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A complex [[Pre-measure|pre-measure]] defined on cylindrical sets in the space of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384803.png" />, with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384805.png" /> by the formula
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$#C+1 = 21 : ~/encyclopedia/old_files/data/F038/F.0308480 Feynman measure
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384807.png" /></td> </tr></table>
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A complex [[Pre-measure|pre-measure]] defined on cylindrical sets in the space of functions  $  x ( t) $,
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$  0 \leq  t \leq  T $,
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$  T > 0 $,
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with values in  $  \mathbf R  ^ {n} $,
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$  n = 1, 2 \dots $
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by the formula
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384808.png" /> is a parameter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f0384809.png" />, and
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$$ \tag{1 }
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\mu _ {x, T }
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\{ B _ {\tau _ {1}  \dots \tau _ {k} , x }  ^ {A} \}  = \
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\prod _ {j = 1 } ^ { k }
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[ 2 \pi ia ( \tau _ {j} - \tau _ {j - 1 }  )]  ^ {-} n/2 \times
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848010.png" /></td> </tr></table>
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$$
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\times
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\int\limits _ { A } \prod _ {j = 1 } ^ { {k }  + 1 }  \mathop{\rm exp} \left \{ -
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{
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\frac{1}{2ai }
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}
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\frac{( \xi _ {j} - \xi _ {j - 1 }  )  ^ {2} }{( \tau _ {j} - \tau _ {j - 1 }  ) }
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\right
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\}  d \xi _ {1} \dots d \xi _ {k + 1 }  .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848011.png" /></td> </tr></table>
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Here  $  a > 0 $
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is a parameter,  $  0 < \tau _ {1} < \dots < \tau _ {k} < T $,
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and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848012.png" /></td> </tr></table>
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$$
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B _ {\tau _ {1}  \dots \tau _ {k} , x }  ^ {A\ } =
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848013.png" /> is some Borel subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848014.png" />. Sometimes one also considers the so-called conditional Feynman measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848015.png" /> obtained from the measure (1) by restricting it to the set of trajectories with  "end"  at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848016.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848017.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848018.png" />, and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848019.png" />, was introduced by R.P. Feynman in connection with representing the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038480/f03848021.png" /> is a Sturm–Liouville operator, in the form of a path integral — a [[Feynman integral|Feynman integral]].
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$$
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= \
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\{ x ( t):  x ( 0) = x = \xi _ {0} , \{ x (
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\tau _ {1} ) \dots x ( \tau _ {k} ), x ( T) \} \in A \} ,
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$$
 +
 
 +
$$
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x  \in  \mathbf R  ^ {n} ,\  k  = 0, 1 \dots
 +
$$
 +
 
 +
where  $  A $
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is some Borel subset in $  ( \mathbf R  ^ {n} ) ^ {( k + 1) } $.  
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Sometimes one also considers the so-called conditional Feynman measure $  \mu _ {x, y, T }  $
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obtained from the measure (1) by restricting it to the set of trajectories with  "end"  at the point $  y \in \mathbf R  ^ {n} $:  
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$  x ( T) = y $.  
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The measure $  \mu _ {x, T }  $,  
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and also $  \mu _ {x, y, T }  $,  
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was introduced by R.P. Feynman in connection with representing the semi-group $  \mathop{\rm exp} \{ itH \} $,  
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where $  H $
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is a Sturm–Liouville operator, in the form of a path integral — a [[Feynman integral|Feynman integral]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.P. Feynman,  "Space-time approach to non-relativistic quantum mechanics"  ''Rev. Modern Phys.'' , '''20'''  (1948)  pp. 367–387</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Daletskii,  "Integration in function spaces"  ''Progress in Mathematics'' , '''4'''  (1969)  pp. 87–132  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 83–124</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.A. Albeverio,  R.J. Høegh-Krohn,  "Mathematical theory of Feynman path integrals" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.P. Feynman,  "Space-time approach to non-relativistic quantum mechanics"  ''Rev. Modern Phys.'' , '''20'''  (1948)  pp. 367–387</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Daletskii,  "Integration in function spaces"  ''Progress in Mathematics'' , '''4'''  (1969)  pp. 87–132  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 83–124</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.A. Albeverio,  R.J. Høegh-Krohn,  "Mathematical theory of Feynman path integrals" , Springer  (1976)</TD></TR></table>

Latest revision as of 19:39, 5 June 2020


A complex pre-measure defined on cylindrical sets in the space of functions $ x ( t) $, $ 0 \leq t \leq T $, $ T > 0 $, with values in $ \mathbf R ^ {n} $, $ n = 1, 2 \dots $ by the formula

$$ \tag{1 } \mu _ {x, T } \{ B _ {\tau _ {1} \dots \tau _ {k} , x } ^ {A} \} = \ \prod _ {j = 1 } ^ { k } [ 2 \pi ia ( \tau _ {j} - \tau _ {j - 1 } )] ^ {-} n/2 \times $$

$$ \times \int\limits _ { A } \prod _ {j = 1 } ^ { {k } + 1 } \mathop{\rm exp} \left \{ - { \frac{1}{2ai } } \frac{( \xi _ {j} - \xi _ {j - 1 } ) ^ {2} }{( \tau _ {j} - \tau _ {j - 1 } ) } \right \} d \xi _ {1} \dots d \xi _ {k + 1 } . $$

Here $ a > 0 $ is a parameter, $ 0 < \tau _ {1} < \dots < \tau _ {k} < T $, and

$$ B _ {\tau _ {1} \dots \tau _ {k} , x } ^ {A\ } = $$

$$ = \ \{ x ( t): x ( 0) = x = \xi _ {0} , \{ x ( \tau _ {1} ) \dots x ( \tau _ {k} ), x ( T) \} \in A \} , $$

$$ x \in \mathbf R ^ {n} ,\ k = 0, 1 \dots $$

where $ A $ is some Borel subset in $ ( \mathbf R ^ {n} ) ^ {( k + 1) } $. Sometimes one also considers the so-called conditional Feynman measure $ \mu _ {x, y, T } $ obtained from the measure (1) by restricting it to the set of trajectories with "end" at the point $ y \in \mathbf R ^ {n} $: $ x ( T) = y $. The measure $ \mu _ {x, T } $, and also $ \mu _ {x, y, T } $, was introduced by R.P. Feynman in connection with representing the semi-group $ \mathop{\rm exp} \{ itH \} $, where $ H $ is a Sturm–Liouville operator, in the form of a path integral — a Feynman integral.

References

[1] R.P. Feynman, "Space-time approach to non-relativistic quantum mechanics" Rev. Modern Phys. , 20 (1948) pp. 367–387
[2] Yu.L. Daletskii, "Integration in function spaces" Progress in Mathematics , 4 (1969) pp. 87–132 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 83–124
[3] S.A. Albeverio, R.J. Høegh-Krohn, "Mathematical theory of Feynman path integrals" , Springer (1976)
How to Cite This Entry:
Feynman measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feynman_measure&oldid=46916
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article