A generalized $L_{p}$-analytic conditional Feynman integral over paths in abstract Wiener space
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Let $C_0[0,T]$ denote the Wiener space, the space of continuous real-valued functions $x$ on $[0,T]$ with $x(0)=0$. Let $C^{\mathbb B}[a,b]$ denote an analogue of Wiener space over paths in abstract Wiener space $\mathbb B$, the space of $\mathbb B$-valued continuous functions on $[a,b]$. Let $W(x)=\{x(t_n)\}_{n=0}^\infty\in \mathbb B^{\aleph_0}$ for $x\in C^{\mathbb B}[a,b]$, where $\{t_n\}_{n=0}^\infty$ is a strictly increasing sequence in $[a,b]$ with $t_0=a$ and $\lim_{n\to\infty}t_n=b$. In this paper, we introduce a positive finite measure with scale on $C^{\mathbb B}[a,b]$ which is a generalized analogue of Wiener measure. Then we extend the time integral which is a Riemann integral of functions on $C_0[0,T]$, to a Riemann integral of functions on $C^{\mathbb B}[a,b]$ which is a more generalized time integral. Finally, using a simple formula for calculating a Radon-Nikodym derivative similar to the conditional Wiener integral of functions on $C^{\mathbb B}[a,b]$ given $W$ which has an initial weight, we evaluate a generalized $L_p$-analytic conditional Feynman integral of the extended time integral. The established $L_p$-analytic conditional Feynman integrals are of interest in quantum mechanics, especially in Feynman integration theory.
Abstract
Let $C_0[0,T]$ denote the Wiener space, the space of continuous real-valued functions $x$ on $[0,T]$ with $x(0)=0$. Let $C^{\mathbb B}[a,b]$ denote an analogue of Wiener space over paths in abstract Wiener space $\mathbb B$, the space of $\mathbb B$-valued continuous functions on $[a,b]$. Let $W(x)=\{x(t_n)\}_{n=0}^\infty\in \mathbb B^{\aleph_0}$ for $x\in C^{\mathbb B}[a,b]$, where $\{t_n\}_{n=0}^\infty$ is a strictly increasing sequence in $[a,b]$ with $t_0=a$ and $\lim_{n\to\infty}t_n=b$. In this paper, we introduce a positive finite measure with scale on $C^{\mathbb B}[a,b]$ which is a generalized analogue of Wiener measure. Then we extend the time integral which is a Riemann integral of functions on $C_0[0,T]$, to a Riemann integral of functions on $C^{\mathbb B}[a,b]$ which is a more generalized time integral. Finally, using a simple formula for calculating a Radon-Nikodym derivative similar to the conditional Wiener integral of functions on $C^{\mathbb B}[a,b]$ given $W$ which has an initial weight, we evaluate a generalized $L_p$-analytic conditional Feynman integral of the extended time integral. The established $L_p$-analytic conditional Feynman integrals are of interest in quantum mechanics, especially in Feynman integration theory.