Canonical quantization of Electrodynamics: physical intuition ?

In summary, the conversation is about the canonical quantization of electrodynamics and the search for intuition in understanding the concept. The suggestion given is to follow a specific order of steps to build intuition, starting with the canonical quantization of a single harmonic oscillator and then progressing to more complex systems. The use of analogies and the correspondence principle can also aid in understanding.
  • #1
maajdl
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TL;DR 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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  • #2
maajdl said:
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.
 
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  • #3
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!
 
  • #4
maajdl said:
Would you think one could also recover a story like that abour wave packets ?
In other words, what about the commutation relations?
I'm not sure what's the problem/question here?
 

FAQ: Canonical quantization of Electrodynamics: physical intuition ?

1. What is the concept of canonical quantization?

Canonical quantization is a mathematical procedure used to quantize a classical field theory, such as electrodynamics. It involves replacing the classical fields with operators that satisfy the commutation relations of quantum mechanics, allowing for the description of the system in terms of discrete quantum states.

2. How does canonical quantization apply to electrodynamics?

In electrodynamics, canonical quantization involves replacing the classical electromagnetic fields (electric and magnetic fields) with quantum operators, and expressing the Hamiltonian in terms of these operators. This allows for the description of the electromagnetic field in terms of quantized particles called photons.

3. What is the physical intuition behind canonical quantization?

The physical intuition behind canonical quantization is that the classical fields are made up of discrete, quantized particles, and that the energy of these particles is quantized. This approach allows for a more accurate description of the behavior of particles at the quantum level, where classical mechanics no longer applies.

4. How does canonical quantization relate to the uncertainty principle?

The uncertainty principle states that certain pairs of physical properties, such as position and momentum, cannot both be known to arbitrary precision simultaneously. In canonical quantization, the operators representing these properties do not commute, resulting in a similar uncertainty relation. This means that the more precisely one knows the value of one property, the less precisely they can know the value of the other.

5. What are the limitations of canonical quantization in electrodynamics?

One limitation of canonical quantization in electrodynamics is that it only applies to linear systems, meaning that the fields and particles must obey linear equations. It also cannot fully account for the effects of gravity, as it does not incorporate the principles of general relativity. Additionally, canonical quantization does not take into account the interactions between particles, which can be important in some situations.

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