Quantum Gravity 396.8: P(B|A) vs P(A-->B) Dilemma is Somewhat Analogous to the Rigged Hilbert Space vs Hilbert Space Dilemma
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From: Osher Doctorow on 23 May 2010 10:26 From Osher Doctorow Rigged Hilbert Space (RHS) rather than Hilbert Space turned out to be the correct space for Quantum Mechanics, although physicists now usually refer to RHS as "Hilbert Space" for simplicity. See: 1) Rafael de la Madrid, "The role of the Rigged Hilbert Space in Quantum Mechanics," 31 pages, arXiv: quant-ph/0502053 v1 9 Feb 2005 and Arno R. Bohm (U. Texas Austin) Phys. Rev. D71 2005 085018 among others. It turned out that expectations, uncertainties, commutation relations were not defined on the whole Hilbert Space for unbounded observables with continuous spectrum, including continuous and resonance spectra. The theory of Distributions was used to solve or repair this difficulty, the word "Rigged" referring to "equipped (with additional things)", and it has many relationships to Probability theory. The Dilemma in P(A-->B) - P(B|A) taking on values between 0 and 1 is a somewhat analogous dilemma, because of the following: 2) If u and v are variables in [0, 1] and u - v takes values from 0 to 1 either inclusively (including 0 and 1) or non-inclusively (except for 0 and/or 1), then the "variation" in a generalized sense between u and v is as large as the range of u and of v (inclusively or non- inclusively). It is thus necessary to have two theories: one for P(A-->B) (Probable Causation/Influence) and one for P(B|A) (Conditional Probability), and to compare the predictions from each, rather than leave the question of whether one is using one or the other "undecided" or even "negative" (only using one of the two). Since physics and mathematics already use P(B|A), therefore P(A-->B) needs to supplement the former in both fields and applications. There are 22 papers in arXiv on "Rigged Hilbert Space," from 1995 through 2006. The ending of arXiv contributions in 2006 was because the incorporation of Rigged Hilbert Space into Quantum Mechanics was regarded as completely solved. Wikipedia's article "Rigged Hilbert Space" includes the de la Madrid reference, but surprisingly omits to mention its main uses! Osher Doctorow |