There is a class of problems in mechanics having exact analytical solution.
The basic historical example in classical mechanics is the problem of sun and comet. Newton’s laws give three nonlinear differential equations for relative coordinates of sun and comet. Solution of them for arbitrary initial data is given by conic section in the ecliptic plane – hyperbola, parabola or ellipse.
Corresponding example in quantum mechanics is the Hydrogen atom. Schroedinger equation for electron wave function is a partial differential equation. Stationary wave function is expressed in terms of radial spherical function and angular Legenrde functions. As the result, the Hydrogen atom has the simple harmonic spectrum.
These two basic two-body problems of celestial and quantum mechanics are called solvable or integrable. By “integral” we understand a solution of differential equation or, more widely, a quantity conserved in time. Existence of such quantities (energy, angular momentum and a specific quantity called Runge-Lentz vector) following from symmetry of equations guarantees the analytical solvability.
A three-body problem is in general non-integrable. The celestial system of a comet, sun and a planet can’t be solved in exact analytical form for arbitrary initial data. One known approach to this problem is to solve two two-body problems comet-sun and planet-sun first, and then consider the forces between comet and planet as a small perturbation. The result is an infinite perturbative series. Another approach is machine numerical computations.
However, there does exist a class of many-body and even infinitely many-body solvable problems. In classical mechanics, it is a specific class of mechanical systems with many degrees of freedom, systems described by array of coordinates Φi(t), i=1,2,…,N. Celestial two-body problem is the case of three degrees of freedom. The case of infinitely many degrees of freedom is a classical field theory, Φi(t)→ Φ(x,t), the space position vector x stands for continuous index of field coordinate Φ. This is the class of integrable non-linear partial differential equations, or soliton equations. The famous examples are: Koreteveg-de-Vries, sine-Gordon and nonlinear Schroedinger equations in two-dimensional space-time and three-wave resonant equations in three-dimensional space-time. In all these systems in addition to a field-theoretical Hamiltonian there is an infinite hierarchy of higher time-conserved functionals. Equations of motion can be solved by means of specific non-linear integral change of variables – traditionally this change of variables is called the Inverse Scattering Method.
Classical field-theoretical equations can also be defined on a space-time lattice as partial difference equations. Continuous equations correspond to infinitely small spacing parameter; difference equations are nevertheless integrable for finite spacing parameter. Discrete difference equations have more reach structure of parameters; their soliton and algebraic-geometry solutions are more general than those in continuous limit. Also, the discretization makes the number of mechanical degrees of freedom finite. This is very important for a quantum-mechanical quantization. The limit when the number of degrees of freedom goes to infinity is thus a self-consistent field-theoretical quantization (to my opinion, this is only self-consistent formulation of non-linear quantum field theory).
Quantum-mechanical quantization is a replacement of real fields of classical discrete equations by operator-valued fields such that operator-valued equations of motion constitute Heisenberg evolution. Operator-valued fields at every point of discrete space-like surface form local algebra of observables, a well defined quantum theory is a subject to its representation. In two-dimensional quantum models the algebra of observables is related to quantum groups. In three-dimensional world – quantized discrete three-wave system – the algebra of observables is either Bose or Fermi q-oscillators. Quantum theories essentially depend on a choice of representation.
It is somehow a wrong and misleading idea to separate continuous and discrete equations as well as classical and quantum systems. It resembles an artificial separation of Natural Philosophy to Physics and Mathematics.
The most famous example of quantum theory is the Heisenberg magnet, or XXZ spin chain. Stat-mechanical form of it is Baxter’s six-vertex model. This is the case of quantum group Uq(sl2) However, almost all known two-dimensional quantum integrable models are descendants of the three-dimensional q-oscillator system. Integrability in quantum world means the existence of big enough set of mutually commuting operators – quantum counterpart of classical integrals of motion. A set of eigenvalues of quantum integrals of motion corresponds in general to a complete set of quantum numbers defining an eigenstate. There is a specific way to write the spectral problem for two-dimensional quantum systems as a set of transcendental algebraic equations – the Bethe Ansatz.
A way to produce complete set of quantum integrals of motion is closely related to the self-consistency of the quantization. Algebraic self-consistency equations are called simplex equations: the Yang-Baxter triangle equation in two-dimensional world, Zamolodchikov tetrahedron equation in three-dimensional world and n-simplex equations in n-dimensional world. Simplex equations can be viewed also as self-consistency of non-perturbative definition of path integral measure. Simplex equations guarantee commutativity of transfer matrices – generating functions of quantum integrals of motion. The transfer-matrix method sometimes is called Quantum Inverse Scattering Method.
Note, any linear classical field theory, continuous or discrete, and its quantum counterpart – a quantum theory of free particles – are always integrable for any dimension of space-time.
Three-dimensional q-oscillator system is very sensible as the quantum field theory in 2+1 dimensional space-time. At every point of space we have an elementary Schwinger oscillator or even a set of Bose oscillators (gauge fields) and Fermi oscillators (matter). Relativistic evolution operator describes interaction and propagation of oscillator’s excitations. All these processes can be presented graphically as Feynmann diagrams on a lattice. The q-oscillator system is thus the integrable Quantum Electro-Dynamics in 2+1 dimensional space-time.
There is a huge amount of open yet unsolved problems in the theory of multidimensional quantum integrability. We still do not know a proper form of multidimensional spectral equations except for a few particular cases of spin lattice and two-dimensional non-relativistic Bose gas. For instance, we do not know spectral equation for relativistic evolution operators in 2+1 dimensional discrete space-time, especially for non-compact integral representation of q-oscillators. Even known spectral equations need an essential further study in the thermodynamic (field-theoretical) limit. It is also not known how to equip the q-oscillator scheme by an isotopic symmetry of standard model. And, of course, there is a problem of proper formulation of four-simplex theory. We know a few quite limited solutions of the four-simplex equation. These solutions do not define a proper four-dimensional quantum theory. However, the existence of solutions is a big hope for further study of this subject.
The reader can find some mathematical details here .