  It was shown recently that stochastic quantization can be made into a well
defined quantization scheme on (pseudo-)Riemannian manifolds using second order
differential geometry, which is an extension of the commonly used first order
differential geometry. In this letter, we show that restrictions to
relativistic theories can be obtained from this theory by imposing a stochastic
energy-momentum relation. In the process, we derive non-perturbative quantum
corrections to the line element as measured by scalar particles. Furthermore,
we extend the framework of stochastic quantization to massless scalar
particles.
