Feynman measure

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)