Quantization Doubts: Techniques, P & Functional Derivative

[eljose79]

I have some soubts on quantizying theories..how many techniques are there?..is always as simple as to take p=-ihd/dx ?..or the functional derivative?..thanks.

 

[selfAdjoint]

The aim is to impose the uncertainty principle on your theory by requiring the operators in it to obey the equal time commutation laws. The methods of attaining that aim are various, but that is the purpose of all of them.

 

[Javier]

There are actually many methods of quantization ("canonical", "covariant", "geometric", to name a few popular ones). There are certain axioms of quantization (a la Dirac), and in some methods of quantization one relaxes one or more of the conditions. In canonical quantization, the one used in intro quantum mechanics of point particles, we can just say that we take the phase space coordinates (in our case position and momentum), and turn them into operators on a suitable Hilbert space (you can read about this "suitable Hilbert space" in books as well as I can discuss it here). The Poisson bracket between the phase space coordinates ({p,x}) is mapped to a commutator (this is the handwaving way of explaining this, and a mathematician familiar with this stuff would happily tell you about the subtleties here, but we are in a physics forum :).
Anyway, the "p=-id/dx" you refer to is called a 'representation' of the operator on a 'representation space'. The representation space you have picked (representing the Hilbert space mentioned above) is the space of wavefunctions (in particular, the "coordinate rep"), and in this rep, the operator, p, takes the form you wrote abovem while the operator x is represented as multiplication by the number x.
You could have chosen the 'momentum representation', meaning your rep space is the space of (wave)functions of momentum rather than position. Then the operator p would be represented by multiplication by p, while the operator x would be of the form d/dp.
The wavefunction rep of qm is what Schrodinger used. You could instead use the "state vector" representation (used in the intro QM text of Liboff and by Feynman in his lectures in physics, vol 3). This is how Heisenberg formulated qm.
All of this stuff does not represent what we would call different methods of quantization, rather just different representations of the canonical quantization of non-relativistic point particles.
The issue of different methods of quantization becomes more significant in doing *relativistic* quantum mechanics where we must talk about quantum field theory (QFT). There, it turns out that Feynman's "path integral quantization" is more suitable, but to see why you have to study QFT.
Cheers

 

[selfAdjoint]

Sure, and when you get to gauge theory you have Fadeev-Popov and BRST. But the aim is always to get to a theory where uncertainty is built in.