synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Over a smooth manifold $\Sigma$ of dimension $p+1$, let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle and $\tilde E^\ast \coloneqq E^\ast \otimes_{\Sigma} \wedge_\Sigma^{p+1} T^\ast \Sigma$ the tensor product of vector bundles of the dual vector bundle with the differential (p+1)-form bundle.
(formally adjoint differential operators)
are called formally adjoint if there exists a bilinear differential operator
such that for all $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have
This implies by Stokes' theorem, in the case of compact support, that under an integral $P$ and $P^\ast$ are related via integration by parts.
See also (Vinogradov-Krasilshchik 99, chapter 5, §2.3)
(Klein-Gordon operator is formally self-adjoint differential operator)
Let $\Sigma = \mathbb{R}^{p,1}$ be Minkowski spacetime with Minkowski metric $\eta$ and let $E \coloneqq \Sigma \times \mathbb{R}$ be the trivial line bundle. The canonical volume form $dvol_\Sigma$ induces an isomorphism $\tilde E^\ast \simeq E$.
Consider then the Klein-Gordon operator
This is its own formal adjoint (def. ) witnessed by the bilinear differential operator (1) given by
(Dirac operator on Dirac spinors is formally anti-self adjoint)
The Dirac operator on Dirac spinors is a formally anti-self adjoint (def. ):
In brief, the point is that when the Clifford generators themselves are formally self-adjoint, as they are (this Prop.) with respect to the Dirac conjugate (this Def.), then (only) the single derivative operator picks up a sign under passing to adjoints (i.e. under integration by parts).
In more formal detail:
Regard the Dirac operator as taking values in the dual spin bundle by using the Dirac conjugate $\overline{(-)}$:
Then we need to show that there is $K(-,-)$ such that for all pairs of spinor sections $\Psi_1, \Psi_2$ we have
But the spinor-to-vector pairing is symmetric (see at spin representation), hence this is equivalent to
By the product law of differentiation, this is solved, for all $\Psi_1, \Psi_2$, by
Peter Olver, chapter 5.3, around p. 328-330 of Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.
Alexandre Vinogradov, I. S. Krasilshchik (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. (pdf)
Igor Khavkine, Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv:1402.1282)
Last revised on March 3, 2021 at 17:48:27. See the history of this page for a list of all contributions to it.