tags: physics

Being confused for a long time about the main aspects of quantum mechanics and quantisation, I decided now to summarise my current knowledge.

So, dear diary, non-relativistic classical mechanics essentially equals symplectic geometry. To be totally honest, while a symplectic manifold captures all the relevant information about a mechanical system, it only appears as result of a complete understanding of the system. If one wishes to derive the equations of motion from fundamental principles then one proceeds as follows.

Take a manifold (probably a Riemannian manifold) which describes possible positions of elements of your mechanical system. Trajectories are just curves in this manifold. Knowing that your system is in a state \(x_0\) we need to understand how it will evolve in time. So we have a functional called action on the space of possible trajectories obtained by integrating something called Lagrangian density over the curve. The principle of least action says that the system will evolve according to the path which minimizes the action; the differential equations of motion are called the Euler-Lagrange equations. One can prove a theorem that for mechanical systems of a certain kind (? not sure what are the precise requirements here) the trajectories can be described in a particularly nice way as curves parallel with respect to Hamiltonian flow on a symplectic manifold. The translation from Riemannian to symplectic picture is called Legendre transform.

Once in symplectic setting, the evolution is described as follows. There is a distinguished function \(H\) on the sympletic manifold, called Hamiltonian, and since symplectic form defines a one-to-one correspondense between vector fields and 1-forms, \(dH\) gives rise to a vector field. If the system is in a state \(x_0\) then it will evolve along a curve starting at \(x_0\) and touching the vector field.

These two approaches to mechanics are called respectively Lagrangian and Hamiltonian.

For idealization, it wouldn’t hurt to assume that all manifolds we work with are \(C^\infty\). Note that the \(C^\infty\)-algebra of functions of a manifold determines this manifold. From a physical point of view, the elements of this algebra are physical quantities one can measure on the system’s particular state, or observables.

A quantum mechanical system is described by two pieces of data: a Hilbert space and an algebra of self-adjoint operators acting on it, the first playing the role of the manifold and the second playing the role of the algebra of observables — now they are not the same. A state of the systemsis a 1-dimensinal subspace in the Hilber space, one usually talks about their representatives which are called wave functions. Contrary to the classical situation, one cannot simply find out the value of an observable at a given state, what one gets instead is a measure. If \(\varphi\) is a wavefunction and \(A\) is an operator corresponding to a quantity we would like to measure then \[ \langle A \psi, \psi \rangle \] defines the density of the probability measure for the observable \(A\) to take a particular value when the quantum system is in the state \(\psi\).

Typically, introductory expositions of quantum mechanics start with appealing to analogies with the Hamiltonian approach to classical mechanics (the symplecic picture). From this point of view the evolution of a quantum system is described by the Schroedinger quation \[ i \hbar \frac{d}{dt} \psi = H \psi \] where \(H\) is a distinguished observable (the Hamiltonian).

The most basic quantum system is a quantum harmonic oscillator which has the Hamiltonian of the form \[ H \psi = \Delta \psi + V(x-x_0) \] where \(\Delta\) is a Laplace operator and \(V(x)\) is a quadratic form.

In the classical situation the Hamiltonian is constant along a trajectory and can take a continuum of different values along different trajectories. In the case of quantum harmonic oscillator, there are only countably many trajectories along which the Hamiltonian takes a constant value with 100% certainty: they correspond to different eigenspaces of \(H\). It seems that the quadratic potential takes a special place among the physical models; I am not sure if this is because little is known for potential of more general form or because this example is deemed more suitable for expository purposes.

The corresponding simplest classical mechanical system is described by the symplectic manifold \(T^* \mathbb R\). Its \(C^\infty\)-algebra is ``generated’’ by two functions \(p, q\), coordinate functions using which the symplectic form can be written as \(\omega = dp \wedge dq\).

The algebra of observables of the quantum harmonic oscillator is the Weyl algebra \(\langle x, \partial | [x, \partial] = 1\rangle\) where \(x\) corresponds to the classical position observable, and \(\partial\) corresponds to the classical momentum observable. There are higher-dimensional versions of Weyl algebra too. Incidentally, the Hamiltonian of the quantum harmonic oscillator happens to belong to this algebra. By Stone-Weierstrass theorem the Weyl algebra has a unique (up to unitary transformations) representation on a Hilbert space which is a reflection of a possibility of reconstruction of a manifold from the algebra of observables.

Now when we have the classical and quantum chunks of data, what can one do mathematically with them? Well, there is a problem of quantisation, i.e. given a classical mechanical system (a symplectic manifold) produce a Hilbert space and the algebra of observables that give back the classical system in some sense.

The deformation quantization is only interested in the algebra of observables. Given a symplectic (or more generally poisson) manifold, we are interested in a sheaf of algebras \(\mathcal A\) on this same manifold over formal power series \(k[[\hbar]]\) such that \(\mathcal A/\hbar\) gives back the structure sheaf. Since every symplectic manifold locally looks like a model manifold, the problem reduces to finding a way to glue the Weyl algebras consistently. There is a quantization procedure due to Fedosov. Kontsevich has proved that any Poisson manifold admits quantization.

In geometric quantization one seeks to construct the Hilbert space as the space of sections of certain line bundle on the symplectic manifold (which can be made canonical (?) given some extra data on the manifold). The notable names here are Kostant, Kirillov and Soriau. The geometric procedure is a generalization of the folowing observation on how one can make the Weyl algebra appear.

Let \((V,\omega)\) be a symplectic vector space. Consider the Heisenberg group \[ 1 \to \mathbb{R} \to H \to V \to 1 \] which is a group extension of \(V\) by \(\mathbb R\) given by a group cohomolgy 2-cocycle constructed from \(\omega\). Then the universal envelopping algebra of the Lie algebra of \(H\) is the Weil algebra.

What about the Lagrangian approach ? Due to probabilistic nature of quantum mechanics, we have to understand better what evolution is. Suppose that in moment \(0\) the system was in a state represented by wave function \(\psi_0\) and after a period of time \(t\) it has evolved into a state represented by wave function \(\psi_t\). Schroedinger equation tells us that \[ \psi_t = e^{-\frac{i}{\hbar}tH} \psi_0 \]

Assume that the system invoves only one particle so that the time evolution operator on the right is an integral operator \[ e^{-\frac{i}{\hbar}tH} \psi_0 = \int K(x', t', x, t) \psi_0(x) dx \] The kernel, \(K(x',t',x,t)\) can be interpreted as probability density of the particle being at \(x'\) on moment \(t'\) if it were at \(x\) on moment \(t\). The Feynman path integral formulation of quantum mechanics allows to compute \(K\). Namely, one takes the classical action functional \(S\) and integrates over the ``space of all paths’’ between \(x\) and \(x'\) \[ K(x',t', x, t)=\int_{paths} e^{-S(\gamma)} D\gamma \] which means that paths with smaller action contribute more (a reflection of the principle of least action).

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