Why do we need creation and annihilation operators in QFT? 2. Why do we need creation and annihilation operators?
Main point is that a particle can be created by creation operator and destroyed by the annihilation operator. But to destroy a particle is to change from one vector in Hilbert space to another, right? So these operators act on states and actually change them. In a definite way. So why do we need them? We can just write the states as we want...if I know I want a state with two particles of definite momenta I can just write it, why use these operators to create particles? Obviously, I am missing something here. What is the real purpose of creation and annihilation operators? Is it because through these operators, fields are represented better than through momentum or energy operators? So, combinations of creation and annihilation operators give us our observables?
 A: "Need them?"  We don't need anything.  They arise naturally.  If you express the field equations in momentum space you get a set of harmonic oscillator equations and the algebra of that system applies to the field.  These states form a basis for Hilbert space, but are not the only ones.  Certainly one can use any number of orthonormal bases but the occupation number basis is the easiest one to use.  The solution is easy to arrive at and easy to use and they map nicely to the processes we are trying to study.  I wouldn't say we need them I would say the field equations have given them to us through the process of quantization. 
A field theorist would start with a field equation for some A(x, t) as a scalar, vector, spinor, or tensor and then apply QM to it.  I guarantee that if you do this for a "free field" you will arrive at a harmonic oscillator formalism.  Particle theorists are more pragmatic.  They tend to model processes using existing paradigms and then look for a field theory that fits that model.  There is nothing wrong with this.  this is how science works and is a pragmatic approach to particle phenomenology.  If you are reading a text or paper by a particle phenomenologist they may start with the harmonic oscillator formalism and just start building up processes.  This can look a little informal to a mathematician or pure theorist but it is just as valid or an approach as starting from a classical field theory and trying to quantize it.   
