Path Integral Quantization and Riemannian-Symplectic Manifolds

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a proper (normalizable, σ-additive) measure on the path space which is covariant under general coordinate transformations. Since the discovery of the Hamiltonian path integral by Feynman, the indisputable fact of its coordinate dependence has generally complicated any straightforward use of Hamiltonian path integrals in applications. Indeed, to justify any formal manipulation regarding the path integral measure, the operator formalism has generally been invoked. The aim of this letter is to resolve this long-standing problem and to establish a rigorous, coordinate-free path integral formalism for general Hamiltonian systems. In [2] it was shown that the formal phase-space path integral measure can be made covariant under general coordinate transformations by means of a special regularization that involves an auxiliary Brownian motion on a flat phase space. The desired quantum mechanics is restored by taking the diffusion constant to infinity. The modified Hamiltonian path integral determines a stochastic process on the phase space in which the original first-order Lagrangian plays the role of the external potential. The stochastic process on a flat phase space remains flat under coordinate transformations. Thus one may change the variables in the regularized path integral according to the rules of the stochastic integral On leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia.