Nelson's stochastic quantization of a free linearized gravitational field and its Markovian structure

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It is shown that by applying Nelson's stochastic quantization scheme to a free linearized gravitational field tensor, one can associate with the resulting stochastic system a stochastic tensor field which coincides with the ‘space’ part of the Riemannian tensor in Euclidean space-time. However, such a stochastic field fails to satisfy the Markov property. Instead, it satisfies the reflection positivity. The Markovian structure of the stochastic fields associated with the electromagnetic field is also discussed.