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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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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Set of 4 anticommutative matrices

How would you go about showing that there cannot be a set of four 2 by 2 matrices that satisfy the anticommutative relation $AB + BA = 0 $ or $2I$ if $A=B$? i.e minimum order has to be 4. I know that ...
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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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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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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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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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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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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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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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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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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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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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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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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 pushforward under the left action in the group of units of a Clifford algebra

The following I know to be true: let $A$ and $B$ be elements of $GL(m,\mathbb{R})$ and let $X \in T_BGl(m, \mathbb{R})$ and let $L_A:Gl(m, \mathbb{R}) \to GL(m, \mathbb{R})$ be the left multiplication ...
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Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or ...
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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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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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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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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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Interpretation of a line integral in complex analysis

$\newcommand{\C}{\mathbb{C}}$ Suppose $f\colon \Omega\subset \C\to\C$ is a holomorphic function and $\gamma:[0,1]\to\Omega$ is a continuous path. If $\Omega=\C\setminus\{0\}$, $\gamma(t):= e^{2\pi i ...
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Solving equation in Clifford algebra

Let $g_i$ denote the $i$'th orthonormal basis of a Clifford algebra $G_{p,q}$, then any number in Clifford domain can be represented as follows: $$a=\sum_{i=1}^{2^{p+q}}(a_ig_{i})$$ where $a_i$ is ...
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Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
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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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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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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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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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204 views

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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What are the “irreducible factors” in an algebra?

What are the irreducible factors in an algebra? In the book "Spin Geometry" by Lawson and Michelsohn, page 35, line 1, there occurs "take irreducible factors of...", but I don't know what irreducible ...
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An $SU(3)$ isomorph in Clifford $G(5,0)$?

I am a computer scientist using the geometric (Clifford) algebras $G(n,0)$ over $\mathbb{Z}_3 = \{0,1,-1\}$ to describe distributed computations. My question concerns $G(5,0)$ with generators ...
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Hermitian conjugation and representations of the Lorentzian Clifford algebras

The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
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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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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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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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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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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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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 ...