# Is there one wavefunction per field? [closed]

Is the big picture of quantum field theory that:

• There are fields (EM, electron, Higgs, gravity, etc.)
• A field can be described by a wavefunction indicating the probability density of 1 or more 'particles,' which are just localizations/quantizations of the field
• One field affecting another is an 'interaction'
• Would that mean that photons, as the force-carrier of the EM field, have their own field? I thought that photons were just localizations of the EM field.
• Why aren't electrons similarly localizations of the EM field?

I know this is a very broad question for a Stack Exchange site, but I have yet to find a treatment that covers these questions that isn't either uselessly general (i.e. 'popular') or extremely, extremely specific (i.e. textbooks). Even the few in-between treatments seem to avoid these very fundamental concepts.

• "A field can be described by a wavefunction" nope. No wfs in QFT. And "One field affecting another is an 'interaction'" is a rather vague and imprecise definition of "interaction". If a textbook is "extremely specific" for you, then I don't think we could do any better. – AccidentalFourierTransform May 26 '16 at 19:27
• @AccidentalFourierTransform, thank you—I am using some textbooks, what I should have said is "I can't ask textbooks broad questions" and/or "I'd like to keep to just a few textbooks" :) – T3db0t May 26 '16 at 19:42
• No, it's more complicated than that. A classical particle's position can be described by a number, while a quantum particle's state is described by a wavefunction, i.e. a function from positions to numbers. A classical field's configuration can be described by a function, and a quantum field configuration can be described by a 'wavefunctional', i.e. a function from functions to numbers. So the state of a quantum field is much, much 'larger'. – knzhou May 26 '16 at 20:13
• Electrons are a separate quantum field. Why would you think they should be part of the EM field? That isn't even true in classical mechanics. – knzhou May 26 '16 at 20:14
• @knzhou I think the role of charge in an EM field was throwing me off. – T3db0t May 26 '16 at 21:36

In the quantum mechanical description of any physical system, including a quantum field or a collection of interacting quantum fields, there is always one state vector – one collection of numbers (probability amplitudes) that generalizes what is referred to as the "wave function" in quantum mechanics of particles.

In quantum field theory, a better name is a "wave functional". Just like for the wave function, it is a complex function on the configuration space. For example, for a coupled scalar and Maxwell field, the state vector is the wave functional $$\Psi [ \vec A(x,y,z), \Phi(x,y,z) ]$$ For every configuration of the field $\vec A$ and the field $\Phi$, there is one complex number. Usually, different bases than this "functionally continuous" bases are used to describe the wave functional.

At any rate, this wave functional is a function of infinitely many variables. It has enough information to "include" the particle-based wave functions $\psi(\vec x_1,\vec x_2,\dots \vec x_n)$ for an arbitrarily large number $n$ of particles.

For fermions such as electrons, the wave functional must also be a function of Grassmann (anticommuting) variables $\eta(x,y,z)$ describing the Dirac or Weyl or Majorana fields.