$ a Clifford-algebra if: $C$ is an assoziative algebra with 1, ...
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Clifford algebra and Spin group of 4-dimensional Euclidean space

I’m seeking for a straightforward construction of well-known $\mathrm{Spin}(4) = \mathrm{Spin}(3)\times\mathrm{Spin}(3)$ isomorphism using geometric algebra-based definition of “Spin”, without ...
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56 views

Clifford algebra over non-Archimedean field

Usually the Clifford algebra is defined over the Reals $\mathbb{R}$ or the Complex $\mathbb{C}$ numbers. Can the definition be extended over non-Archimedean fields, such as the hyperreal numbers ...
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Maps and Wedge Products

I'm currently trying to prove Cramers rule with the wedge product definition of a determinant. The proof hinges on the fact that if M is some map, then we can write: $$ M\left(\vec a \wedge \vec ...
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123 views

About the definition of norm in Clifford algebra

I have seen two definitions for the norm in the Clifford algebra $\mathrm{Cℓ}_{p, q, r}$. According to one of them $\Vert x\Vert = ⟨x. x^\dagger⟩_0$, where the dagger stands for the reversal of the ...
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62 views

Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
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40 views

Ideal of $TV$ which trivially intersects $V$

Let $V$ be a vector space over a fied $\mathbb{K}$ and let $TV = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ be its tensor algebra. Let $b \colon V \times V \rightarrow \mathbb{K}$ be a bilinear form ...
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Struggling with connection between Clifford Algebra (/GA) and their matrix generators

As I thought I understood things, the Gamma matricies behave as the 4 orthogonal unit vectors of the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb C)$, (also the Pauli matricies are for the 3 of ...
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Is there a deeper meaning behind the “determinant” formula for the cross product?

We all know that for all vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R^3}$, if $(a_x,a_y,a_z)^\top$ is the component form of $\mathbf{a}$ and similarly $(b_x, b_y, b_z)^\top$ is the component form of ...
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Aren't these linear transformations only orthogonal with respect to *some* inner product?

On p.154 in Husemoller's Fibre Bundles, during his introduction of Clifford algebras, I found a claim which seems questionable to me (highlighted in red): You can click here for some context ...
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Clifford Algebras for Projective and Conformal Geometry

According to Clifford Algebra: A Visual Introduction, A Clifford Algebra over $\mathbb{R}^3$ may describe the rigid motions in space (namely, conjugation acts as a reflection by a plane). A ...
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(Geometric algebra) Acceleration of a particle with constant speed as a bivector-vector inner product

I've been working on (self-studying) Geometric Algebra for Physicists which, sadly, has no solutions manual. This is not a problem in general, but I feel like one of my solutions for a question asked ...
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Basis for Clifford algebra $Cl^2 (W)$ and quotient space $Cl^3(W)/Cl^2(W)$

Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^*$. I need help to write the basis for Clifford ...
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How can I derive the back propagation formula in a more elegant way?

When you compute the gradient of the cost function of a neural network with respect to its weights, as I currently understand it, you can only do it by computing the partial derivative of the cost ...
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Why is the grade of the wedge product of two arbitrary blades the sum of the two blades' grades independently?

I'm reading Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry and it says that this is true of any two arbitrary blades. $\ grade( \textbf{ A} \wedge \textbf{B})=grade( ...
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Computing the centre of the Clifford Algebra

I'm currently looking at a $8$-dimensional $\mathbb{R}$-algebra denoted either by $C_n$ or $C(a_1, a_2, a_3)$. After some looking around, I think this is called a Clifford algebra? If not, I ...
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Does the determinant of a complex-valued matrix have a geometric interpretation?

The determinant of a real-valued matrix can be seen as the volume of the parallelotope with the column vectors as the sides. Is there an analogous interpretation for complex-valued matrix ...
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Space time algebra isomorphic to matrix algebra

i have the following problem: I already know that there exists representation of the Clifford Algebra of the Minkowski space $\mathcal{C}l(M,\eta)$. Here $M$ denotes the Minkowski space and $\eta$ ...
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Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
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Geometric algebra and quantum field theory

How does the reformulation of QFT with GA look like? I read that GA can be applied to almost every kind of physics, but QFT is rarely mentioned. Is there a lot of research going on in this direction ...
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I would like to see a De Morgan's Law example from Geometric Algebra

thanks for your time and effort. I really appreciate your help. In the documents "www.cs.bham.ac.uk/~fauserb/pdf/theses/Habilschrift.pdf" page 25 by B. Fauser and "Grade Free Product Formulae from ...
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Element of the spin group

I've got the following question: why it true, that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined via ...
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Finding quaternion that transforms to particular basis

I want to find a quaternion $x \in{\mathbb{H}} $ that transforms (rotates) the $ i,j,k $ basis to a particular basis. In equations: $$ x i x^{-1} = a_1 $$ $$ x j x^{-1} = a_2 $$ $$ x k x^{-1} = a_3 ...
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Clifford Algebras in Characteristic Two

A Clifford algebra $Cl(V)$ for a vector space $V$ is defined to be the quotient of the tensor algebra $T(V)$ with the ideals generated by elements of the form $v \otimes v - (v,v)1$, where $(,)$ is ...
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What is the adjoint of the connection operator on a Clifford bundle?

From Elliptic Operators, topology and asymptotic methods, John Roe, page 43-45. Let $M$ be a Riemannian manifold. Let $S$ be a Clifford bundle over $M$, such that each $S_{m}$ over $m\in M$ is a ...
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Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
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A question on the dual relationship between the regressive product and the exterior product

I am trying to understand the following sentence, which I came across in a book: The underlying beauty of the Ausdehnungslehre is due to this symmetry [the duality between the regressive and ...
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Injection of vector space in clifford algebra

Let $K$ be a field (of characteristic $\neq 2$ if that matters) and $V$ a finite dimensional vector space over $K$, $dim(V)=n$. Let $g$ be a symmetric bilinear form on $V$. By $Cl(V,g)$ I denote the ...
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Infinite dimensional Clifford algebras?

Do infinite dimensional Clifford (and/or Grassmann) algebras exist/makes sense? Do you know good references about them?
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What are some simple examples I can use to demonstrate the power of geometric algebra?

What are some simple examples I can use to demonstrate the power of geometric algebra over "everyday" vector algebra? An alternative way of thinking of this question might be: what example ...
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What *is* the working form of a 1-vector in geometric algebra?

Consider the geometric algebra definition of 0-vectors (scalar), 1-vectors (vector), and the inner product. Let $a$ and $b$ be 1-vectors. Then $a + b = c$, where $c$ is another 1-vector. Now, ...