Magnetic Monopoles in Quantum Mechanics

Abstract

In this chapter we examine the dynamics of a two-dyon system in the framework of non-relativistic quantum mechanics. In quantum mechanics, the description of a physical system requires the introduction of a Hilbert space $\mathcal{H}$, whose elements $\psi$ represent the states of the system, and of a self-adjoint Hamiltonian operator H which determines the time evolution of the states. The operator H is obtained from the classical Hamiltonian via the recipe of canonical quantization, and the classical Hamiltonian descends, in turn, from the classical Lagrangian L of the system, via a Legendre transformation. Finally, the Lagrangian is uniquely determined by the classical equations of motion, modulo total derivatives. A prototypical example of this procedure is provided by a non-relativistic charged particle in the presence of an external electromagnetic field $F^{\mu \nu }=(\mathbf {E},\mathbf {B})$, whose equation of motion is thus the Lorentz equation

$$\begin{aligned} m\mathbf {a}=e\left( \mathbf {E}+{\mathbf {v}\over c}\times \mathbf {B}\right) \!. \end{aligned}$$

If the tensor $F^{\mu \nu }$ satisfies the Bianchi identity $\partial _{[\mu }F_{\nu \rho ]}=0$, we can introduce a vector potential $A^\mu =(A^0,\mathbf {A})$ such that $F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu$, and in this case we know that there exists a Lagrangian giving rise to the equation of motion (21.1), see formula (21.4) below. With this Lagrangian as starting point, the above process of canonical quantization then proceeds without meeting any obstacle, see Sect. 21.1.