From: (Hugh Miller)
Newsgroups: sci.crypt

Subject: Re: Are there truly random phenomena?
Date: 5 Aug 91 20:38:48 GMT
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Sender: (NewsMistress)
Organization: University of Chicago
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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

       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

   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:
  "Power, not reason, is the new currency of this court's decision-making."
    -- Hon. Thurgood Marshall (dissenting), Tennessee vs. Payne, 6/27/91