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and
: <math> p = \frac{\partial L}{\partial \dot{q}},</math>
where the [[partial derivative]] with respect to <math>\dot q</math> holds {{math|''q''(''t'' + ''ε'')}} fixed. The inverse Legendre transform is
: <math> \varepsilon L = \varepsilon p \dot{q} - \varepsilon H,</math>
where
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== 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.
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: <math>K(x - y; T) \propto e^{ -\frac{(x - y)^2}{2T}}.</math>
The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a [[stochastic process]].
The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the [[central limit theorem]], which can be interpreted as the first historical evaluation of a statistical path integral.
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* {{cite journal |author-link=Cécile DeWitt-Morette |last=DeWitt-Morette |first=C. |title=Feynman's path integral: Definition without limiting procedure |journal=Communications in Mathematical Physics |volume=28 |issue=1 |year=1972 |pages=47–67 |mr=0309456 |doi=10.1007/BF02099371 |bibcode=1972CMaPh..28...47D |s2cid=119669964}}
* {{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1933 |title=The Lagrangian in Quantum Mechanics |journal=Physikalische Zeitschrift der Sowjetunion |volume=3 |pages=64–72 |url=http://www.hep.anl.gov/czachos/soysoy/Dirac33.pdf}}
* {{cite journal |first1=İ. H. |last1=Duru |author-link1=İsmail Hakkı Duru |first2=Hagen |last2=Kleinert |author-link2=Hagen Kleinert |title=Solution of the path integral for the H-atom |year=1979 |journal=Physics Letters |volume=84B |issue=2 |pages=185–188 |url=http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf |access-date=2007-11-25 |bibcode=1979PhLB...84..185D |doi=10.1016/0370-2693(79)90280-6 |archive-date=2008-03-09 |archive-url=https://web.archive.org/web/20080309160840/http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf |url-status=dead }}
* {{cite web |last=Etingof |first=P. |author-link=Pavel Etingof |title=Geometry and Quantum Field Theory |publisher=MIT OpenCourseWare |year=2002 |url=http://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/index.htm}} <small>This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.</small>
* {{cite book |last=Feynman |first=R. P. |author-link=Richard Feynman |editor-last=Brown |editor-first=L. M. |title=Feynman's Thesis — A New Approach to Quantum Theory |year=2005 |orig-year=1942/1948 |publisher=World Scientific |doi=10.1142/5852 |bibcode=2005ftna.book.....B |url=https://cds.cern.ch/record/910611 |isbn=978-981-256-366-8}} <small>The 1942 thesis. Also includes Dirac's 1933 paper and Feynman's 1948 publication.</small>
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* {{cite book |first1=Gerald W. |last1=Johnson |first2=Michel L. |last2= Lapidus |title=The Feynman Integral and Feynman's Operational Calculus |series=Oxford Mathematical Monographs |publisher=Oxford University Press |year=2002 |isbn=978-0-19-851572-2}}
* {{cite book |author-link=John R. Klauder |last=Klauder |first=John R. |title=A Modern Approach to Functional Integration |place=New York |publisher=Birkhäuser |year=2010 |isbn=978-0-8176-4790-2}}
* {{cite book |author-link=Hagen Kleinert |last=Kleinert |first=Hagen |year=2004 |title=Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets |edition=4th |place=Singapore |publisher=World Scientific |isbn=978-981-238-107-1 |url=http://www.physik.fu-berlin.de/~kleinert/b5 |archive-date=2008-06-15 |access-date=2005-02-16 |archive-url=https://web.archive.org/web/20080615134934/http://www.physik.fu-berlin.de/~kleinert/b5 |url-status=dead }}
* {{cite arXiv |last=MacKenzie |first=Richard |year=2000 |title=Path Integral Methods and Applications |eprint=quant-ph/0004090}}
* {{cite book |last= Mazzucchi |first= S. |title=Mathematical Feynman path integrals and their applications |publisher=World Scientific |year=2009 |isbn=978-981-283-690-8}}
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