# Questions tagged [clifford-algebras]

The clifford-algebras tag has no usage guidance.

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### Definition of Clifford super-connections

I have some questions concerning the definition of Clifford super-connections in Heat Kernels and Dirac Operators:
Definition 3.39. If $A$ is a super-connection on a Clifford module $E\to M$, we say ...

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### Can any Clifford module bundle be extended to a Dirac bundle?

I assume that the question in the title is clear, so let me talk about its relevance:
According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem
\begin{equation}\tag{1}
\mathrm{ind}...

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### "The index is independent of the Dirac operator"

Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem ...

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### Classification of real Clifford algebras

$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...

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### On Dirac/ Clifford matrices

Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$.
The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition
$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$
where $\{A,B\}=...

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### Unitary operators with the same inner product as vectors

Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...

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### Uniqueness of spinor representation

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$I asked a similar question on math stack exchange here, but I wonder if it may be better received here.
Let $n$ be ...

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### Why Representation of Clifford algebra are constant for an orthonormal frame?

Let $e_\alpha$ be a basis of the tangent bundle $TM$ and $ \rho: T_x M \rightarrow \operatorname{End}\left( W\right)$ a representation of a Clifford algebra.
In this text Field theory from a bundle ...

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### Shouldn't $\mathrm{End}_{C(TM)}(E)$ be defined differently in Heat Kernels and Dirac Operators?

The first four chapters of the book lead up to the proof of theorem 4.1. Its main consequence is that it provides the local index theorem for Dirac operators. The statement of theorem 4.1 involves a ...

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### Clifford modules - How is the grading on $\mathrm{Hom}_{C(V)}(S,E)$ defined?

From the book Heat Kernels and Dirac Operators:
Proposition 3.27. If $V$ is an even-dimensional real Euclidean vector space,
then every finite-dimensional $\mathbb{Z}_2$-graded complex module $E$ of ...

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### Group of invertible elements in a degree 4 central simple algebra with symplectic involution with norm in a center

Let $A$ be a central simple algebra of degree 4 (i.e. dimension 16) over a field $F$ with $\mathrm{char}(F) \neq 2$. It is known that any such algebra is a tensor product $D_1 \otimes D_2$ of two ...

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### Literature on Clifford modules

I encountered Clifford modules in the book Heat Kernels and Dirac Operators. I am particularly interested in the definition of the isomorphism
$$\mathrm{End}(E)\cong C(V)\otimes \mathrm{End}_{C(V)}(E)$...

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### Action of volume form on spinors in odd dimension

We know that for a smooth orientable manifold of dimension $2n, i^n$ times the volume form acts as identity on the positive spinors and acts as minus identity on the negative spinors via Clifford ...

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### Homomorphism from Clifford modules to Stable homotopy

In the paper "Clifford modules" by Atiyah, Bott and Shapiro, a homomorphism $\alpha:A_k\rightarrow \tilde{KO}(S^k)$ from a certain group of Clifford modules to real $K$-theory of spheres is ...

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### Norm of Killing spinor

A Killing spinor on a Riemannian spin manifold is a section of the spinor bundle satisfying the equation:
\begin{align*}
\nabla_X\phi=\lambda X\cdot\phi
\end{align*}
Here $X$ is a vector field and $\...

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### Dual Clifford module

$\DeclareMathOperator\Cl{Cl}$Let $\Cl(V)$ be the Clifford algebra of a quadratic vector space $(V,Q)$ over $k$, and let $M$ be a (left) $\Cl(V)$-module. The dual Clifford module is typically defined ...

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### Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...

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### CCR vs. CAR vs. Clifford algebras, infinite tensor products and type of the corresponding von Neumann algebra

$\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras ...

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### Proving that a product of reflections and an orthogonal matrix is in $\mathrm{SO}_*(V)$

Let $V=(V,b)$ be a finite-dimensional vector space equipped with $b$ a symmetric and positive definite bilinear form. And let $\{e_1,\dotsc,e_n\}$ be a orthonormal basis for the subspace $\ker((P_A)^t)...

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### Bott periodicity in characteristic p via Clifford algebras

I am currently reading Husemoller's wonderful book on fibre bundles, specifically the section on Clifford algebras. He defines these groups $L_k$ as follows. Let $M_k$ denote the free abelian groups ...

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### Example of nice isomorphism between Cl$_{p,q}(\mathbb R)$ and matrix algebras over $\mathbb R,\mathbb C,\mathbb H,\mathbb R^2,\mathbb C^2,\mathbb H^2$

$\DeclareMathOperator\Cl{Cl}$It is known that every Clifford Algebra $\Cl(Q)$ over the real numbers where $Q: \mathbb R^n \to \mathbb R$ is a non-degenerate quadratic form is isomorphic to a matrix ...

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### Clifford Algebra - axiomatic definition by Hestenes

I am reading David Hestenes' book "Clifford Algebra to Geometric Calculus" and am already getting stuck on the first few pages.
My university math is rusty and I've never studied Clifford ...

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### Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$.
My question is if there exists an analog of a Clifford algebra in ...

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### Decomposition of a bivector of a Lorentzian manifold [closed]

In the case of the $(1, -1, -1, -1)$ metric, a bivector (or a 2-form) $F$ decomposes uniquely as the sum of two orthogonal simple bivectors if $F^2 \ne 0$.
I have the impression that it is very little ...

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### Integral representation of cochains and a theorem of Hopf

The classical theorem of Hopf asserts that for any
n-dimensional CW-complex $K$, there is an isomorphism
between homotopy classes from $K$ to the sphere $S^n$ and
the nth singular cohomology group:
$$
...

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### Relationship with between Clifford multiplication and pullback

Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...

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### Problem concerning about an $n$-subspace of $ A_{n}(F) $

Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...

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### Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...

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### Can one extend the norm function to the symmetric square of a (complexified) Clifford algebra?

Let $A = Cl_{r,s} \otimes \mathbb{C}$ be the complexification of the real Clifford algebra $Cl_{r,s}$ associated to a non-degenerate quadratic form on $\mathbb{R}^n$, with $n = r+s$, with signature $(...

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### Identifying a $4$-form on a $6$-dimensional manifold

Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...

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### Finding inverses in Clifford Algebras

Let $C = \operatorname{Cl}(V,q)$ be a Clifford algebra where $V$ is an $N$-dimensional space with basis $B = \{e_1,e_2, \dotsc, e_N\}$. I'm looking for a way to invert elements.
What I've already ...

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### Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and
Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf
Somehow I find some inconsistency with his Tables of Euclidean and ...

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### Clifford multiplication formula for $d\beta$ where $\beta$ is a $3$-form

Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta$...

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### Gamma matrices are irreducible

For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible?
From my previous question, I know ...

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### What is the relationship between the Dirac algebra and the Clifford algebra?

While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is ...

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### How are Clifford algebras and spinors used to study the Ising model?

I've heard Clifford algebras and spinors are useful tools to study the Ising model, but I've never find any good discussion on this matter. Also, as far as I know, in the original solutions of the 2-D ...

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### Hasse invariant and the Clifford algbera

Let
$$q = a_1 x_1^2 + \cdots + a_n x_n^2$$
be a quadratic form over some $p$-adic field $\mathbb{Q}_p$. We thus have its Hasse invariant
$$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...

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### Central extensions of orthogonal group by $C_2$

Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...

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### Efficient computation of scalar part in Clifford algebra

$\DeclareMathOperator\Cl{Cl}$Problem: Let $\Cl(d)$ be the Clifford algebra corresponding to the vector space $\mathbb{R}^d$ with the usual inner product. Given $v_1, \dotsc, v_k \in \mathbb{R}^d$, ...

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### What is the inverse in K-theory represented by Clifford module extensions?

I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X_n]$ where the ...

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### Control and observability of Clifford algebra and quaternion valued systems?

Good evening everybody, I'm asking you if there is a mathematical theory about control and observability of differential systems within the quaternionic and Clifford (geometric) algebras. And if so, ...

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### Explicit computation of spinor norm

I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow.
Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...

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### Atiyah-Singer theorem in heat kernels and Dirac operators

I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne.
I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...

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### Invariant theory for the orthogonal group and Clifford algebras

The first fundamental theorem of invariant theory for
the orthogonal group $O_n(k)$ asserts that the
ring of invariants is generated by the scalar products:
a polynomial function of $m$ vectors $v_1,.....

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### The inner product of a Clifford Algebra

Any Clifford algebra $\operatorname{Cl}(k, p)$ carries an induced inner product, which is the "trace" on its 0-blade: $\langle AB\rangle_0$ for given elements $A, B$ of the algebra.
This inner ...

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### Is there a list of all real unital subalgebras of M(2,C)?

Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...

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### What's "geometric algebra"?

Sometimes one bumps into the name "geometric algebra" (henceforth "GA"), in the sense of this Wikipedia article. Other names appear in that context such as "vector manifold", "pseudoscalar", and "...

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### Maximal orders in Clifford algebras

Let
$$
\mathcal{C}_n(R)=R\langle e_1,\ldots,e_n\rangle/(\{e_i^2+1\}, \{e_ie_j+e_je_i:i\neq j\})
$$
be the Clifford algebra for the negative definite quadratic form $-\sum_ix_i^2$ obtained by adjoining ...

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### Bott periodicity homeomorphisms for spaces of Clifford extensions

I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).
Let $W = \mathbb{R}^{\...

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### Principled construction of the quaternions

Is there a construction of the quaternions that doesn't proceed through generators and relations, and which makes the connection with 3D rotations clear?
I'm not happy with Clifford Algebra as an ...