Exploring the Concept of Quantized Electromagnetic Fields in QFT"

[snoopies622]

What does it mean for an electromagnetic field to be quantized? If I have a proton at point A, then classical physics tells me that at an electron at point B feels a constant electrical force described by Coulomb's law. If the field is quantized, does this mean that sometimes a force is felt there and at other times it isn't? Or that only certain, discrete field strengths are permitted so the distribution of force per unit charge is not continuous? How does the electron behave differently?

 

[Shawakaze]

I do not have an immediate answer to that. but yes, a field is quantized for particular allowed spacetime "points" just like first quantization where one has allowed energies, momenta and coordinates. I bet the electron's field interacts with the em field, though one would have to calculate for that. that's just my thinking anyway!

 

[Sojourner01]

I am not sure that spatial quantisation is really the right way of looking at it. My understanding is that quantisation of a field implies that propagating disturbances in the field, i.e. particles, come in discrete lumps, i.e. you can't have probability distribution of observed field strength multiplied by a fractional value from a given process, you either have a 'full' one-particle probability distribution, or the vacuum probability distribution.

 

[Shawakaze]

Ok you've got a point there, but the way I see it all dynamical variables cannot exist without associating them to a wave or particle thus if permitted to say that in quantum field theory "spacetime" is represented by a species of particles or waves hence the quantisation of the field. I mean after all we quantize observables isn't it?

 

[calhoun137]

In many of my answers I continue to stress the same point: READ LANDAU!

The reference you want is volume 4, section 2, "quantization of the free electromagnetic field". You must however fully understand volume 2, section 51, "The Fourier resolution of the electrostatic field".

Let's briefly review the main idea of quantizing the electromagnetic field.

First of all, in the theory of the quantum harmonic oscillator, the energy levels are quantized; and a very convenient mathematical method is to use the raising (creation) and lowering (annihilation) operators (which raise and lower the energy level by 1).

The Fourier coefficients of the electromagnetic field correspond to the creation and annihilation operators for creating and destroying photons of a given energy.

This physical interpretation of the quantization process distributes the energy of the electromagnetic field in discrete energy packets (photons), just like in a simple quantum oscillator.

When quantizing the electromagnetic field, it's best to work with the Heisenberg picture of quantum mechanics, which is based on Poisson brackets, and not the Schrodinger picture. The idea is to put the field equations in terms of Poisson brackets, and then to apply the rule that a Poisson bracket becomes a commutator for operators.

 

[snoopies622]

...volume 4, section 2, "quantization of the free electromagnetic field". You must however fully understand volume 2, section 51, "The Fourier resolution of the electrostatic field".

Thanks for the references!

I'm still having some trouble with this concept. I'm reading that in an electric (or magnetic) field every point in space is like a quantum harmonic oscillator in that it can only have certain discrete energies. But shouldn't that be discrete energy densities? After all, if I were to integrate over an infinite set of points, each with its own energy, than even for a finite volume of space I'd have too much.

 

[calhoun137]

unfortunately, there zero point energy of the field is infinite (each level has a zero point energy (1/2)h-bar). The energy of a certain energy level is just the integral of the energy density, so it's not a big deal. Actually if you look at the Poyniting vector of the EM field and the Maxwell Stress Energy Tensor, you should be able to figure out how the momentum/energy density of the field behaves under quantization.