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) ...
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 ...
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^...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\eta(0)$ ...
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 ...
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 ...
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]...
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 $M_4(\mathbb R)$.
...
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 ...
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 ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 $...
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 ...
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 ...
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, $\sigma(S(X))\psi$ ...
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 ...
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, ...
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 $k\omega'>0$.
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 ...
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