Euclidean approach for bosons
One can distinguish two basic approaches to quantum field theory. In the more traditional approach, one views the underlying physical Hilbert space equipped with the self-adjoint generator of the dynamics – Hamiltonian or Liouvillean – as the basic object. There also exists a different philosophy, whose starting point is paths (trajectories). The physical space and the physical Hamiltonian or Liouvillean are treated as derived objects (if they can be defined at all).
The second approach is often viewed as more modern and useful by physicists active in quantum field theory. Also from the mathematical point of view, the method of paths has turned out to be in many cases more efficient than the operator-theoretic ...