# Q-Gaussian processes and probability space

It is well known that for isonormal Gaussian processes, one can decompose the space $$L^2(\Omega)$$ into Wiener chaos so that the space $$L^2(\Omega)$$ is isomorphic to the direct sum of the Wiener chaos, which in turn is isomorphic to a symmetric Fock space. The multiplication by a Gaussian random variable (i.e. in the first Wiener chaos) is then unitarily equivalent to a field operator $$a+a^*$$ acting on the Fock space.

Using the definition of $$q$$-Gaussian process (https://en.wikipedia.org/wiki/Q-Gaussian_process), this corresponds to the case $$q=1$$.

For $$q\in[-1,1)$$, the definition of $$q$$-Gaussian is given in terms of $$q$$-Fock space. For these values of $$q$$, is it always possible to construct a probability space $$L^2(\Omega)$$ (and $$\Omega$$) which is isomorphic to the $$q$$-Fock space and so that the multiplication in $$L^2(\Omega)$$ can be written in terms of field operators ?

There is no classical $$L^2$$-space which would do this, as multiplication with the field operators does not commute. However, the $$q$$-Fock space $$\cal{F}_q(\cal{H})$$ is to be considered as this analogue of $$L^2(\Omega)$$. The vacuum $$\Omega$$ corresponds to the constant variable $$1$$ and the field operator $$s=a+a^*$$ can be identified with an element in the $$q$$-Fock space via its action on the vacuum, $$s\Omega$$.