Canonical Quantization: Proving the Theory

[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!

 

[PeterDonis]

I read some books on QM and QFT

What books?

a proof for the canonical quantization

I'm not sure what you mean by this.

 

[Silviu]

What books?
I'm not sure what you mean by this.

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.

 

[PeterDonis]

I don't find it obvious for E to become $i\hbar\frac{\partial}{\partial t}$, for example.

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

 

[PeterDonis]

As far as the position and momentum operators are concerned, the basic argument for the canonical commutation relations is that they correspond to the Poisson bracket of classical mechanics, as described briefly here:

https://en.wikipedia.org/wiki/Canonical_quantization#Single_particle_systems

 

[A. Neumaier]

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

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

 

[vanhees71]

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!

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 textbook by Ballentine, Quantum Mechanics - A modern development.

 

[Silviu]

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

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?