- Nonlocal Operators
The description of many-particle quantum mechanics given till now does not give us much insight into how one may go about computing the Green function of quantum systems with infinitely many coupled degrees of freedom. Apart from the free particle case and perhaps the harmonic oscillator, there are precious few exact computations of Green functions of many-particle systems. Even the approximate methods are found wanting since we have already alluded to the ‘uncontrolled’ nature of most of the approximations that have been proposed to date. There is one method that offers some hope in this regard. This method is (wrongly) called ‘bosonization’. This method will be the main focus of much of this and the next chapter. The main mathematical tool used is the introduction of operators that are ‘non-local’ in a sense to be made precise later, which enables the exact computation of the asymptotic properties of the Green function (G(x− x′, t− t ′) in the regime |x− x′| → ∞ and/or |t − t ′| → ∞) under some further restrictive assumptions. We wish to ease into this subject through the study of quantum vortices in charged bosons where the notion of nonlocality makes its presence felt in a relatively more familiar setting.