CHAPTER 1 Mathematical Shapes of Uncertainty For a non-mathematical audience, the term ‘Uncertainty Principle’ is unequiv- ocally associated with Werner Heisenberg’s 1927 statement, which has become one of the fundamental constituents of quantum mechanics. In his ground-breaking pa- per [63] Heisenberg explained the new principle in physical terms. A mathematical justification quickly followed via the work of Kennard, Pauli and Weyl [75, 148]. The so-called Heisenberg Inequality, see (1.3), contained in Weyl’s work and attrib- uted to Pauli, illustrates the mathematical meaning of the principle in terms of one of the fundamental notions of Harmonic Analysis, the Fourier transform. From 1924 to 1927 Heisenberg held a position of Privatdozent in G¨ottingen, where together with Max Born and Pascual Jordan he worked on the foundations of quantum mechanics. In 1925, ottingen was visited by Norbert Wiener who gave a seminal lecture in Harmonic Analysis. The lecture intrigued the famous G¨ottingen scientists including Hilbert, Courant and Born. In his memoir [149] Wiener writes: “The talk which I gave the ottingen people on my work on general harmonic analysis was very well received. Hilbert, in particular, showed great interest in the subject, but what I did not realize at that time was that my talk was closely keyed to the new ideas of physics which were about to burst into bloom at ottingen in the form of what is now known as quantum mechanics.” Wiener’s lecture concerned the basic question of Harmonic Analysis, the prob- lem of decomposing a wave into a linear combination of simple harmonics. One of the main points of the lecture, according to Wiener’s popular-science account in [149], was the statement that a function and its Fourier transform cannot both be localized. Aiming at a broader audience, Wiener discusses this statement using the example of a musical note which cannot be both short in time and low in frequency. From the point of view of the Fourier analysis, Heisenberg’s inequality expresses a very similar property which looks like the next step in Wiener’s line of reasoning. Although the exact contents of Wiener’s ottingen lecture seem to be lost, it may be viewed as one of the early declarations of the Uncertainty Principle in Harmonic Analysis, which happened two years before Heisenberg’s paper was published. A possible conclusion to this historical remark is that, perhaps, it would be incorrect to think that the Uncertainty Principle came to Harmonic Analysis from Quantum Mechanics, or vice versa. Rather, like many fundamental ideas, it seems to have emerged as a result of interaction between two adjacent areas of science. Mathematical aspects of the Uncertainty Principle were studied by many promi- nent mathematicians throughout the twentieth century. To this day it remains one of the classic, yet active and exciting areas of research. New methods that ap- peared in this area in the last 10-12 years have produced solutions to long standing problems and sparked new interest towards the Uncertanty Principle from analysts worldwide. Samples of this recent activity will be discussed in the present text. 1
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