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 LΠ

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.