Question about understanding quantum fields 1. How do we interpret colisions in QFT formalism?
How did we know, when developing the theory, that we are getting the fields which describe creation of particles? How does one excite the field to get a particle out of it? By smashing particles together? So, are these fields through the particles interacting with some other fields and creating other particles in some other fields? So, if we colide electron and anti-electron, they can give us something, some photons, lets say. So what happend? Did electron interact with photon field? Is it somehow coupled to it? What is then a clasical electrostatic field? Is it something that an electron creates arround itself or something that is there and the electron is coupled to it? So if charged particles attract, they do it because the are both coupled to this photon field? So in violent interactions, we can even disturb electron field and the final result can be creation of some photons? In our formalism electrons were destroyed and photons were created.
2. How does the Hilbert space look like?
So, does our Hilbert space acomodate for different kinds of particles? If so, then we can say that system went from one state to some other state and that there is some amplitude for that. How do we know that our formalism is actually giving us these amplitudes? And even if they do, is it not all a big mass of infinitely many ways of geting from one state to some other state?
Another thing...it looks like that in nature we rarely see superposition of states with some indefinite number of particles, it happens for a brief time and then, after the colision, we get some definite number id est, eigenstate of a number operator...
 A: First of all there is no real classical equivalent to spin.  Before the Dirac equation we described this using Pauli spin as a complete intrinsic property of the electron and other elementary particles.  This was thought to be a completely quantum degree of freedom (and still is).  So trying to visualize the classical nature of a spin field might not be the right thing to do.
To get a better understanding of the information being conveyed in QFT consider reviewing QM.  The QM operators correspond to classical degrees of freedom of what was assumed to be a particle system.  So the wave function describes the probability to find the particle at a given location with a given momentum and other such state variables.  This is a useful paradigm for extending the meaning of QM to QFT.  We start with a configuration space, define the maxima set of independent degrees of freedom (in the classical sense) and their momenta using Hamiltonian theory.  Then we impose the formalism of quantization on this classical system.  In the case of a classical field, any field for example, the classical degree of freedom is a shape of the field.  This can be though of as an infinite (continuously infinite) number of locations each corresponding to an amplitude at each point of space.  The particle has one and only on such variable which can take any real number.  A field has a space of functions that can take an infinite number of shapes over all of space, or some region of space if there are boundary conditions. 
The field equivalent of the position operator would be the shape or amplitude operator, A(x, t), and the corresponding momentum operator would be -ihbard/dA(x, t) where differentiation is elevated to functional derivative.  This is a completely valid formalism and in my humble opinion easier to conceptually connect classical field theory to QM with meaning.  However this is a messy and intractable way to quantize a field.  What we are usually taught is to go into momentum space and use the harmonic oscillator formalism, annihilation creation and number operators.  Now each represents the number of "coherent" states present in the field configuration.  Coherent has slightly different meanings in different fields, maybe I should say stationary.  So it is like describing the spectral density of a configuration using quantum occupation numbers.  In theory one can transform back and find the A(x, t) for a given occupation number state but this usually is of no use in practice.  Especially in particle physics (the mother of QFT) since we are always doing collider experiments and really only want an estimate of the cross section.  
ADDITION:  There is a type of field that obeys different algebra than ordinary real or complex numbers, Grassmann algebra.  One can have a classical Grassmann field and then quantize it.  I think this may be related to spin in some abstract theoretical way.  You can apply QM to anything. You just first need to define the DoF and build a Lagrangian for it.  From there everything follows (it may be a mess).  I should also say that the quantum wave function for a field would yield the probability to observe the field in a given shape.  
I hope this helps a little and expect either a lot of answers or an eventual [close] of this question.  
