Jet Bundles

Prerequisites: Fiber Bundles. Full path (5 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

A Lagrangian density depends on a field and on its derivatives at a point. When we write it down we treat the field value and its derivatives as if they were independent variables, even though for an actual field configuration the derivatives are determined by the field. The jet bundle is the space that makes this precise.

Think of the Taylor expansion of a function around a point. Two functions that agree in their value and their derivatives up to some order at that point have the same Taylor polynomial up to that order; we say they have the same jet of that order there. The space of all possible jets of a given order at a point is just a space of numbers: the value, the first derivatives, the second derivatives and so on, all treated as independent coordinates. Collecting these spaces over all points of spacetime gives the jet bundle.

A Lagrangian is then an ordinary function on the jet bundle, and a differential equation is a condition on points of the jet bundle. An actual field configuration traces out a special surface inside the jet bundle, the one on which the "derivative coordinates" really are the derivatives of the "value coordinate". This separates the algebra of a field theory (which functions of fields and derivatives appear) from the question of which configurations solve it.

Concrete

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 $f : J^\infty_\Sigma E \longrightarrow F$

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

Abstract

Let $\pi:E\to\Sigma$ be a smooth fibre bundle with $\dim\Sigma=n$ and fibre dimension $m$. Two local sections $\phi,\psi$ defined near $\sigma\in\Sigma$ have the same $k$-jet at $\sigma$ if, in some (hence any) adapted coordinates, their values and all partial derivatives up to order $k$ agree at $\sigma$. The equivalence class is written $j^k_\sigma\phi$, and the set of all $k$-jets is the $k$-th jet manifold $J^kE$. It carries adapted coordinates $(x^i,u^a,u^a_I)$, where $I$ runs over symmetric multi-indices with $1\le|I|\le k$, and $j^k_\sigma\phi$ has coordinates $u^a_I=\partial_I\phi^a(\sigma)$. Since there are $\binom{n+k}{k}$ multi-indices of length at most $k$ in $n$ variables (including the empty one), $J^kE\to\Sigma$ has fibre dimension $m\binom{n+k}{k}$.

There are projections $J^kE\to J^{k-1}E$, and each is an affine bundle; for $k=1$, $J^1E\to E$ is affine over $T^*\Sigma\otimes VE$, where $VE$ is the vertical tangent bundle. The infinite jet bundle $J^\infty E$ is the inverse limit of this tower. Every section $\phi$ of $E$ has a prolongation $j^k\phi:\sigma\mapsto j^k_\sigma\phi$, a section of $J^kE\to\Sigma$. Not every section of $J^kE$ is of this form. The holonomic ones are characterized by the contact forms $$\theta^a_I = du^a_I - u^a_{Ii}\,dx^i,\qquad 0\le|I|\le k-1,$$ (with $u^a_I=u^a$ for the empty multi-index), which pull back to zero exactly along prolonged sections.

On functions on $J^\infty E$ act the total derivatives $$D_i = \frac{\partial}{\partial x^i} + u^a_i\frac{\partial}{\partial u^a} + u^a_{ij}\frac{\partial}{\partial u^a_j}+\dots,$$ which satisfy $(D_iF)\circ j^\infty\phi = \partial_i(F\circ j^\infty\phi)$. A Lagrangian of order $k$ is a horizontal $n$-form $L\,d^nx$ with $L$ a function on $J^kE$, and the action of a section is $S[\phi]=\int_\Sigma (j^k\phi)^*(L\,d^nx)$. Its variation gives the Euler-Lagrange expressions $$E_a(L) = \sum_{|I|\le k}(-D)_I\frac{\partial L}{\partial u^a_I} = \frac{\partial L}{\partial u^a}-D_i\frac{\partial L}{\partial u^a_i}+D_iD_j\frac{\partial L}{\partial u^a_{ij}}-\dots,$$ where $(-D)_I=(-1)^{|I|}D_{i_1}\cdots D_{i_r}$, and the equations of motion are $E_a(L)\circ j^\infty\phi=0$. More generally, a differential operator of order $k$ from sections of $E$ to sections of a bundle $F$ is the same thing as a bundle map $J^kE\to F$ over $\Sigma$, and a system of differential equations is a submanifold of $J^kE$. The de Rham complex of $J^\infty E$ splits into horizontal and contact degrees, the variational bicomplex, in which Lagrangians, Euler-Lagrange forms, conserved currents and Noether's theorems all have a natural place.

Why is it interesting?

Given a field bundle […], we know what type of quantities the corresponding fields assign to a given spacetime point. Among all consistent such field configurations, some are to qualify as those that “may occur in reality” if we think of the field theory as a means to describe parts of the observable universe. Moreover, if the reality to be described does not exhibit “action at a distance” then admissibility of its field configurations should be determined over arbitrary small spacetime regions, in fact over the infinitesimal neighbourhood of any point. This means equivalently that the realized field configurations should be those that satisfy a specific differential equation, hence an equation between the value of its derivatives at any spacetime point.

In order to formalize this, it is useful to first collect all the possible derivatives that a field may have at any given point into one big space of “field derivatives at spacetime points”. This collection is called the jet bundle of the field bundle.https://ncatlab.org/nlab/show/field+%28physics%29#AFirstIdeaOfQuantumFields

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Manifold
5. Fiber Bundles