User Tools

Site Tools


Sidebar


Add a new page:

advanced_tools:jet_bundles

This is an old revision of the document!


Jet Bundles

Why is it interesting?

Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

Student

The dynamics of a field theory is specified by an equation of motion, a partial differential equation for such sections. Since differential equations are equations among all the derivatives of such sections, we consider the spaces that these form: the jet bundle $J^\infty_\Sigma E $ is the bundle over $\Sigma$ whose fiber over a point $\sigma \in \Sigma$ is the space of sections of $E$ over the infinitesimal neighbourhood $\mathbb{D}_\sigma$ of that point:

Therefore every section $\phi$ of $E$ yields a section $j^\infty(\phi)$ of the jet bundle, given by $\phi$ and all its higher order derivatives.

Accordingly, for $E, F$ any two smooth bundles over $\Sigma$, then a bundle map

encodes a (non-linear) differential operator $D_f : \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)$ by sending any section $\phi$ of $E$ to the section $f \circ j^\infty(\phi)$ of $F$.https://arxiv.org/abs/1601.05956

Researcher

The motto in this section is: the higher the level of abstraction, the better.

Examples

Example1
Example2:

FAQ

History

advanced_tools/jet_bundles.1510156740.txt.gz · Last modified: 2017/12/04 08:01 (external edit)