Feynman integral

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].

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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.

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