Abstract
Quantum physics and signal processing in the line R are strictly related to Fourier transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A rigged Hilbert space is found and a new discrete basis in R obtained. All operators defined on R are shown to belong to the universal enveloping algebra of io(2) allowing, in this way, their algebraic treatment. Introducing in the half-line a Fourier-like transform, the procedure is extended to R+ and can be easily generalized to Rn and to spherical cohordinate systems.
Highlights
The starting point of Quantum Mechanics in the line R (and its wave functions counterpart L2(R)) are the Weyl-Heisenberg algebra (WHA), X, P, I [1] that defines position and momentum and the Fourier Transform [FT] [2] that relates them each other. This approach is extended to Signal Processing both in Informatics and Optics if, there, the operatorial structure is scarcely taken in consideration
The results obtained on the line R can be rewritten in R+, the vector space defined by the operator Y, with basis {|y } with y into the open set (0, +∞): Y |y = |y y, y | y′ = δ(y − y′)
Rigged Hilbert spaces are shown to be more predictive than Hilbert spaces in Quantum Physics and Signal Processing in Optics and Informatics since operators of different cardinality can be considered together
Summary
The starting point of Quantum Mechanics in the line R (and its wave functions counterpart L2(R)) are the Weyl-Heisenberg algebra (WHA), X, P, I [1] that defines position and momentum (or time and frequency) and the Fourier Transform [FT] [2] that relates them each other. This approach is extended to Signal Processing both in Informatics and Optics if, there, the operatorial structure is scarcely taken in consideration. The extension to Rn and to spherical coordinates will be discussed elsewhere
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