Editing Path integral formulation (section)

== Classical limit ==
Crucially, Dirac identified the effect of the [[classical limit]] on the quantum form of the action principle:
{{blockquote|...we see that the integrand in (11) must be of the form {{math|''e''<sup>''iF''/''h''</sup>}}, where {{mvar|F}} is a function of {{math|''q''<sub>''T''</sub>, ''q''<sub>1</sub>, ''q''<sub>2</sub>, … ''q''<sub>''m''</sub>, ''q''<sub>''t''</sub>}}, which remains finite as {{mvar|h}} tends to zero. Let us now picture one of the intermediate {{mvar|q}}s, say {{mvar|q<sub>k</sub>}}, as varying continuously while the other ones are fixed. Owing to the smallness of {{mvar|h}}, we shall then in general have ''F''/''h'' varying extremely rapidly. This means that {{math|''e''<sup>''iF''/''h''</sup>}} will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of {{mvar|q<sub>k</sub>}} is thus that for which a comparatively large variation in {{mvar|q<sub>k</sub>}} produces only a very small variation in {{mvar|F}}. This part is the neighbourhood of a point for which {{mvar|F}} is stationary with respect to small variations in {{mvar|q<sub>k</sub>}}. We can apply this argument to each of the variables of integration ... and obtain the result that the only important part in the domain of integration is that for which {{mvar|F}} is stationary for small variations in all intermediate {{mvar|q}}s. ... We see that {{mvar|F}} has for its classical analogue {{math|{{intmath|int|''T''|''t''}} ''L dt''}}, which is just the action function, which classical mechanics requires to be stationary for small variations in all the intermediate {{mvar|q}}s. This shows the way in which equation (11) goes over into classical results when {{mvar|h}} becomes extremely small. |source=Dirac (1933), p. 69}}
That is, in the limit of action that is large compared to the [[Planck constant]] {{mvar|ħ}} – the classical limit – the path integral is dominated by solutions that are in the neighborhood of [[stationary point]]s of the action. The classical path arises naturally in the classical limit.