# Tagged Questions

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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### A space more fundamental than Euclidean space

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which ...
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### Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
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### Octonionic formula for the ternary eight-dimensional cross product

A cross product is a multilinear map $X(v_1,\cdots,v_r)$ on a $d$-dimensional oriented inner product space $V$ for which (i) $\langle X(v_1,\cdots,v_r),w\rangle$ is alternating in $v_1,\cdots,v_r,w$ ...
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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 ...
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 $\zeta = \... 1answer 47 views ### Why are “innermorphisms” not useful? I commonly studied type of linear function in geometric algebra is the outermorphism. For reference, here's Wikipedia's definition: Let$f$be an$\Bbb R$-linear map from$V$to$W$. The ... 0answers 21 views ### Clifford Algebra: Expression for$\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 ... 1answer 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$U^ \dagger AU=B$? 0answers 81 views ### 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 ... 1answer 50 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,$j:...
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 ...