# 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)
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