[math-fun] Items from the most recent AMS journals
The January 2003 issue of the Bulletin of the AMS has a review by Steven Krantz of "A new kind of science". This has been thoroughly discussed within math-fun, so I will make only two remarks. Footnote 2 of the review claims: "Wolfram formerly had a contract with Addison-Wesley to publish his book. One sticking point in that relationship was that he demanded that any reviewer sign a nondisclosure agreement and promise not to study math or physics for the ensuing ten years." Footnote 7 repeats an error that I have seen one time too many. If f(x) and F(p) are Fourier transforms of each other, then int(x^2 f(x)^2) * int(p^2 F(p)^2) >= 1 is called a statement of the Heisenberg uncertainty principle (HUP). Does this theorem (with suitable hypotheses appended) have a name? It is not the HUP. Calling this theorem the HUP would falsely imply, for example, that if I am given the exact electric and magnetic fields in an electromagnetic pulse, that I could not calculate precisely both its centroid and total momentum. But this is not so. The energy and momentum densities are given by the field components; there is no need to determine frequencies or wavelengths. This example should make it clear that the HUP is strictly a principle of quantum theory; it does not appear in any manner in classical physics. For the case of an electromagnetic pulse, in order for the Fourier transform theorem above to be applicable, I need to be concerned with individual photons; but then this is quantum theory. The January 2003 issue of the Notices of the AMS has a review of "Indra's Pearls". Severe ongoing constraints on my time have prevented me from reading this book, but it is my intention to do so eventually, and report on it to math-fun. Apparently this book was 20 years in the making. It seems incredible with these two books having been released at about the same time, that any competent mathematician would choose to devote the time to reading Wolfram's tome. Gene Salamin __________________________________________________ Do you Yahoo!? Yahoo! Mail Plus - Powerful. Affordable. Sign up now. http://mailplus.yahoo.com
I've always seen this inequality referred to as the "Uncertainty Principle", even in digital signal processing, where it is the embodiment of the principle "a signal cannot be time-limited and frequency-limited simultaneously". In quantum mechanics, one deals with "conjugate pairs" of variables, such as position-momentum, which in QM are Fourier transforms of each other. In this setting, the inequality below (usually with 1/16 on the RHS), is indeed the mathematical embodiment of the Heisenberg uncertainty principle. DHB int(x^2 f(x)^2) * int(p^2 F(p)^2) >= 1 is called a statement of the Heisenberg uncertainty principle (HUP). Does this theorem (with suitable hypotheses appended) have a name? It is not the HUP.
participants (2)
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David H Bailey -
Eugene Salamin