Gaussian Transforms in 2N-dimensional Phase Space

In order to obtain with simplicity the properties of linear canonical transformations (LCTs), it is introduced the notion of dual couple of operators having commutator identical to unity and the property that all relations between them are valuable for any other dual couple. It follows that from the translation operator exp (a
x) which transforms the dual couple (
x,
) into (
x,
+ al) one obtains the dilatation operator exp (aBA) which transforms the dual couple (A, B) into (e -al A, e -al B) then the operator exp (aA2) exp (aB2) and so all, leading to the construction of general linear and linear canonical transformations in phase spaces. Moreover, are also obtained the LCT transforms of functions. By this way different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again. We also obtain a clear relationship between linear and linear canonical transforms from the formula representing the aforementioned integral realisation. Numerous LCT examples are provided to highlight the method's ease of use.