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.
191
questions with no upvoted or accepted answers
12
votes
0
answers
531
views
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 ...
- 2,122
8
votes
0
answers
143
views
In a Clifford algebra over an $n$-dimensional non-degenerate space, is a product of any number of vectors a product of at most $n$ vectors?
In a Clifford algebra over an $n$-dimensional vector space $V$ with a non-degenerate quadratic/bilinear form, can any product of vectors $a_1,a_2,\cdots,a_m\in V$ be written as
$$a_1a_2\cdots a_m=a_1'...
- 5,374
7
votes
0
answers
317
views
Clifford algebra is isomorphic to exterior algebra, proof in Lawson's
This Proposition 1.2, pg 10 of Lawson's Spin Geometry.
Context: We have a homomorphism of graded algebra
$$ \wedge^*(V) \rightarrow G^*$$
induced on each component from the map $$\wedge^r(V) \...
- 9,518
7
votes
0
answers
913
views
Clifford Multiplication and Spinor Bundles
I am trying to follow the discussion of Clifford multiplication on page 384 of The Wild World of 4-Manifolds, by Alexandru Scorpan (link, although I hope this will be totally self-contained), and I'm ...
- 2,655
6
votes
0
answers
217
views
Clifford algebra and Spin
I am reading Riemannian Geometry and Geometric Analysis by Jurgen Jost. Let me mention the notations and what I have learned from the book.
In the book, the Clifford algebra $Cl(V)$ of a real vector ...
- 637
6
votes
0
answers
121
views
Problems in understanding a sentence from Garling's book "Clifford Algebras: An introduction"
The sentence comes from pag. 18 of Garling's book "Clifford Algebras: An introduction".
If $A$ is unital, then a subalgebra $B$ is a unital subalgebra if the identity element of $A$ belongs ...
6
votes
0
answers
181
views
Natural irreducible modules over Clifford algebras?
Let $V$ be a complex vector space with a nondegenerate quadratic form $\langle -, -\rangle$. Let $\mathrm{Cl}(V)$ be the Clifford algebra: the quotient of the tensor algebra $\mathrm{T}(V)$ by the ...
- 5,836
6
votes
0
answers
331
views
Is there a relationship between the trace and the Clifford/geometric product?
In what follows, let $V=\mathbb{R}^n$ (although the following probably applies also to a larger number of finite-dimensional spaces).
We assume throughout that we have made a choice for an inner ...
- 21.1k
5
votes
0
answers
98
views
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 (...
- 2,024
5
votes
0
answers
96
views
Clifford algebra quotient definition authorship
Given a vector space, $V$, with a quadratic form $Q$, one possible definition for its Clifford algebra (and perhaps the one I like the most) is to take it to be the quotient of the tensor algebra of $...
- 131
5
votes
0
answers
79
views
How far (for which $p$) can we generalize $\int x^p \mathrm{d}x=\frac{x^{p+1}}{p+1} + C,p\neq-1 $?
Referring to trivial form, sometimes called Cavalieri's formula:
$$
\int x^p \mathrm{d}x=\frac{x^{p+1}}{p+1} + C, \qquad p\neq-1
$$
I was wondering what other restrictions on $p$ exist besides $p \...
- 934
5
votes
0
answers
749
views
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 ...
- 1,096
5
votes
0
answers
72
views
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 \...
- 709
5
votes
0
answers
803
views
Clifford algebra - Gamma matrices
Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition
$$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
- 51
5
votes
0
answers
1k
views
Geometric algebra and quantum field theory
What 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 ...
- 523
4
votes
0
answers
142
views
Definition of the twisting space of a Clifford module
Let $V$ be an even-dimensional euclidean vector space, $S$ the spinor module and $E$ a module of the Clifford algebra $C(V)$. Then
$$W=\mathrm{Hom}_{C(V)}(S,E)$$
is called the twisting space of $E$. ...
- 3,536
4
votes
0
answers
112
views
For an even multivector $A$, if the map $X\mapsto AX\tilde A$ preserves grade, must $A$ be a product of vectors?
We're working in a Clifford algebra over a non-degenerate $n$-dimensional vector space $V$, and considering various properties a multivector $A$ could have:
(0) $A$ is invertible.
(1) $\tilde AA$ is a ...
- 5,374
4
votes
0
answers
54
views
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 ...
- 918
4
votes
0
answers
146
views
Lemma 4.14 Heat Kernels and Dirac Operators
I am trying to work out Lemma 4.14 of the book "Heat Kernels and Dirac Operators" by Berline and Getzler. I am stuck with the proof. For the sake of brevity I am uploading a picture from the book ...
- 4,092
4
votes
1
answer
677
views
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 ...
- 135
4
votes
0
answers
169
views
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 ...
- 9,800
3
votes
0
answers
46
views
On triality, $Cl_{0,6}(\mathbb{R})$, and the outer automorphism of $S_6$
I ran across this paper recently, which explores Conway's proof that $L_4(2) = A_8$: https://archive.maths.nuim.ie/staff/jmurray/Preprints/jmurrayalt8.pdf . A central part of the argument involves ...
3
votes
0
answers
115
views
Examples of Clifford algebras over finite fields
Yesterday I was in a discussion about solving an applied problem using clifford algebras over a finite field. While this is not (seemingly) disallowed by the definition of a clifford algebra (which ...
- 131
3
votes
0
answers
120
views
Geometric understanding of "Pauli vector" determinant?
If we define the Pauli matrices as
$$\sigma_0 =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}\quad
\sigma_1 =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}\quad
\sigma_2 =
\begin{...
- 6,585
3
votes
0
answers
83
views
Spin$^c$-structure necessary for K-orientation
In Atiyah, Bott, and Shapiro's Clifford Modules, Theorem 12.3, they prove that a Spin$(k)$-structure (resp. Spin$^c(2k)$)-structure gives a KO(K)-orientation on the associated vector bundle of rank $k$...
- 373
3
votes
0
answers
86
views
Notes on Spinors by Deligne
I was reading section 2 (Clifford Modules) of the Notes on Spinors by Deligne, and am a little puzzled by Proposition 2.2. In this proposition, it is assumed $V$ is a complex vector space and $Q$ a ...
- 242
3
votes
0
answers
125
views
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 ...
- 1,786
3
votes
0
answers
63
views
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 ...
- 417
3
votes
0
answers
57
views
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: ...
- 4,025
3
votes
0
answers
149
views
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 ...
- 171
3
votes
0
answers
70
views
How can I bend a complex number (and multivectors in general)?
Let me use the following notation for orthonormal basis $\{\sigma_0,\sigma_1,\dots\}$ and this one for a general curvilinear basis $\{\mathbf{e}_0,\mathbf{e}_1,\dots\}$. The basis are constrained by ...
- 2,589
3
votes
0
answers
43
views
Indefinite Quadratic Forms and their Geometry
I'm looking for a book that would included an elementary introduction to the groups $O(p,q)$ and $SO(p,q)$ that would be suitable to an undergraduate with a background in Linear Algebra and Group ...
- 896
3
votes
0
answers
160
views
What's the problem with Clifford's algebra?
I've been searching trough the internet for more information about Clifford's algebras, and for what I've been reading, Clifford's vector system appears to be "unifying" vector system, in particular I'...
- 445
3
votes
0
answers
87
views
Rotation around a whole sphere by multiplying a single hypercomplex number forever?
In quaternion number system, any unit quaternion $\mathbf{q}\in\mathbb{H}$ can be written as
$$
\mathbf{q} = \cos \theta + (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k})\sin \theta
$$
for some $\...
- 31
3
votes
0
answers
361
views
Lie derivative in geometric algebra/Clifford algebra
What is the form of the Lie derivative in Clifford algebra?
Context:
Consider the Clifford algebra $\mathcal{C}l (p,q) $ with basis $\{e_i \}$.
The geometric derivative following Hestenes is defined ...
- 1,132
3
votes
0
answers
82
views
How to "see" quaternion structure inside the Clifford algebra in dimensions 2, 3 , 4 mod 8?
More precisely, from the classification of real Clifford algebras we see that in dimensions 1 and 5 mod 8 they are actually complex matrix algebras. However this natural complex structure can be ...
- 131
3
votes
0
answers
527
views
The pushforward under the left action in the group of units of a Clifford algebra
The following I know to be true: let $A$ and $B$ be elements of $GL(m,\mathbb{R})$ and let $X \in T_BGl(m, \mathbb{R})$ and let $L_A:Gl(m, \mathbb{R}) \to GL(m, \mathbb{R})$ be the left multiplication ...
- 223
3
votes
0
answers
489
views
Proof: Clifford-Algebra representations are semisimple / completely reducible
There is a theorem:
Every finite-dimensional Clifford-Algebra representation $V$ is semisimple / completely reducible, which means that it's a direct sum of irreducible subrepresentations.
How this ...
- 127
2
votes
0
answers
60
views
On the dot product of a vector with a bivector
I have this identity
$$a\cdot (b \wedge c) = (a \cdot b)c - (a \cdot c)b$$
which I can prove pretty straight forward, using that $ X\cdot Y = XY_\parallel$ and $ X \wedge Y = XY_\perp$ (Where I break ...
- 41
2
votes
0
answers
80
views
Is this statement about quaternion function true?
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying
$\overline{\partial_c}f=0,$
where
$$\overline{\partial_c}=\frac{1}{2}\...
2
votes
0
answers
94
views
What is the intuition behind the regressive product and its axioms?
I'm self-studying Grassmann algebra and I find it hard to visualize the regressive product, specifically the magnitude and orientation of the outcome.
My main soure is John Brown's Grassman Algebra ...
- 41
2
votes
0
answers
42
views
Can you show an example of conformal geometric algebra being applied to synthetic geometry?
I have been studying conformal geometric recently where points, lines, etc are described as $n$-blades in geometric algebra. The notes I am studying claim that conformal geometric algebra can be ...
- 31
2
votes
0
answers
111
views
Intersection of *segments* in 3D using projective geometric algebra?
There is a relatively common technique to find the intersection of two segments in 3D, which can be found in page 304 of Graphics Gems
Note that a segment is a compact subset of a line.
The technique ...
- 3,339
2
votes
0
answers
102
views
extended Clifford algebra
I've been implementing real Clifford algebras and have successfully extended the implementation to include basis vectors that square to $1$, $-1$, and $0$ as well as to include a mix of basis vectors ...
- 21
2
votes
0
answers
98
views
Spin Group is closed subgroup of the multiplicative group of units
Let $V$ be a vector space over $\mathbb{C}$. Clifford algebra $Cl(V,Q)$ is defined by the tensor algebra $T^{\boldsymbol{^*}}(V)$ modulo the ideal $I$ generated by $v\otimes v -Q(v,v)\cdot1$.
Let $Cl^{...
- 111
2
votes
0
answers
47
views
How much novelty is there in spectral theory over Clifford algebras?
Asking here seems like the most efficient way of getting an answer. Maybe I'll get an opinion from a relevant expert.
A demonstration of my idea is to take the Takagi decomposition:
Given a complex-...
- 8,185
2
votes
0
answers
118
views
Connections between Weyl, Clifford, Exterior, and Symmetric algebras
The Clifford algebras are analogous to the Weyl algebras in the same way that the exterior algebras are analogous to the symmetric algebras.
How are the Clifford and Weyl algebras analogous?
The ...
- 1,167
2
votes
0
answers
326
views
How is Grassmann Algebra better than just using determinants?
I am reading Roger Penrose's road to reality. On page 209 ,210 he introduces the idea of Clifford algebra and how we can get back quarternion algebra by consideration of the 'second order' quantities ...
- 11.6k
2
votes
0
answers
40
views
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{...
- 2,024
2
votes
0
answers
118
views
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 ...
- 363