PATH INTEGRAL ON THE EXTREME FIELD CONFIGURATIONS

Abstract

The canonical Hamiltonian path integral measure obeys certain rule which relates such measure on the paths defined on the whole time axis to the measures on the paths defined on the regions constituting the time axis. We show that this "gluing" rule can be reproduced without referring to Hamiltonian formalism, by substituting field configurations with arbitrarily fast change of the fields at the boundary points of these regions into action and viewing the path integral in the sense of generalized function. Now the coordinate along which gluing proceeds can be not only the time. The piecewise-flat (simplicial) minisuperspace gravity system is considered. Arbitrarily fast change of the (tangential component of) metric between the two 4-simplices with common 3-face is studied. That is, we generalize piecewise-flat anzats by allowing tangential metric to be function of the distance from the 3-face in the neighborhood of this 3-face. The action is nondegenerate (nonsingular) with respect to these additional generalized coordinates. The rule for gluing the path integral measures on separate 4-simplices is found. The resulting general expression covers a large variety of the measures including those usually used in numerical calculations and allows one to specify the measure in some applications.