Sp(2,$\mathbb{Z}$) invariant Wigner function on even dimensional vector space

We construct the quasi probability distribution $W(p,q)$ on even dimensional vector space with marginality and invariance under the transformation induced by projective representation of the group ${\rm Sp}(2,\mathbb{Z})$ whose elements correspond to linear canonical transformation. On even dimensional vector space, non-existence of such a quasi probability distribution whose arguments take physical values was shown in our previous paper(Phys.Rev.A{\bf 65} 032105(2002)). For this reason we study a quasi probability distribution $W(p,q)$ whose arguments $q$ and $p$ take not only $N$ physical values but also $N$ unphysical values, where $N$ is dimension of vector space. It is shown that there are two quasi probability distributions on even dimensional vector space. The one is equivalent to the Wigner function proposed by Leonhardt, and the other is a new one.

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