# A Canonical quantization of Electrodynamics: physical intuition ?

#### maajdl

Gold Member
Summary
Going from Poisson brakets to commutators is very easy but not very intuitive.
Hello,

I am freshly retired and enjoy going back to the fundamentals.
I followed the wonderful courses by Alain Aspect on Coursera on Quantum Optics 1 and 2 .
The quantization of Electrodynamics is really easy stuff.
Just follow the correspondence between Poisson brakets and Commutators ... and start counting photons !!!

This correspondence principle comes with some intuition when it is applied to electrons or particles.
This is because the conjugated variables x and p are intuitive from classical mecanics,
and because of the packet wave view which brings its own intuition.

However, when discovering that q and p are conjugated variables for a mode in Electrodynamics,
there is apparently no bonus intuition that come together.
At least because q and p are not the position and momentum of a particle.
Quantization remains easy, but looks a bit like magics!
(specially with the creation and annihilation operators!)

Therefore I am looking for some intuition about the Canonical quantization of Electrodynamics!
Would you have some suggestion?

Thanks

Michel

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#### Demystifier

2018 Award
Therefore I am looking for some intuition about the Canonical quantization of Electrodynamics!
Would you have some suggestion?
To build the missing intuition, do the following in the prescribed order:
1. Do canonical quantization of a single harmonic oscillator, you will find it intuitive because it's just a special case of canonical quantization of a particle.
2. Do canonical quantization of a chain of $N$ coupled harmonic oscillators, it's intuitive because it's still about $N$ particles.
3. In 2. the distance between the neighboring particles in the chain is $a$. Consider the limit $a\rightarrow 0$ and $N\rightarrow\infty$, with $Na$ kept finite. This limit corresponds to the continuum.
4. Do canonical quantization of a scalar field in 1 spatial dimension and observe that it looks exactly like 3. The conceptual relation between 3. and 4. is the most important step in this intuition building.
5. Do canonical quantization of a scalar field in 3 spatial dimensions, it's just a straightforward generalization of 4.
6. Do canonical quantization of a massive vector field in 3 spatial dimensions, it's just a straightforward generalization of 5.
7. Do canonical quantization of a massless vector field in 3 spatial dimensions, which is nothing but canonical quantization of electrodynamics.

#### maajdl

Gold Member
Good idea.
The analogy with a chain of oscillator, is very intuitive indeed, since it is pure mechanics.
I should also remember the analogies Maxwell and other at that time had in mind when building electrodynamics.
A canonical change of variable and the usual correspondence principle will do the job.

Would you think one could also recover a story like that abour wave packets ?
In other words, what about the commutation relations?

Thanks!