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 approach. This chapter is devoted to a brief description of a certain version of this method, called often the Euclidean approach.

Let us first explain the origin of the word Euclidean in the name of this approach. Originally the Euclidean approach amounted to replacing the real time variable t by the imaginary is, an operation called the Wick rotation. Under this transformation, the Minkowski space ℝ1, d becomes the Euclidean space ℝ1+d. After the Wick rotation, the unitary group generated by the Hamiltonian eitH becomes the self-adjoint group of contractions e-sH. One can then study e-sH from the point of view of the so-called path space. In particular, it is sometimes easier to construct or study interacting models of quantum field theory on the Euclidean space than on the Minkowski space.