On the Path Integral in the Curved Space

Abstract

An attempt is made to determine the coefficient of the scalar curvature appeared in the quantal Lagrangian and Hamiltonian for a system of harmonic oscillators in the curved space with non-vanishing Riemann-Christoffel curvature tensor. The c-number Hamiltonian appear­ ing in the path integral in the phase space is derived from the quantal one of Kawai and Kama by the correspondence rule of Weyl ordering. In connection with the consideration of point transformations in the normal coordinate system, the arbitrary scalar curvature in the Hamiltonian is determined so that the additive vacuum energy due to two-loop bubble Feynman diagrams is eliminated. The transformation function characterized by the Hamil­ tonian with the determined scalar curvature satisfies the equation obtained by Feynman's theory due to Pauli and DeWitt, which is based on Van Vleck's work. §I. Introduction Gervais and Jevicki have shown in a simple example that point canonical transformations cannot be performed by simply changing integration variables in the path integral formalism.n They show how a formal treatment o£ point trans­ formations leads to erroneous results by employing the Feynman diagram technique at the approximation of two-loop level. To get rid of the difficulty they presented a correct discussion of point transformations in path integrals and obtained an addi­ tional potential terms in the action of the path integral as compared with the formal treatment. They also showed that their correct (midpoint) path integral corresponds to the operator Hamiltonian where the non-commuting factors are or­ dered by the rule of Weyl. The preference of W eyl ordering in the path integral formalism was also stressed by Mizrahi.) The model used by Gervais and Jevicki is a kind of quantum mechanics with velocity dependent potential described by the classical Lagrangian