Why is symplectic geometry so important in modern PDE ?

Question

First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds is called symplectic geometry. In Hörmander's classic book ALPDO (The analysis of partial differential operators Ⅰ-Ⅳ) he wrote: symplectic geometry pervades a large part of the modern theory of linear partial differential operators with variable coefficients. And he had devoted the entire chapter ⅩⅩⅠ to discuss it.

Now, with some basic background (its origins in the Hamiltonian formulation of classical mechanics), I want to know further that why it would make such a important role in modern analysis (such as fourier integral operators)?

Thanks in advance.

Answer 1

Linear partial differential operators (or, in the language of quantum mechanics, quantum observables) on, say, ${\bf R}^n$, are (in principle, at least) generated by the position operators $x_j$ and the momentum operators $\frac{1}{i} \frac{\partial}{\partial x_j}$, which are then related to each other by the basic commutation relations $$ \frac{1}{i} [x_j, \frac{1}{i} \frac{\partial}{\partial x_k}] = \delta_{jk}$$ where $[,]$ here is the commutator $[A,B] = AB-BA$.

Meanwhile, classical observables on the phase space $T^* {\bf R}^n$ are (again in principle) generated by the position functions $q_j$ and momentum functions $p_j$, which are related to each other by the basic commutation relations $$ \{ q_j, p_k \} = \delta_{jk}$$ where $\{,\}$ is now the Poisson bracket (I may have the sign conventions reversed here).

One of the great insights of quantum mechanics (or, on the mathematical side, semi-classical analysis) is the correspondence principle that asserts, roughly speaking, that the behaviour of quantum observables converges in the high-frequency limit (or, after rescaling, the semi-classical limit) to the analogous behaviour of classical observables. The correspondence is easier to see on the observable side than on the physical space side, for instance by connecting the von Neumann algebra of bounded quantum observables with smooth symbol with the Poisson algebra of smooth classical observables. The former is connected to linear PDE and the latter to symplectic (or Hamiltonian) geometry.

Another way to see the connection is to investigate what happens when one applies a linear partial differential (or pseudodifferential) operator to a high-frequency function (or "quantum state"), when viewing that function through its Wigner transform, which can be viewed as approximately describing the quantum state by a classical one. A standard calculation shows (under Weyl quantisation) that the top order contribution of the operator on this transform is given by its symbol, and the next order term is basically given by the Hamiltonian vector field associated to that symbol. (This is discussed for instance in Folland's "Harmonic analysis on phase space".) This suggests that the dynamics of linear PDE at high frequencies are going to be driven by the associated Hamiltonian dynamics of the symbol of that PDE.

Answer 2

It is the non-commutativity of the algebra of (pseudo-)differential operators which makes symplectic geometry so important in the modern theory of linear PDE. The principal symbol of a commutator is (up to a constant factor) a Poisson bracket of principal symbols. The Poisson bracket encodes the symplectic structure of the cotangent bundle. The principal symbol of an $m$-th order differential operator $P$ is defined by $$\sigma(P)(d\varphi)=\lim_{\omega\to\infty}\omega^{-m}e^{-i\omega\varphi}Pe^{i\omega\varphi},$$ which makes it evidently a function on the cotangent bundle. It is a very important fact that non-commutativity is not too bad in that the order of a commutator $[P,Q]$ is strictly less than the sum of the orders of $P$ and $Q$. Microlocal analysis and its tools such as the (general) Friedrichs lemma or FIO calculus depend on this.

The above point of view on the role of symplectic geometry is expressed by Kashiwara in section 2.1 of his book "$D$-modules and microlocal analysis".

Answer 3

Symplectic forms also appear as one tries to give a description of selfadjoint (also dissipative etc) boundary conditions for a differential operator. Already for ordinary differential operators, the standard Green-Lagrange formula defines a symplectic form on appropriate collections of boundary values of functions and their derivatives. Selfadjoint boundary value problems can be identified with Lagrangian linear manifolds. Modern theory of selfadjoint extensions is a far-reaching generalization of this observation and can be applied to some classes of partial differential equations. For an introduction see

M. L. Gorbachuk and V. I. Gorbachuk, Boundary value problems for operator differential equations. Dordrecht, Kluwer, 1991.

A review of recent results is given in the paper

J. Bruening, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, No. 1, 1-70 (2008); http://arxiv.org/pdf/math-ph/0611088.pdf

Answer 4

Fourier Integral Operator is an operator which has its Schwartz kernel as a distribution whose singularities are on a Lagrangian submanifold. In fact, we can associate a FIO with an amplitude and a Lagrangian submanifold in a unique manner. Lagrangian submanifold is a topic of symplectic geometry.

Answer 5

As the cotangent space has a natural symplectic structure, we can also phrase the question as "Why is cotangent space so important in modern pde?". A nice answer for this question is given in http://math.berkeley.edu/~mjv/WhereforeCot.pdf.