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We present an interpolation between the bosonic and fermionic relations. This interpolation is given by an object which we call `generalized Brownian motion' and which is characterized by a generalization of the pairing rule for the calculation of the moments of bosonic and fermionic fields. We develop some basic theory for such generalized Brownian motions and consider more closely one example, which turns out to be intimately connected with Voiculescu's concept of `free product'.

Interpolations between bosonic and fermionic relations given by generalized Brownian motions