On \(^{\ast}\)-representation of Wick CCR (Q2755272)

Recall that the classical \(CCR\)-algebra can be defined by the generators \(\{ a_{i}, a^{\ast}_{i}\), \(i=1,\ldots, d\}\) with relations \(a^{\ast}_{i}a_{i}=1+a_{i}a^{\ast}_{i}\), \(i=1, \ldots ,d\), \(a^{\ast}_{i}a_{j}=a_{j}a^{\ast}_{i}\), \(i\not= j,\) \(a_{i}a_{j}=a_{j}a_{i}\), \(i\not= j\). From the viewpoint of Wick algebras it is sufficient to consider the relations between \(a_{i}\), \(a^{\ast}_{j}\), \(i, j=1, \dots ,d\), and also to consider a \(^{\ast}\)-algebra WCCR which is generated by the above mentioned generators connected by means of the two relations indicated above. The maximal cubic Wick ideal \(I_{3}\) is generated by the following system of elements: \(\{ A_{ij}a_{k}-a_{k}A_{ij}\), \(i\not= j\), \(k=1, \ldots ,d\}\) where \(A_{ij}=a_{j}a_{i}-a_{i}a_{j}.\) In the paper under review, a classification (up to unitary equivalence) of \(^{\ast}\)-representations of Wick algebras is obtained for which the image of the maximal cubic Wick ideal equals zero. The main theorem states that for arbitrary \(\{ \lambda _{ij}\in C\}\) there exists a unique (up to unitary equivalence) representation of relations for which \(A_{ij}=\lambda _{ij}1\), \(i<j\). Explicit formulas are obtained.