What is Semiclassical Analysis?

The field of semiclassical analysis originated in the early 20th century as a way to translate between the mathematical formulations of classical and quantum mechanics. Motivated by the various similarities and parallels between classical systems and their corresponding quantum analogues, it attempted to formalize the correspondence principle, which states that the behavior of quantum systems should reduce to the laws of classical mechanics (in essence, Newton’s laws) at large enough physical scales. The scale of the system can be controlled the semiclassical parameter $h>0,$ and classical behavior is recovered in the semiclassical limit $h \to 0^+.$

By this point, classical mechanics had been already effectively described using the Hamiltonian formalism, which interprets the laws of motion for the position $x(t)$ and velocity $\xi(t)$ of a system as a flow $(x(t),\xi(t)) \in \mathbb{R}^n \times \mathbb{R}^n$ in phase space associated to the total energy or Hamiltonian $H.$ This flow is generated by the Hamiltonian vector field of $H$ and is described by Hamilton’s equations

$x'(t) = \xi(t), \quad \xi'(t)=-x(t).$

In contrast, the laws of quantum mechanics are best formulated using linear algebra, where a quantum state is a square-integrable wavefunction $\psi \in L^2$ and its evolution is governed by the Schrödinger equation

$\psi_t = (-h^2 \Delta + V(x))\psi.$

In this picture, a classical system is measured using functions on phase space, known as classical observables, while a quantum state is measured using which quantum observables, which are linear operators on $L^2.$ One of the major insights was to relate the evolution of classical observables (known as symbols) and the corresponding quantum observables through a process known as quantization. The most well-known such map is the semiclassical Weyl quantization, defined for a symbol $a$ as the Fourier integral operator

$a^w(x,hD)u(x) = \frac{1}{(2\pi h)^n} \int_{\mathbb{R}^{2n}} e^{\frac{i}{h}(x-y)\cdot \theta} a\left(\frac{x+y}{2},\theta\right)u(y)dy d\theta.$

Best thought of as a Fourier multiplier with symbol $a(x,\xi)$ the operator $a^w(x,hD)$ replaces every instance of $\xi$ with the semiclassical derivative $hD, D=\frac{1}{i}\partial_x,$ and is hence known as a pseudodifferential operator. This definition directly links the classical and quantum pictures — for instance, the Weyl quantization of the harmonic oscillator $|\xi|^2+|x|^2$ becomes the famous quantum harmonic oscillator $-h^2 \Delta +|x|^2,$ and the Schrödinger operator $-h^2 \Delta +V(x)$ itself becomes the quantization of the classical Hamiltonian $|\xi|^2+V(x).$ This allows us to study eigenvalue and evolution problems for partial differential equations, including those outside quantum theory, using the geometric and dynamical behavior of their corresponding symbols.