20 votes

Quantum corrections to geometry

Physicist chiming in; "quantum corrections of a geometry" represents the vague idea that gravity should be quantum. Classically, gravity is a geometric theory: "reality" is modelled as a (Lorentzian) ...
AccidentalFourierTransform's user avatar
19 votes

Is Witten's proof of the positive mass theorem rigorous?

You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3). 6. Conclusion From this ...
Francesco Polizzi's user avatar
11 votes

Quantum corrections to geometry

I presume this refers to quantum corrections to the metric tensor. These are expected to be important in general relativity when the curvature of space-time approaches the Planck scale $\sqrt{G\hbar/c^...
Carlo Beenakker's user avatar
10 votes

What's "geometric algebra"?

The label "geometic algebra" was William Cifford's name for the algebra he discovered (invented). This bit of history is recounted by David Hestenes, who has perhaps been most responsible ...
David Jones's user avatar
9 votes
Accepted

Is Witten's proof of the positive mass theorem rigorous?

The positive mass theorem is more or less to do with the geometry of a type of positive scalar curvature condition. Witten's work considers harmonic spinors, which are solutions to a certain linear ...
Quarto Bendir's user avatar
8 votes

Action of the spin covariant derivative on gamma matrices?

The question is not clearly stated, but if appropriately interpreted it does make sense. Let $S$ be a spinor bundle over a pseudo-Riemannian manifold $(M,g)$, whose bundle of Clifford algebras is ...
Bilateral's user avatar
  • 3,064
7 votes
Accepted

Path integral presentation of solutions of Dirac equation

There are several relevant papers: Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation (1987) Path Integral for Relativistic Equations of Motion (1997) Path ...
Carlo Beenakker's user avatar
7 votes

What's "geometric algebra"?

The essence of "geometric algebra" (better known as Kähler-Atiyah algebra) is the classical Chevalley-Riesz isomorphism, which presents the Clifford algebra of a quadratic space $(V,h)$ as a ...
amathematician's user avatar
7 votes

One question about the $\eta$ invariant

Summary: The integral $\int_0^\infty t^{\frac{s-1}{2}} \operatorname{Tr}[ D^u (\exp(-t D^2) ] \, dt$ has a simple pole at $s=0$ with residue $2 C_{-1/2}$, from which your expression for $\eta(0)$ ...
Phillip Andreae's user avatar
6 votes

General questions on the eigenfunctions of Laplacian and Dirac operators

No, we cannot (completely) hear the shape of a drum, even if it is spinorial. Two metric fields with the same collection of eigenvalues are called isospectral. There exist Dirac isospectral ...
Carlo Beenakker's user avatar
6 votes
Accepted

McKean-Singer formula in Heat Kernels and Dirac Operators book

The assertion is supposed to be that $d(e^{-tD^2})/dt$ has the same smooth kernel as $-D^2 e^{-tD^2}$, i.e. they are the same operator. This is because $e^{-tD^2}$ is the solution operator to the ...
Paul Siegel's user avatar
  • 28.8k
6 votes
Accepted

Proof that $[[D^2,f],f]=2[D,f]^2$

Note that for an odd operator $B$ we have $[B, B] = BB - (-1)^{|B||B|}BB = 2B^2$. Therefore, we can deduce $(2)$ as follows: \begin{align*} [[D^2, f], f] &= [[D, [D, f]], f]\\ &=-[f, [D, [D, f]...
Michael Albanese's user avatar
5 votes
Accepted

Classification of real Clifford algebras

The classification is described in quite some detail in the Wiki on the topic. In particular, the case $(p,q)=(1,3)$ is isomorphic to $M_2(\mathbb H)$ and the case $(p,q)=(3,1)$ is $M_4(\mathbb R)$. ...
rschwieb's user avatar
  • 1,593
5 votes
Accepted

Dirac operator on a Morita equivalent algebra

Your question is entirely covered by Section 2 of Brain–Mesland–Van Suijlekom, but the fgp case is simple enough to ultimately boil down to folklore proved by Chakraborty–Mathai. Let me summarise what ...
Branimir Ćaćić's user avatar
5 votes

What's "geometric algebra"?

This answer is my opinion. I don't have references with me at the moment. Feel free to suggest some by making edits. Geometric algebra is a school of thought towards linear algebra, geometry, and ...
wlad's user avatar
  • 4,823
5 votes

What's "geometric algebra"?

I am unsure how appropriate this answer is. I hope to provide some explanation for the recent interest in geometric algebra within computer graphics. My understanding of this viewpoint comes from the ...
Siddharth Bhat's user avatar
4 votes

What's "geometric algebra"?

Apart from any one else's hijacking of the phrase "Geometric Algebra", my own professional sense of this (from early 1970s) was more based in the sense of it in E. Artin's "Geometric ...
paul garrett's user avatar
  • 22.5k
4 votes

What's "geometric algebra"?

This is too long for a comment. I found the article Lazaroiu, Calin Iuliu; Babalic, Elena Mirela; Coman, Ioana Alexandra, Geometric algebra techniques in flux compactifications, Adv. High Energy ...
AlexArvanitakis's user avatar
4 votes
Accepted

Bochner formula in different forms

Here's a reference to a paper; Theorem 3 gives a general formula for differential forms with no constraints on the boundary values. I'm not sure if this shows up in a book yet. Raulot, S.; Savo, A. ...
Jeffrey Case's user avatar
  • 1,483
4 votes

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

You haven't quoted all the definitions, so I'm not sure what level of generality you are interested in. However, for usual spinor bundles over a Riemannian manifold $M$, every such bundle has a unique ...
Igor Khavkine's user avatar
4 votes
Accepted

Partial derivative in terms of Kronecker delta and the Laplacian operator

No, the operator $\partial_i\partial_j$ cannot be expressed in terms of the Laplacian $\nabla^2$. Indeed, take any real $a$ and $b$. If $\phi(x_1,\dots,x_n)=ax_1x_2+b(x_1^2+\dots+x_n^2)/(2n)$ for all ...
Iosif Pinelis's user avatar
3 votes
Accepted

Yamabe operator, conformal transformations and square of the Dirac operator

The discrepancy comes from the fact that the square of the Dirac operator is in general not conformally covariant. There are ways to modify powers of the Dirac operator to get a conformally covariant ...
Jeffrey Case's user avatar
  • 1,483
3 votes
Accepted

On Dirac/ Clifford matrices

If you require that the matrix $C$ in \eqref{2} preserves the involutions \eqref{3}, then you get the consequence $\tilde{\gamma}^\mu = C C^* \tilde{\gamma}^\mu (C C^*)^{-1}$, which then implies that $...
Igor Khavkine's user avatar
3 votes

The first eigenfunction of Dirac operator for surface

First some historical comments: The first sufficiently clear reference that I know for the spinorial Weierstrass is a preprint by Rob Kusner and Nick Schmitt link. From the point of view of the ...
Bernd Ammann's user avatar
2 votes

General questions on the eigenfunctions of Laplacian and Dirac operators

Most of the questions raised above are answered in the article "The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles" by Christian Bär and myself published in Annals of Global Analysis and ...
Bernd Ammann's user avatar
2 votes
Accepted

Question on a paper by U. Krähmer ("Dirac operators on quantum flag manifolds")

The map $\sigma$ is a representation of $U_q(\mathfrak{l})$ on the vector space $\Sigma_{2m}$, let's say for the moment just any such representation. To answer the second question, $\sigma(S(X))\psi$ ...
Nicola Ciccoli's user avatar
2 votes

Induced action by an involution on spinor bundle and Dirac operator

The action $T\colon S\to S$ is compatible with the Clifford-multiplication, hence the decomposition into $\pm1$ eigenspaces is preserved. Moreover, the spin connection is also preserved, and by the ...
Sebastian's user avatar
  • 6,715
2 votes
Accepted

"The index is independent of the Dirac operator"

First, for a Dirac operator $D$, it extends to a Fredholm operator \begin{equation} D^+\colon H^1(M,E^+)\to L^2(M,E^-), \end{equation} see Lawson-Michelsohn "Spin Geometry", Page 193, ...
Local's user avatar
  • 96
2 votes

An integral similar to the Delta function

$$\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau=\frac{2 k \omega'}{\left(k^2+1\right) \omega'^2+\omega^2-2 \omega\omega'},$$ for $k\omega'>0$.
Carlo Beenakker's user avatar
1 vote

Dirac operator on Kähler manifold

You are considering a Kähler manifold of real dimension 4. The bundle $(\Lambda^{0,2}\oplus\Lambda^0)\oplus\Lambda^{0,1}$ is a $\mathbb Z_2$-graded Clifford module with compatible hermitian metric and ...
Sebastian's user avatar
  • 6,715

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