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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 ?

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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, $s\Omega$.

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