Confirming Uncertainty: Electromagnetic Fields in Hilbert Space

[snoopies622]

I think this is right, but could someone confirm (or deny) this for me?

While a particle like an electron - or a finite set of particles for that matter - is represented by a single normed vector in Hilbert space which is acted on by operators such as ones for energy, position and momentum, an electromagnetic field is represented by an uncountably infinite number of such vectors, one at every point in space and time, each of which are acted on by operators such as energy, electric field strength (and direction) and magnetic field strength (and direction).

 

[bapowell]

Not quite. According to quantum field theory, each particle is an excitation of a more fundamental object -- the quantum field. In this view, photons are seen as the excitations of the electromagnetic field. The same is true for electrons -- they are excitations of the electron field that carries an infinite number of degrees of freedom, just like the EM field.

 

[snoopies622]

Are the EM field and the electron field represented by Hilbert spaces? Fock spaces?

 

[tom.stoer]

They are represented by rays in Fock space, where a Fock space is just an infinite product of Hilbert spaces.

 

[snoopies622]

Is a ray in Fock space the same thing as a Fock state, as defined here

http://en.wikipedia.org/w/index.php?title=Fock_state&printable=yes'

?

 

[tom.stoer]

basically yes;

a ray {v} is the equivalence class of all vectors v, w, ... with w=cv where c is a non-zero but otherwise arbitrary compex number; if you can normalize all vectors (e.g. for the harmonic oscillator) "state" is certainly enough