Basics of quantum field theory

So far, we have been able to compute the large scale amplitude of the cosmological perturbations as a function of some quantity C which now has to be fixed by the initial conditions, which in our context are given by quantum held theory. 

As a warm-up, let us recall the formalism to describe a particle evolving in a potential V. One describes the particle through the action S and the Lagrangian 

The equation of motion of the particle are deduced by varying the action with respect to the position x  

When quantizing the theory, the variable x and is conjugate momentum p become operator obeying the commutation rule  

One can do the same thing for a scalar field, except that since the field is defined everywhere in space, one uses the Lagrangian density L, 

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