Is there a relationship between Fourier transformations and cotangent spaces? Maybe a trivial question but I can't seem to find it treated anywhere.
Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function $F:Q\to\mathbb{R}$ induces a map $q\mapsto dF_q \in T^*_qQ$ which induces the Legendre involution when $F$ is convex/concave. I think of $(q,dF_q)$ as determining a Lagrangian submanifold $L_F\subset T^*Q$, and the Legendre transformation as lifting functions on $Q$ to $L_F$ and subsequently projecting them to a fibre. The geometric construction shows it is a symplectomorphism. In coordinates, this changes the variables from $q$ to $p=\partial F/\partial q$. 
My question is how can I understand the Fourier transform in this picture. By the stationary phase approximation (Wiki) or similarly Laplace's method (also on Wiki) we have something like $\mathcal{F}(\exp(i f)) \approx \exp(i\mathcal{L}f)$ where $\mathcal{L}$ is the Legendre transform.
Furthermore, how does this relate to (quantum) field theory? Path integrals always seem to have the form $\int \exp(\frac{i}{h}S[\phi]) F(\phi) \mathcal{D}\phi$ where $S$ is the action functional and integration is over all paths $\phi$ with given endpoints. This action functional ``is'' the tautological one-form (Wiki yet again), so is path integration just some kind of Fourier transform?
Any reference will help, I am not an expert.
 A: Since the question asked for a reference for this relation between the Fourier and Legendre transforms, here's one:

Guillemin & Sternberg, Geometric Asymptotics (AMS, 1990). See in particular the discussion on pp.403-404 in section VII.1.

Writing the path integral with an external source, $\int \exp(\frac{i}{h}(S[\phi]+J\cdot\phi)) F(\phi) \mathcal{D}\phi$, where $J\cdot\phi = \int J(x) \phi(x) dx$, like I suggested in the comments, makes it formally look precisely like a Fourier integral (mapping functions of $\phi$ to functions of $J$).
A: There is indeed a deep relation between Lagrangian submanifolds, the Fourier transformation and microlocal analysis. This is extensively discussed in Bates and Weinstein: Lectures on the Geometry of Quantization.
Let me illustrate the basics idea in the simplest case. Assume we are given an Lagrangian immersion $\iota: L \to T^* R = R^2$. Projecting on the $p$-component yields a map $\pi_p: L \to R$. If $\pi_p$ is a diffeomorphism then, for every half-density $a$ on $L$, we can define a function $B$ on the momentum line $R$ by
$$ B |dp|^{1/2} = \pi_p^* \, e^{i \tau} a .$$
Here $\tau: L \to \
R$ is a "generalized phase function" satisfying $d \tau = \iota^* \theta$ with $\theta$ being the canonical 1-form on $T^* R$. Taking the inverse Fourier transformation of $B(p)$ yields a function $F(q)$ living on the configuration space. Hence, we see that indeed the Fourier transformation of $F$ is related to the Lagrangian immersion $\iota$. However, your formula $\mathcal{F}(\exp(i f)) \approx \exp(i\mathcal{L}f)$ is a little bit to naive and just works under certain non-degeneracy assumptions. (If $\pi_p$ fails to be a diffeomorphism, then you need to include further corrections - these include the Maslov index of the Lagrangian immersion).
So what is the relation to quantum mechanics? If your $F: Q \to R$ satisfies the Hamilton-Jacobi equation, then it may serve as the phase function of a first-order approximate solution of Schrödinger’s equation. In other words, the WKB ansatz $e^{i / \hbar F(q)}$ is then a first-order approximation. More generally, you could also consider approximate solutions of the form $a\,e^{i / \hbar F}$, where the amplitude $a$ is essentially the half-density $a$ living on $L$ as above.
