The core principle is that quantum mechanical and field-theoretic time evolution operators can be represented by a functional integral over all possible paths in configuration or phase space, weighted by the exponential of $i$ times the action divided by $\hbar$. This formalism eliminates explicit operator ordering ambiguities present in canonical quantization by utilizing Weyl-symmetrized Hamiltonians for general theories and Gaussian integration techniques to derive specific forms like those involving kinetic momentum squared plus potential energy. The mechanism relies on discretizing time, inserting complete sets of states, and taking the continuum limit where operators automatically emerge as Schrodinger-picture observables strictly ordered by their temporal insertion points in correlation functions. This concept bridges classical Lagrangian mechanics with quantum probability amplitudes within Quantum Field Theory, serving as the foundational tool for calculating scattering matrix elements (S-matrix) and vacuum expectation values of time-ordered products via functional differentiation.
The core principle is that quantum mechanical and field-theoretic time evolution operators can be represented by a functional integral over all possible paths in configuration or phase space, weighte…