Nonstandard analysis of linear canonical transformations on a Fermion Fock space with an indefinite metric (Q1191267)

The authors extensively use nonstandard analysis and Berezin calculus to investigate the linear canonical transformations in an infinite dimensional Fermion Fock space with an indefinite metric, especially their implementability by a \(\Lambda\)-unitary operator. In the same time, the authors show how the Berezin calculus on a finite dimensional superspace can be applied to analysis on an infinite dimensional Fermion Fock space by using nonstandard analysis. In the indefinite metric case, even if the standard part of a nonstandard \(\Lambda\)-unitary operator exists, it is not necessarily a bounded operator, and therefore the authors introduce the notion of weakly \(\Lambda\)-unitary operators. The main result of the paper (Theorem 7.7) states that under certain conditions the linear canonical transformation is weakly \(\Lambda\)- unitarily implementable.