Boson Condensation and Collective Modes

Quantum field theory provides a two-level description for a system  

As a next step, the set of the physical states, or Fock space, of the system is introduced. It is denoted by 0 and contains all the states obtained by repeated application of the quasiparticle creation operators on the ground state, or vacuum, 0 . The quasiparticle creation operators are functions of the fields (x). 

At the first level of description the forces acting among the system components appear through the functional while, at the second 

Equation (3.3) is the bridge between the deep underworld where the forces are at work and the apparent world where observations are performed. In Eq. (3.3) a and fe are wave-packet states, which belong to the Fock space 0 and [ (x)] is a functional of (normally ordered) products of the (x) fields. 

Let us now investigate how a symmetry at the dynamical level would affect the phenomenological description. Consider a transformation which leaves Eq. (3.1) invariant, namely, let g be an element of the symmetry group of Eq. (3.1)  

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