Questions tagged [clifford-algebras]

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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18 views

What is the proof of the inner product formula between a vector and an r-blade?

I would like to prove the formula $ a\cdot A_r = \frac{1}{2}(aA_r - (-1)^r A_r a)$ for a grade-1 vector $a$ and a simple r-blade $A_r$. How? In Geometric Algebra for Physicists (Doran & Lasenby), ...
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How to we square unit vectors to 1,-1, or 0? What does ℜ( p, q, r) means?

My Geometric Algebra book starts with the following: Which I did not understand a bit, and went on with the book hoping for it to be further explained. It turns out that doesn't happen! I'm reading ...
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The bases of Clifford algebra

In order to get the bases of Clifford algebra, I want to understand the proof of the following proposition. Proposition: Let $Cl(V)$ be the Clifford algebra of a $\mathbb{R}$-vector space $V$ with ...
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algebraic structure of $ \otimes_{\Bbb R}$ in $\text{Spin}^c(V)\subset Cl(V)\otimes_{\Bbb R} \Bbb C$

I hope to know a clear explanation on the tensor product notation: $\otimes_{\Bbb R}$: Suppose we write $A \otimes_{\Bbb R} B$, what is the required algebraic structure of $A$ and $B$? What is the ...
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Rewriting the Geometric Product in 3D Space?

I'm reading a passage from Geometric Algebra for Physicists, and in section 2.4.1 there's a train of logic I'm not quite understanding. I'll copy it word for word, with italics being my comments/...
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Why are Clifford Algebras Superalgebras?

In Mathematical Gauge Theory With Applications to the Standard Model of Particle Physics Hamilton on page 332 proves that Clifford algebras are superalgebras by the following: First he defines two ...
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Embedding $SU(2)\times SU(2)\to Cl(\Bbb R^4)$

Consider the spin group $\text{Spin}(4)$. By definition it is contained in the Clifford algebra $Cl(\Bbb R^4)$ (https://en.wikipedia.org/wiki/Spin_group#Construction). Since it is known that $\text{...
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Exponential of a 3D multivector in Clifford's Geometric Algebra

An arbitrary 3D multivector can be written as the superposition: $M=Z+F$, where $Z=a+bi$ with $a,b\in\mathbb R$ and $i$ been the pseudo scalalar of $(Cl_3)$. The remaining term $F=v+iw$, where $v$ and ...
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The action of $SL(2, \mathbb R)$ on $\mathbb R^4$ though Clifford algebra

My question is about the realization of $SL(2, \mathbb R)$ as the group of unit quaternions in the even-dimensional Clifford algebra corresponding to an anisotropic quadratic form on $\mathbb R^3$ (...
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Grading of Clifford algebra [duplicate]

Given a vector space $V$ and a quadratic form $q$ on it, we know that the Clifford algebra $Cl(V,q)$ is a graded algebra, which is a property inherited by the Tensor algebra $T(V)$. Unless in the ...
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An embedding of $U(n)\to \text{Spin}^c(2n)$

Let $(V,\langle,\rangle)$ be an $2n$-dimensional real inner product space and consider its Spin$^c$ group $\text{Spin}^c(V)\subset Cl(V)\otimes_{\Bbb R} \Bbb C$. Suppose there is a compatible (...
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Is there an analogous concept for De Rham cohomology in the framework of Clifford algebras?

I’ve recently read about Clifford’s geometric algebra being a more general framework for differential geometry than differential forms, simpler for the study of spaces with a metric tensor, and ...
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Matrices that are both unitary and hermitian are real?

I am following a book on relativistic quantum mechanics and during some proof related to charge conjugation the author assumes that $\gamma_0$ is real (where $\gamma_0$ is one of the four 4x4 matrices ...
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How do I use duals in operations in Projective Geometric Algebra?

I've been looking at a lot of PGA stuff, particularly from bivector but I'm having a real hard time understand how to A: actually calculate a dual, and B: actually use the result in any operation. ...
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Is a normed vector space with an isometry group acting transitively on its unit ball an inner product space?

In section 2.3 of The Octonions, Baez discusses a proof for the Hurwitz theorem on normed division algebras via Clifford algebras that goes through the following argument. Supposing that $\mathbb K$ ...
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Addition in Geometric and Exterior Algebra

I was trying to define in a mathematically precise way the constructions of Geometric Algebra does in terms of exterior algebra, and I found myself stuck in the following problem. If one tries to ...
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Spin groups and Clifford Algebra

Consider $\Bbb R^n$ with standard norm and its Clifford algebra $Cl(\Bbb R^n)$. It decomposes as $Cl(\Bbb R^n)=Cl_0(\Bbb R^n)\oplus Cl_1(\Bbb R^n)$ (even part and odd part). The group $\text{Spin}(n)...
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Conjugate of a matrix exponential?

Say I have a multi vector of real geometric algebra of dimension 2. I define the Clifford conjugate as follows: $$ (A+\mathbf{X}+\mathbf{B})^\ddagger=A-\mathbf{X}-\mathbf{B} $$ where $A$ is a scalar, $...
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Deriving geometric product of basis vectors from outer product

I was exploring geometric algebra in general, when I came across these two formulas relating the inner and outer products of two objects with their geometric products: $$\begin{align} \mathbf{u} \cdot ...
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Isomorphism between exterior algebra and associated graded algebra (of the Clifford algebra)

I am looking for an isomorphism of $\mathbb{K}$-algebras between $\Lambda(V)$ and the associated graded algebra $\mathcal{G}$ of $\mathcal{C}l(V,q) = T(V) / \mathcal{I}_q$ with $\mathcal{I}_q$ ...
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$\operatorname{Spin}^+(s,t)$ is a group

I'm reading materials on spin groups. Let $Cl(s,t)$ be the Clifford algebra of $\mathbb{R}^{s+t}$ with standard bilinear form $\eta$ with signature $(s,t)$. The book then defines, $$\operatorname{Spin}...
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Decomposition of bivectors in four dimensions

In Clifford Algebras and Spinors by Pertti Lounesto it is written the following about bivectors in four dimensions: Now, I am wondering, how many different decompositions there are in the non-unique ...
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Is it possible to derive the spinor for $Cl_{1,3}(\mathbb{R})$?

Formally we have $$Cl_{1,3}(\mathbb{R})\otimes_\mathbb{R} \mathbb{C} \cong Cl_4(\mathbb{C})\cong M_4(\mathbb{C})\cong \text{End}(S)$$ and on a more explicit level (see wiki) we can exhibit the spinor ...
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Definition of Clifford group.

I know that this question is trivial for some, but everywhere I look, the authors already assume that the reader knows what it is about. The definition of Clifford group is a set of invertible ...
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Density of an even algebra

I’m trying to prove that if a set $S$ is dense in the even subalgebra $A^+$ of a $\mathbb{Z}_2$-graded algebra $A$ then $S$ is dense in the algebra $A$. But since density is not transitive, I don’t ...
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Conflicting definitions of a spinor

I've come across two definitions of "spinors" that I'm having a hard time reconciling: Spinors are the "square root" of a null vector (see here, and also Cartan's book "The ...
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Clarification of example of Clifford Algebra

I just started studying Clifford algebras and I am puzzled by the following example. Let $X$ be a Hilbert space with $\mathrm{dim}\ X = 1$. Let $\{e_1\}$ be the basis for $X$. Then the Clifford ...
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Does $h(1) = e$ hold?

Let $V$ be a vector space over a field $\mathbb{K}$ with characteristic zero and $Q$ be a quadratic form on $V$. Let $\mathcal{C}l(V,Q)$ be the associated Clifford algebra, with Clifford map $\varphi: ...
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Spin group for the complexified Clifford algebras

The $Spin(p,q)$ group is usually defined as $$ Spin(p,q) = \{s \in C\ell^{+}_{p,q}: s \tilde{s}=1\} $$ where $s$ belongs to the Lipschitz group and $C\ell^{+}_{p,q}$ denotes the even element subset of ...
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Generalizing norms: "$\lambda$-pseudonorms"

The study of norms is generally limited to those that are most interesting, most notably the Euclidean $2$-norm (and rarely the Manhattan $1$-norm, and supremum $\infty$-norm). However, I can ...
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Clifford map is injective for $B=0$.

Here's my definition of a Clifford algebra: Definition: Let $B(\cdot,\cdot)$ be a symmetric bilinear form on a vector space $V$ over $\mathbb{K}$ and $Q$ its associated quadratic form. The Clifford ...
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Triality with even and odd spinors

The spin group with mixed metric signs ($p$ for positive and $q$ for negative) is usually defined as $$ Spin(p,q) = Pin(p,q) \cap C\ell^{+}_{p,q}\,, $$ and in the special case of $Spin(8,0)$ one has ...
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Definition of the twisted spin covariant derivative for twisted spinor bundles

Let $S\to M$ be the spinor bundle and \begin{equation} \nabla^S\colon\Omega^0(M,S)=\Gamma(M,S)\to\Omega^1(M,S)=\Gamma(M,T^*M\otimes S) \end{equation} the spin covariant derivative. In addition, let $E\...
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Why is $C\ell$ functorial (why can we apply the appropriate universal property)?

I'm struggling to understand how a map $f:(V,q)\to (V',q')$ with $q'(f(v))=q(v)$, namely $f^*q'=q$ gives a algebra homomorphism of Clifford algebras $C\ell(V,q)\to C\ell(V',q')$. I understand this is ...
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Construction of $C\ell(V,q)$ for a quadratic space $(V,q)$ and proving an injection $V\hookrightarrow C\ell(V,q)$ exists.

I've been reading Lawson & Michelsohn's Spin Geometry, and struggled particularly with their (apparently wrong) proof that a quadratic space embeds into its Clifford algebra. I did a bit of ...
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Geometric algebra from the vector space of $N\times N$ Hermitian matrices

I am a physicist working in quantum mechanics, and I am trying to learn geometric algebra in order to get a different perspective on the same thing. In particular, I am interested in the possibility ...
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Do those Hermitian and unitary matrices form a basis for the underlying complex vector space?

I conjecture that in the complex vector space $\mathbb{C}^{2^N \times 2^N}$, where $N$ is a positive integer, there is a basis $\mathcal{B}$ whose elements are Hermitian and unitary. However, I don't ...
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Connections between Clifford algebra and polynomials?

The multi-vector idea in Clifford algebra really reminds me of polynomial multiplication. Can we treat every basic vector as a degree one polynomial, and multi-vectors as simply high-degree ...
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Isomorphism of representation spaces

The 4-dimensional spin group $Spin(4)=SU_{+}(2) \times SU_{-}(2)$, denote a typical element as $(A_+,A_-)$. We have for the 4-dimensional Euclidean space $V=\mathbb{R}^4 \simeq \mathbb{H} $ we can ...
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Ideal generated by $v\otimes v - \Phi(v)1$

I'm trying to understand the proof of the existence of a Clifford algebra. Let $V$ be a vector space over $\mathbb{K}$, $\varphi: V\times V \to \mathbb{K}$ a bilinear form and $\Phi(v) = \varphi(v,v)$ ...
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Definition of Clifford Algebras

I'm currently studying algebras, and in particular Clifford Algebras. Primarily I've been looking at this paper, and on page 7 and 8 the definition of a Clifford Algebra is given. I have a broad ...
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Geometric Algebras

As far as I can tell there are several extensions of linear algebra which can be used to do geometry on $\mathbb{R}^n.$ There are: the Clifford Algebra, the Grassman Algebra, the Exterior Algebra, ...
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Pseudovectors in Geometric / Clifford Algebra

I am studying geometric algebra and I am confused about why pseudovectors are written as single vectors with an $i$ in front. In other words, for basis vectors $\gamma_{\mu}$ where $\mu = 1,2,3,4$, ...
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Is it possible to replace the basis of 3D Clifford Multivector with Pauli matrices?

I was reading about Clifford isomorphisms (Cl3 to Mat(2,0)) here. Let's say we have a multi vector: $$\tag{1} 240 + (3{e1}) + (35{e2}) + (159{e3}) + (625{e12}) + (127{e13}) + (71{e23}) + (6{e123}) $$ ...
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Understanding pseudovectors and pseudoscalars

In doing an exercise about pseudovectors How do the components of a cross product transform under inversion? I got stuck. I didn't know how to compute the inversion of a cross product. My naive ...
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Invertability of a Multivector

I am writing a general geometric algebra library (dealing exclusively with euclidean bases) for Clojure to gain a better understanding on the topic, but how to invert a general multivector continues ...
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Quaternions +Geometric (Clifford) Algebra: What Is the Proper Prerequisite Sequence Before Learning These Subjects

What is the systematic prerequisite sequences of learning that must be mastered before approaching the subject of learning Quaternions, and then Clifford Algebra? My ultimate goal is to learn and ...
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Velocity gradient in vector and geometric calculus

While geometric calculus is seen as a success in the description of electromagnetism, a topic I see little discussion regarding this framework is continuum mechanics. I was wondering how useful it ...
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An issue identifying Clifford Algebra with Endomorphism ring.

I am trying to work through pages 303-305 of Fulton and Harris and have ran into a problem. I will first give a little bit of setup to try and make this post self contained. Let $V$ be an even ...
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What is the mathematical equivalent of an axle?

I am talking about axles in machines, on which we can put gears, belts and other devices that help us convert the motion. So this structure has to have (at least) the following properties: distance - ...

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