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Path integral formulation (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrodinger), and Variational Mechanics (Due to Dirac). DO NOT USE THIS TAG for line/contour integrals.
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path integrals: how/why can the phase be identified with the action?
In Peskin & Schroeder, chapter 9 introduces the functional methods.
The idea, to recall, is simply to sum over all the possible paths:
$U(x_a,x_b;T) = \sum_{\text{all paths}} e^{i . \text{phase} …