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THE UNCERTAINTY PRINCIPLE MICHAEL G COWLING The uncertainty principle states that a function f on Rn and its Fourier transform fˆ can not both be too highly localized. In the Schrödinger formulation of quantum mechanics, this becomes the statement that both the position and the momentum of a quantum particle cannot be prescribed too accurately, and if we think of the Fourier transform as giving a spectral version of the function, then the principle states that a function and its spectrum cannot both be too highly localized. One variation of the principle, sometimes ascribed to Heisenberg, Pauli and Weyl, states that Z 2 Z Z 2 2 2 |f (x)| dx ≤ C |x − a| |f (x)| dx |ξ − α|2 |fˆ(ξ)|2 dξ R R R and another, due to Hardy, states that if |f (x)| ≤ Ce−a|x| 2 and |fˆ(ξ)| ≤ Ce−α|ξ| 2 where a, α > 0, and if aα is large enough, then f has to be zero. These and similar results have been generalised into different areas, such as spectral analysis of functions on manifolds and the analysis of operators. This talk reviews the original theorems, and discusses recent developments of the theory. 1