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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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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Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
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Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I ...
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Commutation relation between su(N) and clifford algebra generators.

Why does the $ \gamma_5 $ matrix commute with the generators of the $su(N)$ algebra? In the case of the chiral symmetry from physics, [$Q_a$, $Q_b^5$] = $i \epsilon_{abc}Q^5_c$ where the $Q_a, Q^5_a$ ...
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Correspondence between tensors and multivectors

In one of its definitions the Clifford algebra is defined as a quotient space of a (infinite dimensional) Tensor algebra. The question is: Given the metric signature and the tensor $T$ and its ...
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Confusion about division in Clifford Algebra

On page 202 of The Road to Reality, Penrose claims that if we want to generalize Quaternions to n dimensions using Clifford Algebra, we must abandon the division property. I have a hard time believing ...
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Coordinate free Geometric Algebra vs. Linear Algebra

I think I know what coordinate free means. But I never found in ANY text a good explanation of it or something like: This is the problem solved with coordinates and this is the problem solved without ...
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What is a multivector?

I know how to visually interpret single parts of a multivector. But what do they look like as a whole? Making an analogy with complex numbers doesn't work.
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61 views

Why is the pseudoscalar called pseudoscalar in Geometric Algebra

It makes sense to call it a pseudoscalar in odd dimensions, because it commutes with all other objects. But in even dimensions it anticommutes, why is it still called pseudoscalar? Further I don't ...
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50 views

Anisotropic scaling in geometric/Clifford algebra

Take the geometric algebra over $\Bbb R^n$. Suppose we have a blade multivector in this algebra. Now we want to anisotropically scale this multivector. Is there a general closed-form expression for ...
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Good introductory book on geometric algebra

The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems' proofs. ...
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Intuition behind definition of spinor

Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't ...
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Geometric product of two bivectors

First the definition of the inner and outer product in terms of grade projection: $A_{r} \cdot B_{s} = \langle A_{r}B_{s} \rangle_{|r-s|}$ $A_{r} \wedge B_{s} = \langle A_{r}B_{s} \rangle_{r+s}$ So ...
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Why do Clifford bivectors represent the orthogonal Lie algebra?

It takes a long, painful, but straightforward calculation to see that the commutators of grade one elements $[\mathbf e_i,\mathbf e_j]$ of a Clifford algebra $\mathrm{Cl}(p,q)$ have exactly the same ...
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528 views

What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
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117 views

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, ...
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Clifford Algebra Isomorphic to Exterior Algebra

Let $E$ be a vector space over a field $k$ and $Q$ be a quadratic form, that is, $$Q:E\to k$$ such that $$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$ and such that $P_Q:E^2\to k$ is ...
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507 views

Solving a geometric algebra equation

I'm reading Hestenes' book "New Foundations of Classical Mechanics" as an introduction to Geometric (Clifford) Algebra. Don't worry, no physics mentioned here :) An exercise asks to solve a vector ...
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67 views

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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43 views

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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42 views

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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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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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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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 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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58 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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45 views

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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28 views

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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An expression for $e_1e_2e_3e_4$

In a real Clifford algebra $\mathbb{Cl}(2,2)$ over ${\mathbb R}^4$ with the quadratic form defined on the orthogonal basis $e_1,e_2,e_3,e_4$ by $e_1^2=1, e_2^2=1, e_3^2=-1$, and $e_4^2=-1$, find an ...
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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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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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What is the “grade” in geometric algebra

I'm reading a book (Linear and Geometric Algebra, by Alan Macdonald) where the author uses the term grade without ever defining it. I have a murky sense of what the grade of a blade may be (a ...
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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 $(v+\xi , ...
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What operation is “$\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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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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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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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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Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
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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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Opposite Clifford-Algebra

for a symmetric bilinearform $\beta$ on a $\mathbb{K}$-vectorspace $V$ the associated Clifford Algebra $Cl(\beta)$ is the associative algebra with unit subject to the relations $$v\cdot ...
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Seeking intuitive explanation of Clifford Algebra

Is there a simple intuitive graphical explanation of Clifford Algebra for the layman? Since Clifford Algebra is a Geometric Algebra, surely the best way to present those concepts is with graphical ...
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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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138 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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Looking for a clear definition of the geometric product

In brief: I'm looking for a clearly-worded definition1 of the geometric product of two arbitrary multivectors in $\mathbb{G}^n$. I'm having a hard time getting my bearings in the world of ...