Editing Path integral formulation (section)

== Feynman's interpretation ==

Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the [[canonical commutation relation]]s from this rule. This was done by Feynman.

Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates:
# The [[probability]] for an event is given by the [[squared modulus]] of a complex number called the "probability amplitude".
# The [[probability amplitude]] is given by adding together the contributions of all paths in configuration space.
# The contribution of a path is proportional to {{math|''e''<sup>''iS''/''ħ''</sup>}}, where {{mvar|S}} is the [[Action (physics)|action]] given by the [[time integral]] of the [[Lagrangian mechanics|Lagrangian]] along the path.

In order to find the overall probability amplitude for a given process, then, one adds up, or [[integral|integrates]], the amplitude of the 3rd postulate over the space of ''all'' possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes elaborate [[curlicues]], curves in which the particle shoots off into outer space and flies back again, and so forth. The '''path integral''' assigns to all these amplitudes ''equal weight'' but varying [[phase (waves)|phase]], or argument of the [[complex number]]. Contributions from paths wildly different from the classical trajectory may be suppressed by [[Interference (wave propagation)|interference]] (see below).

Feynman showed that this formulation of quantum mechanics is equivalent to the [[Quantization (physics)|canonical approach to quantum mechanics]] when the Hamiltonian is at most quadratic in the momentum.  An amplitude computed according to Feynman's principles will also obey the [[Schrödinger equation]] for the [[Hamiltonian (quantum mechanics)|Hamiltonian]] corresponding to the given action.

The path integral formulation of quantum field theory represents the [[transition amplitude]] (corresponding to the classical [[correlation function]]) as a weighted sum of all possible histories of the system from the initial to the final state. A [[Feynman diagram]] is a graphical representation of a [[perturbative]] contribution to the transition amplitude.