Classical Field Theory for a system of particles

[LarryS]

In classical field theory, the field, φ, is usually constructed from a very large number of coupled harmonic oscillators. Let's say our φ consists of just electrons.

What does φ best represent physically, a very large number of electrons or can it represent just a few electrons? Which is the best fit, or does it matter?

As always, thanks in advance.

 

[MrRobotoToo]

In classical field theory, the field, φ, is usually constructed from a very large number of coupled harmonic oscillators. Let's say our φ consists of just electrons.

What does φ best represent physically, a very large number of electrons or can it represent just a few electrons? Which is the best fit, or does it matter?

As always, thanks in advance.

In QFT the fields are position-dependent operators that create/annihilate particles at different points in space. They're formally quite analogous to the creation and annihilation operators that one encounters in the quantum mechanical treatment of the harmonic oscillator.

 

[hilbert2]

A quantized field can also have states that are not eigenstates of the particle number operator, e.g. don't have an exactly specified number of electrons or other particles.

An unquantized classical field doesn't usually represent any kind of particles (unless it's some kind of classical probability density field), the particle-wave duality is a completely quantum mechanical thing.

Only fields that have a linear field equation can be represented as a system of harmonic oscillators.

 

[LarryS]

A quantized field can also have states that are not eigenstates of the particle number operator, e.g. don't have an exactly specified number of electrons or other particles.

An unquantized classical field doesn't usually represent any kind of particles (unless it's some kind of classical probability density field), the particle-wave duality is a completely quantum mechanical thing.

Only fields that have a linear field equation can be represented as a system of harmonic oscillators.

Leonard Susskind has an internet video course on "Classical Field Theory" in which he discusses how a classical field of charged particles becomes gauge invariant by coupling the field (covariant derivative) to the EM four-vector. But he never really goes into any detail regarding the physical nature of the "system of charged particles". So, my question is kind of within that context.

Are you saying that the classical field that he describes has no physical purpose other than a field that is to eventually be quantised?

 

[hilbert2]

I've seen some kind of hydrodynamical models where an electrically charged gas that has a density field $\rho$ and a current density field $\mathbf{j}$ is coupled with electric and magnetic fields, but those models assume that matter is continuous and doesn't consist of particles. I can't see how anything useful could be done with fields like Klein-Gordon or Dirac fields without quantizing them, but it's possible that I'm wrong.