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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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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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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32 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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19 views

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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12 views

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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37 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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42 views

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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20 views

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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25 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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47 views

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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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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320 views

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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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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60 views

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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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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22 views

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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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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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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38 views

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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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 ...
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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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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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53 views

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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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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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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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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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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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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201 views

How do you show that the Laplacian is the square of the (Euclidean) Dirac operator?

If I understand correctly, the Euclidean Dirac operator is given by $$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$ where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional ...
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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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(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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67 views

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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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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62 views

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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43 views

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 $\eta$ ...