Path: cactus.org!milano!cs.utexas.edu!samsung!olivea!uunet!midway!gargoyle. + uchicago.edu!hugh From: hugh@gargoyle.uchicago.edu (Hugh Miller) Newsgroups: sci.crypt Subject: Re: Are there truly random phenomena? Message-ID:Date: 5 Aug 91 20:38:48 GMT References: <44901@cup.portal.com> <15218@ulysses.att.com> <148@mtnmath.UUCP> Sender: news@midway.uchicago.edu (NewsMistress) Organization: University of Chicago Lines: 91 In <148@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes: >Physicists have a pretty bad record on this point. In the 1930s von Neuman >published a famous proof that claimed no more complete model could produce >predictions consistent with quantum mechanics. This was widely accepted >until the mid 60's when Bell published a refutation. There is no basis >for the widely accepted belief that randomness is fundamental to quantum >mechanics. Roger Penrose: "Regarding as describing the `reality' of the world, we have none of this indeterminism that is supposed to be a feature inherent in quantum theory -- so long as is governed by the deterministic Schroedinger evolution. Let us call this the evolution process _U_. However, when we `make a measurement', magnifying quantum effects to the classical level, we change the rules. Now we do _not_ use _U_, but instead adopt the completely different procedure, which I refer to as _R_, of forming the squared moduli of quantum amplitudes to obtain classical probabilities! It is the procedure _R_, and _only R_, that introduces uncertainties and probabilities into quantum theory... "The descriptions of quantum theory appear to apply sensibly (usefully?) only at the so-called _quantum level_ - of molecules, atoms, or subatomic particles, but also at larger dimensions, so long as energy differences between alternative possibilities remain very small. At the quantum level, we must treat such `alternatives' as things that can _coexist_, in a kind of complex-number-weighted superposition. The complex numbers that are used as weightings are called _probability amplitudes_. Each different totality of complex-weighted alternatives defines a different _quantum state_, and any quantum system must be described by such a quantum state. Often, as is most clearly the case with the example of _spin_, there is nothing to say which are to be the `actual' alternatives composing a quantum state and which are to be just `combinations' of alternatives. In any case, so long as the system _remains_ at the quantum level, the quantum state evolves in a completely _deterministic_ way. This deteministic evolution is the process _U_, governed by the important _Schroedinger Equation_. "When the effects of different quantum alternatives become magnified to the _classical level_, so that difference between alternatives are large enough that we might directly perceive them, then such complex-weighted superpositions seem no longer to persist. Instead, the squares of the moduli of the complex amplitudes must be formed (i.e., their squared distances from the origin in the complex plane taken), and these _real_ numbers now play a new role as actual _probabilities_ for the alternatives in question. Only _one_ of the alternatives survives into the actuality of physical experience, according the the process _R_ (called reduction of the state-vector or collapse of the wave-function; completely different from _U_). It is here, and only here, that the non-determinism of quantum theory makes its entry.... "The deterministic process _U_ seems to be the part of quantum theory of main concern to working physicists; yet philosophers are more intrigued by the non-deterministic _state-vector reduction R_. Whether we regard _R_ as simply a change in the `knowledge' available about a system [Bohr], or whether we take it (as I do) to be something `real', we are indeed provided with two completely _different_ mathematical ways in which the state-vector of a physical system is described as changing with time. For _U_ is totally determinisitc, whereas _R_ is a probabilistic law; _U_ maintains quantum complex superposition, but _R_ grossly violates it; _U_ acts in a continuous way, but _R_ is blatantly discontinuous. According to the standard procedures of quantum mechanics there is no implication that there can be any way to `deduce' _R_ as a complicated instance of _U_. It is simply a _different_ procedure from _U_, providing the other `half' of the interpretation of the quantum formalism. All the non-determinism of the theory comes from _R_ and not from _U_. _Both_ _U_ and _R_ are needed for all the marvelous agreements that quantum theory has with observational facts." How would you propose to use, say, a Geiger counter strapped to a beta-emitter or the hiss of a hot resistor without performing a state reduction? And, if so, you introduce randomness. >If you think about this as a problem in the theory of recursive functions >you can see how difficult it would be to come up with a theoretical argument >or experimental results that prove this. You are asking is there a recursively >enumerable set that includes all the predictions of quantum mechanics (a >recursively enumerable set) and all the observed results of experiments >(a finite set). Since the union of a finite and recursively enumerable set >is a recursively enumerable set the answer is yes. Why is `the set of all predictions of quantum mechanics' (?) a recursively denumerable set? Hugh Miller | Dept. of Philosophy | Loyola University of Chicago Voice: 312-508-2727 | FAX: 312-508-2292 | UUCP: hugh@gargoyle.uchicago.edu "Power, not reason, is the new currency of this court's decision-making." -- Hon. Thurgood Marshall (dissenting), Tennessee vs. Payne, 6/27/91