The side issue of the provisional distribution

Whereas zealous adherents of the Everett doctrine - and a fortiori of the naive version of the Copenhagen interpretation that was discussed above -would have it that some sort of objective reality can be attributed to the state vector on a sufficiently large scale, and hence to the probability operator that would be relevant on a more local scale, most other schools of thought, including less naive versions of the dualistic Copenhagen interpretation, would concur with the supposition adopted here to the effect that such entities are essentially of a subjective nature. This contrasts with the status of the Hilbert space operator algebra of eventualities and observables, which have a more objectively well defined nature. According to this principle, the amplitudes (and corresponding 'weightings') of Everett-type branches should be considered as ultimately subjective, whereas the branches themselves can be considered to be objective. This does not, of course, entail that such mathematical structures are 'real' in any ontological sense.

Before leaving the subject of the 'branching' process (misnamed because the number of branches involved in the description of a subsystem need not increase, and might even decrease, when an interaction occurs), it is worth commenting further on the nature of the process whereby an a priori probability operator P(0) is replaced by the corresponding provisional probability operator P given by Eqs. (19.21) and (19.22). The original discussions of this process were formulated in terms of what Dirac [7] referred to as the Schrodinger picture, wherein states are considered to have a time dependence in which the evolution from an initial time t(0) to a later time t is given by an operator transformation P(0) m P that will be given, in the special case of a pure state for an isolated system, by a corresponding vector transformation l^(o)) m |*>. In the special case of an isolated system, such a transformation will be given by a unitary operator U (that is continuously generated by some Hermitian Hamiltonian) according to prescriptions of the standard form |^> = U|^(o)) and P = UP(0)U_1. However, the transformation will in general be of a less simple (non-unitary) type when interaction with an external system is involved. The idea, as discussed by von Neumann, was that the preparation of an actual experimental observation should involve an arrangement whereby a transformation of this latter (non-unitary) type produced a provisional probability operator P of the required form. This is given by the Luder formula, Eq. (19.21).

As originally pointed out by Dirac [7], a representation in terms of such a Schroodinger picture can be translated into an equivalent representation in terms of the kind of Heisenberg picture that has been implicitly adopted throughout the present discussion. In this kind of representation, the relevant state vector |^> or probability operator P is considered to be time-independent, and the effect of Schrodinger-type time translations is allowed for by corresponding transformations of the relevant observables and their constituent eventualities. In the special case of an isolated system, these transformations will be of the standard unitary type, so that, for example, if e(0) is the projection operator corresponding to some particular eventuality at a time t(0), then the corresponding time-transposed eventuality at a later time t will be given by e = U_1e(0) U. (19.24)

The essential advantage of using a picture of this kind is that there is no impediment to its extension to (general relativistic and other) applications, for which no globally well defined Newtonian-type time parametrization may be available, so that the concept of a time translation relation of the form e(o) ^ e might make sense only for very particular locally related eventualities.

As seen from this Heisenberg (as opposed to Schrodinger) point of view, the process of preparation of an experimental observation in the manner prescribed by von Neumann should be thought of not as the replacement of an a priori probability operator P(0) by a different provisional probability operator P, but as the replacement of an initially envisaged (perhaps maladapted) observable {e(0)} by an appropriately adjusted observable {e}, with respect to which the probability distribution already has the required Luderian form.

From this point of view, there is no need to bother about any distinction between a priori and provisional probability operators (which - in view of the possibility of using Eq. (19.23) instead of Eq. (19.22) - were, in any case, equivalent for the practical observational purpose under consideration). What matters for the purpose of making what von Neumann would consider to be a satisfactory observation is the choice of a suitably adjusted observable e. However, the main point I wish to emphasize at this stage is that -although it may be of technical interest in particular applications - the importance of the issue of obtaining a satisfactory observation in the sense specified by Luder's rule has been greatly exaggerated, in so far as its relevance to the ultimate interpretation of the observations process is concerned. To start with, there is the consideration that the Luderian desideratum is obtainable not only by the non-trivial process described above, whereby {e} is adjusted to a previously chosen probability operator, but also by the trivial process whereby the subjective a priori choice of P is adjusted ad hoc to fit a prescribed observable {e}, an adjustment that in no way diminishes the credibility of its implications, as can be seen from the equivalence of the prescriptions given in Eqs. (19.22) and (19.23).

A more fundamental reason why the question of the Luderian transition is irrelevant is that, when an observation has been actually carried out (not merely planned), one will be left just with a single confirmed eventuality ei. Such a single eventuality might be incorporated with others to constitute a complete observable set (spanning the entire Hilbert space) in many different ways, whose substitution in the Luder formula, Eq. (19.21), would provide many different results. Nevertheless, however that might be, and regardless of any distinction that may or may not have been made between an a priori probability distribution P(0) and a provisional probability P, one will be left with an unambiguously specified a posteriori probability distribution P-], which is all that matters for the purpose of subsequent predictions one may wish to make.

The upshot is that someone (such as Wigner) concerned about Schrodinger's cat should use the a posteriori distribution when the relevant information has become available, and until then should just continue to use the ordinary a priori distribution. One should avoid getting sidetracked (as so many of Everett's followers have been) by intermediate Luderian technicalities, whose analysis is of little relevance to the two outstanding issues that remain. In addition to the question of interpretation, which will be addressed from an anthropic point of view below, the other outstanding issue is the usual practical Bayesian dilemma of how to decide quantitatively what a priori distribution should be used in a particular context -something that can sometimes be resolved just by symmetry considerations (as in the coin-tossing example described above).

19.10 Perceptions and perceptibles

An important idea that was latent in much of the preceding discussion is that some privileged eventualities and observables are more naturally significant than others. In the discussion of Luder's rule, it was remarked that this can be interpreted as selecting a privileged class of observables, but I should emphasize before continuing that privilege of that kind is not what I am concerned with here, because it is ultimately dependent on an arbitrary subjective choice of the relevant a priori probability distribution. The kind of privilege I am concerned with is something that depends on the essential nature of the system under consideration, in a manner that is independent of the choice of the probability distribution. This is something that could be said about Bohm's idea [13] of privileging position with respect to its dynamical conjugate, namely momentum, but that particular choice is something that would not seem very natural to the numerous physicists whose mental life is based in Fourier space.

The kind of privilege that seems to me more relevant for the interpretation question is something that would be rather generally recognized as being imposed by the circumstances in particular cases. It is exemplified most simply by the existence of a privileged choice (determined by the background magnetic field) for the the particular spin eventualities characterized as 'up' and 'down' in the Stern-Gerlach experiment discussed above. It is also exemplified by many familiar kinds of apparatus, such as can be found in scientific laboratories, and increasingly in ordinary homes, whose output is typically presented in terms of what - at the highest resolution - usually turns out to consist of simple integer valued observables, such as the alternative eventualities in the range from 0 to 9 for a digit in a counter output, or the binary alternatives for a particular pixel on a screen to be 'on' or 'off'. It is mathematically possible to use other bases for a Hilbert space description of such systems, for example by working with eventualities defined as linear superpositions of 'on' and 'off' states of screen pixels, but that is evidently not the kind of treatment for which such an apparatus was intended by its designer.

Although the degree of complexity of the systems involved is very different, it seems to me that there is a rather strong analogy between the special role of the 'on' and 'off' states for a pixel on a screen and the 'awake' and 'sleeping' states of Schrodinger's cat. The privileged status of the particular eventualities in question can be accounted for as the result of a process of design that is attributable in the first case, not just to an individual engineer, but to the collective activity of a scientific community, while it is attributable in the second case to a very long history of biological evolution by Darwinian selection. Having said this about the cat, the next thing to be said is that the same applies to Schroodinger and Wigner, for whom the relevant privileged eventualities are states of mind corresponding to the realization of whether or not the cat is awake.

Whatever doubts we may have about the status of the cat, we must recognize that Schrodinger and Wigner are closely analogous to ourselves (i.e. the author and presumed readers of this essay), which means that insight into the working of their minds can be obtained from our own experience. The only eventualities about whose reality we can be sure are the conscious perceptions in our own minds (of which some, namely those occurring in dreams, are evidently uncorrelated with anything outside). These correspond to the 'mind states' whose essential role has been recognized by authors such as Donald [22], Lockwood [23] and particularly Page [24], whose line of approach is followed here. It seems reasonable to postulate the validity of Page's principle, according to which conscious perceptions are the only eventualities that can be considered actually to happen. It also seems reasonable to make the concomitant postulate that these perceptions must belong to some restricted class of privileged eventualities of the kind discussed in the preceding paragraph. I shall refer to the eventualities of this subclass as perceptibles.

In his 'sensible quantum theory' [24], Page has attributed a privileged role to a class of observables that he refers to as 'awareness operators', which I interpret to mean observables whose individual constituent eventualities are the perceptibles introduced in the previous paragraph. Page has used these particular operators to develop a refined version of the Everett interpretation, in which the branches - or as I would prefer to say, channels - that matter are specified with respect to these awareness operators. Thus, whereas Everett's original version might attribute 'actuality' to branches defined with respect to observables of a rather arbitrary kind, Page's more refined version would attribute 'actuality' only to branches of an appropriately restricted kind, namely the channels that are specified by perceptibles. Having thus provided a much clearer idea of which channels are actually needed, Page was still left with the problem of interpreting what, following Everett's evasive example, he referred to as their 'weighting'. The point at which Everett stumbled was in trying to reconcile his recognition that the weighting was needed with his preceding claim that all the branches were equally real. Page came up against the same problem with respect to the claim to the effect that all the perceptibles are actually perceived.

19.11 The anthropic abstraction

A corresponding paradox is reached from a rather different angle in the approach I am developing here, which is in agreement with that of Page [24] in so far as the special role of perceptions is concerned, but differs in affirming that the weighting in question must be considered to have an essentially subjective and probabilistic nature. The intrinsically probabilistic nature of models of the kind advocated here raises the problem of what it can mean to attach a probability to the actuality of an eventuality in the mind of someone else if the only events one can actually observe are those occurring in one's own mind.

Before presenting what I think is the only acceptable way of dealing with this paradoxical problem, I would mention two less satisfactory ways of resolving the issue that have been suggested in the past. The first way is that of the solipsist, who would deny the existence of any conscious perceptions other than his (or her) own [20], with the implication that the apparent analogy between oneself and others (such as Schrodinger) is merely a superficial illusion. The second way -which (unlike that of the solipsist) has been followed up by many physicists, starting with de Broglie - is to revert to a deterministic description of the world, providing a theoretically well defined answer to the question of what really happens by denying the (experimentally well established) validity of the essentially probabilistic description provided by orthodox quantum theory. Neither the first nor the second of these ways can be said to resolve the paradox; they merely evade the issue by dropping one or other of the essential (experimentally motivated) elements of the problem, which is that of providing an inherently probabilistic treatment of perceived reality that respects the apparent symmetry between different people.

A historical analogy is provided by the incompatibility between Maxwellian electromagnetism and Newtonian gravity, which was ultimately resolved by their unification in Einstein's General Relativity. The problem to be dealt with here is that of reconciling subjective probability with objective reality. The only way that I know of solving this problem in a satisfactory manner is the anthropic approach, which faces the issue head on [3] without denying the validity of the considerations that lead to the paradox.

It is worth emphasizing, by the way, that the problem is not specifically a problem of quantum theory, but also arises in probabilistic versions of classical theory, as was recognized, I suspect, by many of those who were hostile to anything associated with the name of Bayes. The importance in this context of the quantum revolution is that it changed the status of Bayesian theorists from that of radicals (because they were willing to abandon determinism) to that of reactionaries (because they continued to use old fashioned Boolean logic).

The situation, as I understand it, is as follows. Suppose that to describe a system that includes ourselves (but, for the sake of finiteness, perhaps not the whole of the Universe) we have set up some (classical or quantum) theory that provides probabilities for an extensive class of eventualities. This class includes a specially privileged subclass of eventualities that I shall refer to as perceptibles, which are the only ones that can be actually observed as conscious perceptions. The set of such perceptions (not just yours and mine, but also those of everyone else) can be described as objective, and it is the only thing in the theory that can be considered to be real

We thus have an objective model attributing probabilities to perceptibles. But what sense can it make to attribute a probability to an observation you cannot make? If you are Wigner, what sense can it make - even in a classical theory - to use an objective distribution attributing probability to something that can only be known by Schroodinger? The contradiction arises when Schrodinger makes the Bayesian transition to the relevant a posteriori distribution, while Wigner continues, for the time being, to use the a priori distribution. How in these conditions can either of these distributions be considered to be objective?

The resolution to this paradox is provided by what may be called the anthropic abstraction (so called because it underlies that which I designated - perhaps inaptly - as the anthropic principle [25]). The paradox that arises in this case (as in many others) can be attributed to an unnecessary assumption that has been consciously or subconsciously taken for granted. The unnecessary assumption is that of knowing in advance who one is. The anthropic abstraction consists in refraining from assuming in advance that one has the identity of some particular sensorial observer in the model, so that one's status a priori is that of what I shall refer to as an abstract perceptor. It is not until the actual happening of the perception that one can know whether one is Schroodinger, Wigner or whoever else may be included in the model.

It is, of course, to be understood that the perceptible eventualities that are involved in this anthropic approach cannot just be of the elementary type exemplified by the observation that someone else is awake, but that they need to include eventualities of the more complicated kind known as consistent histories [26]. The sort of eventuality that needs to be envisaged is not simply that of finding oneself to be Schroodinger, but that of finding oneself to be Schrodinger at a particular instant in his life, with all the memories he would have had at that moment.

The use (which I see no satisfactory way of avoiding without reverting to determinism) of the anthropic abstraction entails the need to adopt some kind of anthropic principle, by which I mean some kind of prescription for attributing appropriate probabilities to the relevant perceptible eventualities. The rather crude kind of anthropic principle that I have put forward on previous occasions [2] was concerned with the attribution of probability to entire observer systems (such as those associated with the names of Schroodinger or Wigner) without getting into the details of particular moments in their lives. For the applications I was then considering, it was sufficient to use a crude statistical treatment attributing equal weight to all terrestrial or extraterrestrial observers who can be considered to be sufficiently like ourselves to be describable as 'anthropic'. However - as several authors have already remarked [24,25,27] - the more detailed applications I have been considering here (particularly those involving quantum effects) require the use of a more refined kind of anthropic principle [3], which will distinguish not just between anthropic individuals, but also between different instants in the lives of such individuals.

The question that naturally arises at this point is whether it can suffice to use just the probability weightings that are directly provided by orthodox quantum theory (such as has been discussed above), in conjunction with some prescription for deciding which of the many mathematically defined eventualities in the model should be considered to have the privileged status of perceptibility.

Was this article helpful?

0 0
Positive Thinking As The Key To Success

Positive Thinking As The Key To Success

Download this Guide and Discover How To Find And Monetize on Your Expertise And Strengths. Inside this special report, you'll discover: How positive thinking is one of the key factors in a successful life. Five ways and tools to help you stay positive. Use these to help you keep on track. Case studies that'll inspire you to stick to your dreams. Plus much, much more.

Get My Free Ebook


Post a comment