# I Canonical Quantization

1. Nov 25, 2017

### Silviu

Hello! I read some books on QM and QFT but I didn't really noticed (or I missed it?) a proof for the canonical quantization. For example, for energy and momentum it makes sense to have opposite signs, due to Minkowski metric, be related to the variation of space and time, due to Noether theorem or have an $i$ in order to be hermitian, but this is not a proof. Can someone explain to me how can you derive this or point me towards a derivation (not only energy and momentum, but all this theory in general)? And as a side note, why isn't this proved (or at least given some rough clues) in the physics books (at least not in all of them)? Thank you!

2. Nov 25, 2017

### Staff: Mentor

What books?

I'm not sure what you mean by this.

3. Nov 25, 2017

### Silviu

Griffiths and Liboff for QM and Pesking for QFT. And I mean why do p and E in classical physics, take the form they do in QM (and all the rest, x and Poisson brackets etc.). I don't find it obvious for E to become $i\hbar\frac{\partial}{\partial t}$, for example. So I assume there is a mathematical motivation for this, and hence a proof of how one classical variable turns into an operator i.e. I guess these were not just guessed and as they worked they were just used but they have a mathematical motivation.

4. Nov 25, 2017

### Staff: Mentor

It doesn't. Why do you think it does?

5. Nov 25, 2017

### Staff: Mentor

6. Nov 25, 2017

### A. Neumaier

It does in the Klein-Gordon equation for a single relativistic particle.

7. Nov 25, 2017

### vanhees71

Canonical quantization is a heuristic idea how to motivate the rules of quantum mechanics in introductory QM lectures. It's not a mathematically well defined procedure and as such only works well for position and momentum in Cartesian coordinates.

The true way from both a physical and a mathematical point of view are symmetries (related strongly with conservation laws according to Noether's theorems) and their realization in quantum mechanics in terms of unitary ray representations on a Hilbert space. The commutator relations of observable operators (or in mathematical terms the "algebra of observables") follow from the mathematics of the underlying Lie symmetry groups and their corresponding Lie algebras. You can derive the specific way of non-relativistic quantum theory (why is there a mass, why is there spin, why are there mass and spin superselections rules etc. etc.) from a careful study of the unitary ray representations of the Galileo Lie algebra and the resulting quantum version of the Galileo symmetry of classical mechanics. For a very good introduction, see the excellent text book by Ballentine, Quantum Mechanics - A modern development.

8. Nov 25, 2017

### Silviu

It actually does. When you derive the KG equation, you write the relativistic energy of the particle, and then you do the transformation that I just mentioned. My question is why do you do that transformation. What is the motivation behind it?