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, x^ ) into (∂x, 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.
Author(s) Details:
Tan Si Do,
Ho Chi Minh-City Physical Association, Ho Chi
Minh City, Vietnam and Universite libre de Bruxelles, UEM, Bruxelles, Belgium.
Please see the link here: https://stm.bookpi.org/RHMCS-V8/article/view/10190
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