4 1. MATHEMATICAL SHAPES OF UNCERTAINTY If one extends all functions from L2(−a, a) as 0 to the rest of the line, one can apply the Fourier transform to all such functions and obtain the Paley-Wiener space of entire functions PWa = { ˆ|f L2(−a, a)}. By the Paley-Wiener theorem, an alternative definition of PWa is that it is the space of all entire functions F such that F has exponential type at most 2πa, i.e satisfies |F (z)| Ce2πa|z|, z C, for some C 0, and F L2(R). In addition to the Lebesgue measure on R that is infinite, the standard finite measure on the real line is the Poisson measure Π, dΠ(x) = dx 1 + x2 . We call functions from 1 = L1(R,dx/(1 + x2)) Poisson-summable and measures satisfying d|μ|(x) 1 + x2 Poisson-finite. In our estimates we write a(n) b(n) if a(n) Cb(n) for some positive constant C, not depending on n, and large enough |n|. Similarly, we write a(n) b(n) if ca(n) b(n) Ca(n) for some C c 0. Some formulas will have other parameters in place of n or no parameters at all. For instance, may be put between two improper integrals to indicate that they either both converge or both diverge. If I is an interval on R and C 0 we denote by |I| the length of I and by CI the interval with the same center as I of length C|I|. If A is a subset of Rn we write |A| for the Lebesgue measure of A. 2. Variety of mathematical forms of UP In this section we discuss several mathematical statements stemming from the general formulation of UP given in the introduction. The purpose of this section is to give the reader an idea of the variety of mathematical reformulations of UP. As was mentioned before, a full survey of such results would require a much larger text. The selection presented here is somewhat random and only represents the results closely related to the topics discussed in the main body of the text. To see a more complete picture the reader may look at a number of excellent existing books and papers, such as [10, 11, 39, 50, 82, 58, 60, 83]. 2.1. Heisenberg’s inequality. If f L2(R) is a function of unit norm, then |f(x)|2dx is a probability measure on R. In physical settings, f may appear as a wave function, in which case |f|2 is the probability density of the position of the corresponding wave/particle, in the sense that the probability of finding the particle in a Borel set E is E |f(x)|2dx. The Fourier transform ˆ of the wave function f is the representation of the wave in the momentum space. Since by Parseval’s theorem ˆ has the unit L2-norm as well, | ˆ(x)|2dx is another probability measure.
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