Operator-valued Feynman integral via conditional Feynman integrals on a function space

Dong Cho

Open Mathematics (2010)

  • Volume: 8, Issue: 5, page 908-927
  • ISSN: 2391-5455

Abstract

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Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional Γ t x = e x p ∫ 0 t θ s , x s d η s ϕ x t x ∈ C r 0 , t , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral transform as an analytic operator-valued Feynman integral over C r [0, t], and evaluate the integral transform for the function Γt via the conditional analytic Feynman integral as a kernel.

How to cite

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Dong Cho. "Operator-valued Feynman integral via conditional Feynman integrals on a function space." Open Mathematics 8.5 (2010): 908-927. <http://eudml.org/doc/269685>.

@article{DongCho2010,
abstract = {Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional \[ \Gamma \_t \left( x \right) = exp \left\lbrace \{\int \_0^t \{\theta \left( \{s,x\left( s \right)\} \right)d\eta \left( s \right)\} \} \right\rbrace \varphi \left( \{x\left( t \right)\} \right) x \in C^r \left[ \{0,t\} \right] \] , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral transform as an analytic operator-valued Feynman integral over C r [0, t], and evaluate the integral transform for the function Γt via the conditional analytic Feynman integral as a kernel.},
author = {Dong Cho},
journal = {Open Mathematics},
keywords = {Analogue of Wiener measure; Conditional analytic Feynman integral; Conditional analytic Wiener integral; Operator-valued Feynman integral; Simple formula for conditional expectation; Wiener space; analogue of Wiener measure; conditional analytic Feynman integral; conditional analytic Wiener integral; operator-valued Feynman integral; simple formula for conditional expectation},
language = {eng},
number = {5},
pages = {908-927},
title = {Operator-valued Feynman integral via conditional Feynman integrals on a function space},
url = {http://eudml.org/doc/269685},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Dong Cho
TI - Operator-valued Feynman integral via conditional Feynman integrals on a function space
JO - Open Mathematics
PY - 2010
VL - 8
IS - 5
SP - 908
EP - 927
AB - Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional \[ \Gamma _t \left( x \right) = exp \left\lbrace {\int _0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\rbrace \varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right] \] , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral transform as an analytic operator-valued Feynman integral over C r [0, t], and evaluate the integral transform for the function Γt via the conditional analytic Feynman integral as a kernel.
LA - eng
KW - Analogue of Wiener measure; Conditional analytic Feynman integral; Conditional analytic Wiener integral; Operator-valued Feynman integral; Simple formula for conditional expectation; Wiener space; analogue of Wiener measure; conditional analytic Feynman integral; conditional analytic Wiener integral; operator-valued Feynman integral; simple formula for conditional expectation
UR - http://eudml.org/doc/269685
ER -

References

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