A functional integral with respect to a countably additive measure representing a solution of the Dirac equation

A family $\{M_{p}\}_{p\in\mathbb{R}^d}$ of cylindrical measures is constructed on the space of functions ($=$trajectories) $f: [0,+\infty)\to\mathbb{R}^d$ such that for every $t\geq0$ the formula

$\psi(t,p)= \int M_{p}(df)\,\psi_0(f( t))$

(1)

represents a solution of the Cauchy problem

$\frac\partial{\partial t} \psi = i \widehat H \psi \quad (t\geq0),\quad \psi(0,\cdot)=\psi_0$

(2)

(with respect to the required function $\psi: [0,\infty)\times\mathbb{R}^d\to\mathbb{C}^s$, ${d,s\in\mathbb{N}}$), for the general evolution equation from some class containing the classical Dirac equation and the Schrödinger equation in its impulse representation, with ``model'' potentials that are independent of $t$, and are Fourier transforms of countably additive (and in general matrix-valued) measures in the space variables.

The images of the measures $M_p$ obtained by restricting trajectories to finite intervals $[0,T]$ have bounded variation and are countably additive.

The integral kernels (``Green's functions'') of the corresponding solution operators, which can be approximated (using Trotter's formula) by integrals of finite multiplicity of the expressions explicitly defined by the ingredients of the original equation, are (matrix-valued) transition measures that give cylindrical measures $M_p$ similarly to the way Markov transition probabilities give the distribution of a Markov process.