Stochastic Quantisation in a Riemannian Manifold
There is an ambiguity in quantising a classical system in a curved background. It concerns the arbitrariness of the coefficient multiplying the curvature scalar in the Hamiltonian. This problem has somehow been confused, in the method of stochastic quantisation, with the difficulty of defining forward and backward derivatives of tensor fields in a Riemannian space. Two arguments are offered to show that the ambiguity is not avoided in stochastic methods. In the first we compare the kernel of the Feynman path integral with that of the Markovian semi-group transformations and in the second we randomise the metric itself keeping terms up to second order in Wiener differentials.
KeywordsRiemannian Manifold Curvature Scalar Riccati Equation Tensor Field Flat Space
Unable to display preview. Download preview PDF.