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''Feynman path integral''
 
''Feynman path integral''
  
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Suppose that one is given an equation
 
Suppose that one is given an equation
  
<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/f038470/f0384701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
 
 +
\frac{du }{dt }
 +
  = Hu,
 +
$$
 +
 
 +
where  $  0 \leq  t \leq  T $,
 +
$  T > 0 $,
 +
and  $  u ( t, \omega ) $
 +
is a function defined on  $  T \times \Omega $,
 +
where  $  \Omega \ni \omega $
 +
is some space and  $  H $
 +
is a linear operator acting in a suitable way on a selected space of functions on  $  \Omega $.  
 +
In a number of cases the transition function  $  G ( \omega _ {1} , \omega _ {2} , t) $
 +
of equation (1) (that is, the kernel operator of the semi-group  $  \mathop{\rm exp} \{ tH \} $,
 +
$  t \geq  0 $)
 +
can be represented in the form of a path integral
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384703.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384704.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384706.png" /> is some space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384707.png" /> is a linear operator acting in a suitable way on a selected space of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384708.png" />. In a number of cases the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f0384709.png" /> of equation (1) (that is, the kernel operator of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847011.png" />) can be represented in the form of a path integral
+
$$ \tag{2 }
 +
G ( \omega _ {1} , \omega _ {2} , t)  = \
 +
\int\limits _ {\begin{array}{c}
 +
\omega : \\
 +
\omega ( 0) = \omega _ {1} \\
  
<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/f038470/f03847012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\omega ( t) = \omega _ {2}
 +
\end{array}
 +
}  \mathop{\rm exp}
 +
\left \{ \int\limits _ { 0 } ^ { t }
 +
W [ \omega ( \tau )]  d \tau \right \}
 +
\mu _ {\omega _ {1}  , \omega _ {2} , t }
 +
( d \omega ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847013.png" /> is some function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847014.png" />, the integration is carried out over the set of  "trajectories"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847016.png" />, with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847017.png" />,  "leaving"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847018.png" /> at time zero and  "arriving"  at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847019.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847020.png" />, and, finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847021.png" /> is some measure (or pre-measure) given on this set of trajectories. The integral is interpreted either in the usual Lebesgue sense or in the sense prescribed by any one of the methods of path integration (see [[#References|[5]]], [[#References|[6]]]). Integrals of the form (2), and also integrals obtained from them by means of certain natural transformations (for example, changing the integration variables, an additional integration over the  "ends"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847023.png" /> or over other parameters appearing in (2), differentiation with respect to these parameters, etc.) are commonly called Feynman path integrals.
+
where $  W ( \cdot ) $
 +
is some function defined on $  \Omega $,  
 +
the integration is carried out over the set of  "trajectories"   $ \omega ( \tau ) $,  
 +
0 \leq  \tau \leq  t $,  
 +
with values in $  \Omega $,   
 +
"leaving"   $ \omega _ {1} $
 +
at time zero and  "arriving"  at $  \omega _ {2} $
 +
at time $  t $,  
 +
and, finally, $  \mu _ {\omega _ {1}  , \omega _ {2} , t } $
 +
is some measure (or pre-measure) given on this set of trajectories. The integral is interpreted either in the usual Lebesgue sense or in the sense prescribed by any one of the methods of path integration (see [[#References|[5]]], [[#References|[6]]]). Integrals of the form (2), and also integrals obtained from them by means of certain natural transformations (for example, changing the integration variables, an additional integration over the  "ends"   $ \omega _ {1} $
 +
and $  \omega _ {2} $
 +
or over other parameters appearing in (2), differentiation with respect to these parameters, etc.) are commonly called Feynman path integrals.
  
The representation (2) was introduced by R.P. Feynman [[#References|[1]]] in connection with the new interpretation of quantum mechanics that he proposed. He considered the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847025.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847026.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847028.png" /> is a Sturm–Liouville differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847030.png" /> is the Laplace operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847032.png" /> is some function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847033.png" /> (a potential) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847034.png" />. Here one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847035.png" /> in the representation (2) for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847038.png" />, and the complex pre-measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847039.png" /> (the [[Feynman measure|Feynman measure]]) is given on cylindrical sets of the form
+
The representation (2) was introduced by R.P. Feynman [[#References|[1]]] in connection with the new interpretation of quantum mechanics that he proposed. He considered the case when $  \Omega = \mathbf R  ^ {n} $,
 +
$  n = 1, 2 \dots $
 +
the operator $  H $
 +
has the form $  H = iL $,  
 +
where $  L $
 +
is a Sturm–Liouville differential operator $  Lu = - a \Delta u + Vu $,  
 +
$  \Delta $
 +
is the Laplace operator in $  \mathbf R  ^ {n} $,  
 +
$  V $
 +
is some function defined on $  \mathbf R  ^ {n} $(
 +
a potential) and $  a > 0 $.  
 +
Here one obtains $  W = V $
 +
in the representation (2) for the function $  G ( x _ {1} , x _ {2} , t) $,
 +
$  x _ {1} , x _ {2} \in \mathbf R  ^ {n} $,  
 +
$  t > 0 $,  
 +
and the complex pre-measure $  \mu _ {x _ {1}  , x _ {2} , t } $(
 +
the [[Feynman measure|Feynman measure]]) is given on cylindrical sets of the form
  
<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/f038470/f03847040.png" /></td> </tr></table>
+
$$
 +
\{ {x ( \tau ) } : {
 +
x ( 0) = x _ {1} ,\
 +
x ( t) = x _ {2} ,\
 +
x ( \tau _ {i} ) \in G _ {i} ,\
 +
i = 1 \dots k } \}
 +
$$
  
 
where
 
where
  
<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/f038470/f03847041.png" /></td> </tr></table>
+
$$
 +
< \tau _ {1}  < \dots < \tau _ {k}  < t,\ \
 +
G _ {i}  \subset  \mathbf R  ^ {n} ,
 +
$$
 +
 
 +
$$
 +
= 1 \dots k,\  k  = 1, 2 \dots
 +
$$
 +
 
 +
by integration over the set  $  G _ {1} \times \dots \times G _ {k} \subseteq ( \mathbf R  ^ {n} )  ^ {k} $(
 +
with respect to the usual Lebesgue measure on  $  ( \mathbf R  ^ {n} )  ^ {k} $)
 +
of the density
  
<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/f038470/f03847042.png" /></td> </tr></table>
+
$$
 +
\prod _ {j = 1 } ^ { {k }  + 1 }
 +
[ 2 \pi ia ( \tau _ {j} -
 +
\tau _ {j - 1 }  ) ]  ^ {-} n/2 \
 +
\mathop{\rm exp} \left \{ -
  
by integration over the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847043.png" /> (with respect to the usual Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847044.png" />) of the density
+
\frac{( \xi _ {j} - \xi _ {j - 1 }  ) ^ {2} }{2ai ( \tau _ {j} - \tau _ {j - 1 }  ) }
  
<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/f038470/f03847045.png" /></td> </tr></table>
+
\right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847049.png" />. The expression (2) was regarded by Feynman as the limit of the finitely-multiple integrals obtained by replacing the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847050.png" /> in the exponent in the integrand by some integral sum of it. But he did not give a rigorous foundation for the validity of this definition of the integral, or of equation (2).
+
where $  \xi _ {0} = x _ {1} $,  
 +
$  \xi _ {k + 1 }  = x _ {2} $,
 +
$  \tau _ {0} = 0 $,  
 +
$  \tau _ {k + 1 }  = t $.  
 +
The expression (2) was regarded by Feynman as the limit of the finitely-multiple integrals obtained by replacing the integral $  \int _ {0}  ^ {t} W [ \omega ( t)]  d \tau $
 +
in the exponent in the integrand by some integral sum of it. But he did not give a rigorous foundation for the validity of this definition of the integral, or of equation (2).
  
Subsequently M. Kac [[#References|[2]]] obtained (2), in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847051.png" /> is the same as the [[Wiener measure|Wiener measure]], with complete mathematical rigour in the case of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847053.png" /> has the form above. Therefore (2) is often called the Feynman–Kac formula.
+
Subsequently M. Kac [[#References|[2]]] obtained (2), in which $  \mu _ {x _ {1}  , x _ {2} , t } $
 +
is the same as the [[Wiener measure|Wiener measure]], with complete mathematical rigour in the case of an operator $  H = - L $,  
 +
where $  L $
 +
has the form above. Therefore (2) is often called the Feynman–Kac formula.
  
 
The Feynman path integral is used as a convenient and deep analytical tool in a variety of questions in mathematical physics ([[#References|[3]]], [[#References|[4]]], [[#References|[6]]]), probability theory [[#References|[7]]] and the theory of differential equations [[#References|[5]]].
 
The Feynman path integral is used as a convenient and deep analytical tool in a variety of questions in mathematical physics ([[#References|[3]]], [[#References|[4]]], [[#References|[6]]]), probability theory [[#References|[7]]] and the theory of differential equations [[#References|[5]]].
Line 35: Line 131:
 
====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">  M. Kac,  "On some connections between probability theory and differential and integral equations" , ''Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950)'' , Univ. California Press  (1951)  pp. 189–215</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Ginibre,  "Some applications of functional integration in statistical mechanics"  C.M. DeWitt (ed.)  R. Stora (ed.) , ''Statistical mechanics and quantum field theory'' , Gordon &amp; Breach  pp. 327–427</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847054.png" /> Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[5]</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">[6]</TD> <TD valign="top">  S.A. Albeverio,  R.J. Høegh-Krohn,  "Mathematical theory of Feynman path integrals" , Springer  (1976)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,  A.V. [A.V. Skorokhod] Skorohod,  "The theory of stochastic processes" , '''3''' , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.A. Golubeva,  "Some problems in the analytic theory of Feynman integrals"  ''Russian Math. Surveys'' , '''31''' :  2  (1976)  pp. 135–202  ''Uspekhi Mat. Nauk'' , '''31''' :  2  (1976)  pp. 135–202</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">  M. Kac,  "On some connections between probability theory and differential and integral equations" , ''Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950)'' , Univ. California Press  (1951)  pp. 189–215</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Ginibre,  "Some applications of functional integration in statistical mechanics"  C.M. DeWitt (ed.)  R. Stora (ed.) , ''Statistical mechanics and quantum field theory'' , Gordon &amp; Breach  pp. 327–427</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038470/f03847054.png" /> Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[5]</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">[6]</TD> <TD valign="top">  S.A. Albeverio,  R.J. Høegh-Krohn,  "Mathematical theory of Feynman path integrals" , Springer  (1976)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,  A.V. [A.V. Skorokhod] Skorohod,  "The theory of stochastic processes" , '''3''' , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.A. Golubeva,  "Some problems in the analytic theory of Feynman integrals"  ''Russian Math. Surveys'' , '''31''' :  2  (1976)  pp. 135–202  ''Uspekhi Mat. Nauk'' , '''31''' :  2  (1976)  pp. 135–202</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:39, 5 June 2020


Feynman path integral

A collective name for representations in the form of a path integral, or integral over trajectories, of the transition functions (Green functions) of some evolution process.

Suppose that one is given an equation

$$ \tag{1 } \frac{du }{dt } = Hu, $$

where $ 0 \leq t \leq T $, $ T > 0 $, and $ u ( t, \omega ) $ is a function defined on $ T \times \Omega $, where $ \Omega \ni \omega $ is some space and $ H $ is a linear operator acting in a suitable way on a selected space of functions on $ \Omega $. In a number of cases the transition function $ G ( \omega _ {1} , \omega _ {2} , t) $ of equation (1) (that is, the kernel operator of the semi-group $ \mathop{\rm exp} \{ tH \} $, $ t \geq 0 $) can be represented in the form of a path integral

$$ \tag{2 } G ( \omega _ {1} , \omega _ {2} , t) = \ \int\limits _ {\begin{array}{c} \omega : \\ \omega ( 0) = \omega _ {1} \\ \omega ( t) = \omega _ {2} \end{array} } \mathop{\rm exp} \left \{ \int\limits _ { 0 } ^ { t } W [ \omega ( \tau )] d \tau \right \} \mu _ {\omega _ {1} , \omega _ {2} , t } ( d \omega ), $$

where $ W ( \cdot ) $ is some function defined on $ \Omega $, the integration is carried out over the set of "trajectories" $ \omega ( \tau ) $, $ 0 \leq \tau \leq t $, with values in $ \Omega $, "leaving" $ \omega _ {1} $ at time zero and "arriving" at $ \omega _ {2} $ at time $ t $, and, finally, $ \mu _ {\omega _ {1} , \omega _ {2} , t } $ is some measure (or pre-measure) given on this set of trajectories. The integral is interpreted either in the usual Lebesgue sense or in the sense prescribed by any one of the methods of path integration (see [5], [6]). Integrals of the form (2), and also integrals obtained from them by means of certain natural transformations (for example, changing the integration variables, an additional integration over the "ends" $ \omega _ {1} $ and $ \omega _ {2} $ or over other parameters appearing in (2), differentiation with respect to these parameters, etc.) are commonly called Feynman path integrals.

The representation (2) was introduced by R.P. Feynman [1] in connection with the new interpretation of quantum mechanics that he proposed. He considered the case when $ \Omega = \mathbf R ^ {n} $, $ n = 1, 2 \dots $ the operator $ H $ has the form $ H = iL $, where $ L $ is a Sturm–Liouville differential operator $ Lu = - a \Delta u + Vu $, $ \Delta $ is the Laplace operator in $ \mathbf R ^ {n} $, $ V $ is some function defined on $ \mathbf R ^ {n} $( a potential) and $ a > 0 $. Here one obtains $ W = V $ in the representation (2) for the function $ G ( x _ {1} , x _ {2} , t) $, $ x _ {1} , x _ {2} \in \mathbf R ^ {n} $, $ t > 0 $, and the complex pre-measure $ \mu _ {x _ {1} , x _ {2} , t } $( the Feynman measure) is given on cylindrical sets of the form

$$ \{ {x ( \tau ) } : { x ( 0) = x _ {1} ,\ x ( t) = x _ {2} ,\ x ( \tau _ {i} ) \in G _ {i} ,\ i = 1 \dots k } \} $$

where

$$ 0 < \tau _ {1} < \dots < \tau _ {k} < t,\ \ G _ {i} \subset \mathbf R ^ {n} , $$

$$ i = 1 \dots k,\ k = 1, 2 \dots $$

by integration over the set $ G _ {1} \times \dots \times G _ {k} \subseteq ( \mathbf R ^ {n} ) ^ {k} $( with respect to the usual Lebesgue measure on $ ( \mathbf R ^ {n} ) ^ {k} $) of the density

$$ \prod _ {j = 1 } ^ { {k } + 1 } [ 2 \pi ia ( \tau _ {j} - \tau _ {j - 1 } ) ] ^ {-} n/2 \ \mathop{\rm exp} \left \{ - \frac{( \xi _ {j} - \xi _ {j - 1 } ) ^ {2} }{2ai ( \tau _ {j} - \tau _ {j - 1 } ) } \right \} , $$

where $ \xi _ {0} = x _ {1} $, $ \xi _ {k + 1 } = x _ {2} $, $ \tau _ {0} = 0 $, $ \tau _ {k + 1 } = t $. The expression (2) was regarded by Feynman as the limit of the finitely-multiple integrals obtained by replacing the integral $ \int _ {0} ^ {t} W [ \omega ( t)] d \tau $ in the exponent in the integrand by some integral sum of it. But he did not give a rigorous foundation for the validity of this definition of the integral, or of equation (2).

Subsequently M. Kac [2] obtained (2), in which $ \mu _ {x _ {1} , x _ {2} , t } $ is the same as the Wiener measure, with complete mathematical rigour in the case of an operator $ H = - L $, where $ L $ has the form above. Therefore (2) is often called the Feynman–Kac formula.

The Feynman path integral is used as a convenient and deep analytical tool in a variety of questions in mathematical physics ([3], [4], [6]), probability theory [7] and the theory of differential equations [5].

References

[1] R.P. Feynman, "Space-time approach to non-relativistic quantum mechanics" Rev. Modern Phys. , 20 (1948) pp. 367–387
[2] M. Kac, "On some connections between probability theory and differential and integral equations" , Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950) , Univ. California Press (1951) pp. 189–215
[3] J. Ginibre, "Some applications of functional integration in statistical mechanics" C.M. DeWitt (ed.) R. Stora (ed.) , Statistical mechanics and quantum field theory , Gordon & Breach pp. 327–427
[4] B. Simon, "The Euclidean (quantum) field theory" , Princeton Univ. Press (1974)
[5] Yu.L. Daletskii, "Integration in function spaces" Progress in Mathematics , 4 (1969) pp. 87–132 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 83–124
[6] S.A. Albeverio, R.J. Høegh-Krohn, "Mathematical theory of Feynman path integrals" , Springer (1976)
[7] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 3 , Springer (1979) (Translated from Russian)
[8] V.A. Golubeva, "Some problems in the analytic theory of Feynman integrals" Russian Math. Surveys , 31 : 2 (1976) pp. 135–202 Uspekhi Mat. Nauk , 31 : 2 (1976) pp. 135–202

Comments

The phrase "Feynman integral" is also used in physics to denote an (ordinary) integral over a closed loop in a Feynman diagram (arising in particle physics when calculating radiative conditions). Instead of Feynman path integral and Feynman integral one also finds the phrases path integral, functional integral and (rarely) continual integral in the literature.

References

[a1] R.P. Feynman, A.R. Hibbs, "Quantum mechanics and path integrals" , McGraw-Hill (1965)
[a2] L.S. Schulman, "Techniques and applications of path integration" , Wiley (1981)
[a3] J. Glimm, A. Jaffe, "Quantum physics, a functional integral point of view" , Springer (1981)
[a4] V.N. Popov, "Functional integrals in quantum field theory and statistical physics" , Reidel (1983) (Translated from Russian)
[a5] J.-P. Antoine (ed.) E. Tirapegui (ed.) , Functional integration. Theory and applications , Plenum (1980)
[a6] B. Simon, "Functional integration and quantum physics" , Acad. Press (1979) pp. 4–6
How to Cite This Entry:
Feynman integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feynman_integral&oldid=46915
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article