The Uncertainty Principle: a Brief Survey
نویسنده
چکیده
The uncertainty principle is a cornerstone in quantum phsysics. However, its principles play an equally monumental role in harmonic analysis. To put it in one sentence: A nonzero function and its Fourier transform cannot both be sharply localized. While Heisenberg gave a clear physical interpretation of the uncertainty principal in 1927 in [7], it contains little mathematical precision. This was remedied later by Kennard ([8]) and Weyl ([11]) in 1928. We still credit the famous inequality (2.1) to Heisenberg. The paper is structured as follows: In Section 2, we present the simplest and most common form of the uncertainty principle in harmonic analysis (Heisenberg’s inequality). We then extend Heisenberg’s inequality to Hilbert spaces (Section 3) and L spaces (Section 4). In Section 5, we present some of the results from local uncertainty inequalities and their ramifications. We end the paper (Section 6) with a grab-bag of other interesting results. Uncertainty principle results and inequalities are bountiful; this paper does not even scratch the surface of the hundreds of results on this topic. The information in this document primarily comes from Folland’s “The Uncertainty Principle: A Mathematical Survey” [6] which is why this document is titled “... : A Brief Survey”. Nonetheless, we present many of the important uncertainty results with full proofs, sketches of proofs, and, otherwise, references for the reader.
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