In this paper we introduce and study a two-parameter family of integral operators on the Fock space \(F^2({\mathbb {C}})\) . We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group \(SL(2,{\mathbb {R}})\) on the Fock space.
