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
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
f of the wave function f is the representation of the
wave in the momentum space. Since by Parseval’s theorem
ˆ
f has the unit
L2-norm
as well, |
ˆ(x)|2dx
f is another probability measure.
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