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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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Inserting a position operator in the path integral in QFT
The path integral itself is as far as I know not well defined as an "integration over all functions" (i.e. a corresponding measure has not been constructed yet). So let us stay in the "discrete" case …