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In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space ε and show in particular that fractional Brownian motion for Hurst index 0 < H ≤ 1/2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0 < H < 1/2. We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL2(ℝ). We relate this to a measure preserving action on a Gaussian L2-Hilbert space L2(ε).

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