- Concept of Functional Integration
So far we have pointed out that any dynamical equation may be thought of as the Lagrange equation of a suitable Lagrangian. The relevance of this is clear when we attempt to make a transition to quantum mechanics or when attempting to study thermal fluctuations of classical statistical systems. In both cases, a term proportional to the exponential of the action (time-integrated Lagrangian) serves as a ‘weight’ or probability distribution for the field in question as we shall see subsequently. Thus, all possible field configurations are allowed, but each comes with a weight which allows the computation of averages of the field (given by the solution of the classical equations) and the correlations between the values of the field at different points in space and time. To do this involves integrating over all possible field configurations with the weight mentioned above. Therefore, we have to now learn how to integrate over function spaces.