Feynman path integrals as analysis on path space by time slicing approximation (Q707262)

A mathematically rigorous consideration is given of the Feynman path integral \[ \int \exp\left(iS[\gamma]/\hbar\right)F[\gamma]D[\gamma] \] for a general class \(\mathcal{F}\) of functionals \(F[\gamma]\) on the path space \(C([0,T]\rightarrow \mathbb{R}^d).\) The class \(\mathcal{F}\) is defined so that the Feynman path integral has a mathematically rigorous meaning for any \(F[\gamma]\in\mathcal{F}\), i.e. its time slicing approximation uniformly converges with respect to \((x,x_0)\) on compact subsets of \(\mathbb{R}^{2d}\). It is shown, under some assumptions about the function \(B\) and the vector-valued function \(Z\), that the class \(\mathcal{F}\) contains (1)~the evaluation functionals \(F[\gamma]=B(\tau, \gamma(\tau)))\), \(\tau\in[0,T]\); (2)~the Riemann-Stieltjes integrals \(F[\gamma]=\int_{T'}^{T''} B(\tau, \gamma(\tau)))\,d\rho(\tau)\); (3)~the analytic functions of the integrals \(F[\gamma]=f\left(\int_{T'}^{T''} B(\tau, \gamma(\tau)))\,d\rho(\tau)\right)\); (4)~the curvilinear integrals along paths \(F[\gamma]=\int_{T'}^{T''} Z(\tau, \gamma(\tau)))\cdot d\gamma(\tau)\).