Peter Teichner. Classical field theory. Goal: Describe a mathematical framework in which all real physical theories can be expressed. A

• classical field theory consists of the following three data:

• space-time M (a Lorentz manifold of dimension (d − 1) + 1 or a Riemannian manifold of dimension d or an arbitrary semi-Riemannian manifold);

• field content Φ(M) (space of sections of a certain bundle over M);

• classical action S: Φ(M) → R.

Roughly speaking, classically one only sees

• extrema of S. In quantum field theory a state is a superposition of all fields with weight exp(iS(Φ)/h̄). For statistical field theory the probability density is exp(−S(Φ)/T). So the equations become the same if T = ih̄.

Examples:

(a) Mechanics: A particle moving in a configuration space N. Here (1) M is one-dimensional (just time); (2) Φ(M) = C^{∞}(M, N); (3) S(ϕ) = ∫_{M} L(ϕ_{r}(m))dm, where ϕ_{r}(m) is the r-jet of ϕ at point m ∈ M and L: J^{r}(N) → R is a Lagrangian, for example: r = 1, J^{1} = T N, L(v) = g(v, v) − f(π(v)). Note that J^{r}(N) is finite-dimensional manifold.

(b) Free boson: (1) M is any oriented semi-Riemannian manifold; (2) Φ(M) = C^{∞}(M, R) (free scalar σ-model); (3) S(ϕ) = ∫_{M} L(ϕ_{r}(m))ω, where ω is a volume form, L(ϕ) = Δ(ϕ)ϕ + m^{2}ϕ^{2} and Δ = d^{∗}d = −(∗d∗)d ≥ 0. Hence L(ϕ)ω = (dϕ) ∧ (∗dϕ) + m^{2}ϕ ∧ ∗ϕ. Key property:

• Locality of the classical action S. We have a map ev_{r}: M × Φ(M) → J^{r}(Φ). We require that S = ∫_{M} L, where L is a form of degree d = dim M on M × Φ(M), which comes from J^{r}(Φ): L is the pullback of some form of degree d on the manifold J^{r}(Φ) along the map ev_{r}.