How are free fields and static fields in QFT interconnected?

[Lapidus]

- a free field has a definite number of photons, which implies that the expectation value of the free field is zero

- a static field has a indefinite number of photons (or none at all, rather), but its (classical) field strength can be measured precisely

How are free fields and static fields related in QFT then? They seem somewhat the opposite if what I just wrote is true. (Which I'm not sure of!)

 

[A. Neumaier]

- a free field has a definite number of photons, which implies that the expectation value of the free field is zero

- a static field has a indefinite number of photons (or none at all, rather), but its (classical) field strength can be measured precisely

How are free fields and static fields related in QFT then? They seem somewhat the opposite if what I just wrote is true. (Which I'm not sure of!)

You shouldn't post the same text in two different threads...
I answered in the other.

 

[Lapidus]

You shouldn't post the same text in two different threads...
I answered in the other.

Sorry, I thought, my post was overlooked in the other thread. Thanks for your answers, which were...

In QED, the Coulomb field is an interaction term in the Hamiltonian, written in terms of the electron field rather than the photon field. But electron fields of course also wiggle (de Broglie waves)!

and referring to my statemennt that a free field has a definite number of photons:

No. One cannot assign photons to a quantum field, except in a very loose sense.
It is the state that may or may not have a definite number of photons. The field has different expectation values in different states, and in most states, it is not zero.

Ok, but does not the number operator commute with the Hamiltonian in the free theory, and in the interaction theory it does not?

And are you saying that static fields are interacting fields?

thank you

 

[Avodyne]

- a free field has a definite number of photons

A "free field" is a field with an equation of motion that is linear and homogeneous in the field or, equivalently, a field whose lagrangian density is quadratic in the field.

The number of particles (photons for the E&M field) is a property of the state, and not whether or not the field is free (though even defining what is meant by a particle for an interacting field is subtle).

For a free field, the number of particles is a conserved quantity.

In QFT, the best analog of a classical field (static or not) is a coherent state, which does not have a definite number of particles. The field strength is subject to quantum uncertainty, and so cannot be measured precisely; for the E&M field, see e.g.
http://quantummechanics.ucsd.edu/ph130a/130_notes/node466.html

 

[Lapidus]

Thanks Avodyne for that crystal-clear answer. And the great link.


So as I understand, quantum fields, free or interacting, have to 'wiggle', have to be a 'mattress of coupled harmonic oscillators'. Static fields do not work in a quantum theory due to quantum uncertainty. (Though, coherent states describe them pretty well)