Six Sigma

Let me begin by expressing my personal worries and beliefs about the quantum theory. First, let it be said without reservation that the basic scheme, for all its philosophical difficulties, is an extraordinarily beautiful mathematical structure. The strength of the theory (and here I refer to standard non-relativistic quantum theory, not to quantum field theory) lies not just in the unbelievable range and accuracy of

its physical predictions, but also in the mathematical elegance of its formalism.

It is here that I have always had difficulties with most hidden-variable theories. To me, it is no help just to improve upon the underlying philosophy of quantum mechanics by the introduction of hidden variables if the price to be paid is the sacrifice of this mathematical elegance. Yet it must also be emphasized that, in my view, the standard theory is indeed quite unsatisfactory philosophically. Like Einstein and his hidden-variable followers, I believe strongly that it is the purpose of physics to provide an objective description of reality.

However, I do not regard indeterminacy, in the ordinary sense of that word, as being necessarily objectionable. We have become accustomed, through classical physics, to a picture of the world whose future evolution is completely determined by data on an initial Cauchy hypersurface. Yet I find nothing a priori appealing about the idea. We know from results of general relativity (Penrose3, Hawking and Ellis4) that there are otherwise seemingly acceptable space-times for which no Cauchy hypersurface exists. More serious are two other objections. In the first place, the accuracy with which one needs to know the initial data in order to be certain about the qualitative development of a system is absurdly unreasonable (Born5, Feynman et al.6) and easily swamped by quantum uncertainties (if that is not begging the question). In the second place, there is always the problem of deciding what initial data one should be choosing. It seems to me to be of little value to know that there is some initial data containing all information if one has no rules for

determining what that initial data is allowed to be or likely to be.

Considering things on the ultimate universal scale, one needs, it would seem, some theory to tell us how the universe actually started off, or at least which provides probability values for the different possible initial data sets. But if it is the latter, it seems to me that one is not really better off than with a non-deterministic theory. I see no reason to be happier about feeding probabilities into the universe’s initial state than peppering them throughout the space-time, as is done in the standard interpretation of quantum mechanics. If, on the other hand, the initial state is to be determined uniquely by some new principle, then we have the problem of understanding the extreme complication and curious interplay between precisely-operating physical laws on the one hand, and the presence of apparent total randomness on the other. The picture is not an entirely hopeless one, however. One is beginning to become accustomed to the great mathematical complication, variety and apparent randomness that can arise with very simple and precise mathematical transformations, especially where iterative procedures are involved (cf. Ott7 _{and references therein). So perhaps all the complication, variety and apparent}

randomness that we see all about us, as well as the precise physical laws, are all exact and unambiguous consequences of one single coherent mathematical structure.

Such a view I would call strong determinism. My guess is that Einstein8 _{was hinting at such a possibility in his famous remark ‘What I’m really}

interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all.’ Perhaps this view would represent the ultimate ‘optimistic’ attitude to the goals of science. Yet strong determinism is quite unlike ordinary determinism. There is not now any question of the future state being determined by data on an initial Cauchy hypersurface. The entire future (and past) is simply fixed by theory once and for all!

I do not propose to take sides on this grandiose issue. We are clearly extremely far from such a ‘theory’, even if eventually a viewpoint of that kind were to turn out to be ‘correct’. My immediate purpose in bringing such matters up is largely to point out that, strong determinism aside, it seems no worse to feed probabilities into the theory ‘peppered’ throughout the space-time than to feed them all into the initial conditions. I shall return to strong determinism later. At this point I should make mention of the many-worlds types of viewpoint9,10,11_{. Here one may be allowed a form of ‘strong}

determinism without determinism’. The totality of all possible universes may be thought of as a single structure—the omnium (this terminology was suggested to me by Peter Derow)—and one might take the view that it is the omnium that is completely fixed by mathematical rules. The probabilities (or randomness) now arise owing to the uncertainties involved in the question of where one finds oneself located within the omnium. This ‘location’ involves not only one’s spatiotemporal location within a particular universe branch, but also the selection of that particular branch itself.

Unsatisfied though I am by such a world-view, I do not have any really fundamental objection to pictures of this general kind. One problem I do have with them, however, is that the continual branching of the world and the threading of my own consciousness through it would seem to result in my becoming separated from the tracks of consciousness of all my friends. This is what I have referred to as the ‘zombie’ theory of the world12. It seems to me that one needs a respectable theory of consciousness before the many-worlds view can hang together as a physical theory and as a viable

interpretation of quantum mechanics. This strikes me as a tall order at the present time. I shall have more to say about the many-worlds viewpoint later. Let us now consider what the standard formalism of quantum mechanics has to say about the evolution of the world. Taking this formalism at its face value, we have a state vector |ψ> which evolves for a while according to the completely deterministic Schrödinger

equation. (If preferred, one could of course use the Heisenberg picture instead, in which case the state itself is considered not to evolve in time. The distinction is not important for my purposes. The two pictures are completely equivalent, but I feel that, at least for a non-relativistic discussion, the Schrödinger picture is less confusing.) Then, at odd times, when an ‘observation’ is deemed to have been made, the Schrödinger-evolved state vector is discarded and replaced by another, which is selected in a random way, with specific probability weightings, from among the eigenvectors of the operator corresponding to the observation. As has been argued on innumerable occasions, this is a wholly unsatisfactory procedure for a fundamental description of the ‘real world’. There are, of course, very many different attitudes to the resolution of this problem, and it is not my purpose here to enter into a discussion of all of them, but some brief remarks will be in order.

In the first place, it is often argued that |ψ> itself should not be regarded as giving an objective description of the world (or of part of it) but as

providing information merely of ‘one’s state of knowledge’ about the world. This view I really cannot accept. Quite apart from the question as to who the ‘one’ might be in this statement (and the ‘one’ is surely not me!), it seems to me to be perfectly clear that there is (if we accept standard quantum

mechanics) a completely objective meaning to |ψ>—at least to the ray determined by |ψ> in the Hilbert space (so that uncertainty is not of significance). For we can, in principle (according to the theory), set up a measurement defined by the operator:

and find that |ψ> (assumed normalized: <ψ|ψ>=1) is the unique state (up to phase) corresponding to the eigenvalue unity for Q. This |ψ> is distinguished from all other states by the fact that it yields the value unity with certainty for the measurement Q. This is an entirely objective property, so we conclude that (if the theory is correct) the property of being in state |ψ> is, indeed, completely objective.

In practice, however, it may well turn out that the actual performing of the measurement Q is quite out of the question and it is in such circumstances that I would myself begin to doubt that |ψ> actually describes ‘reality’. (There are situations, in the context of relativity or in connection with time- symmetry considerations, where the ‘reality’ of |ψ> seems to lead to a paradox; cf. Penrose12_{. I am not concerned with such matters here, but such}

‘paradoxes’ are part of my own doubts about the complete validity of the quantum formalism.) To reject the objective view requires a denial of one of the fundamental tenets of quantum theory: any (bounded) Hermitian operator, such as Q, represents a measurement that could in principle be made.

the state vector is actually intended to represent ‘reality’. The trouble comes, of course, when a measurement is made for which |ψ> is not in an eigenstate. Now |ψ> jumps non-deterministically into some new state—this is state-vector reduction. Because of this lack of determinism it is not possible to consider such behaviour as resulting from the Schrödinger evolution of some larger system which includes the apparatus configuration as part of the state. (Some attempts have been made to pin the blame for this indeterminism on a lack of knowledge about the environment, the claim being that when this random environment is taken into account, Schrödinger evolution might hold always. I have not yet found these attempts to be very believable and I shall ignore this possibility in the discussion which follows.) So it seems that quantum mechanics asserts two quite distinct types of evolution: deterministic Schrödinger evolution and state-vector reduction.

If we accept that state-vector reduction is a real physical process—and the physical objectivity of the state vector itself seems to imply this (leaving the many-worlds viewpoint aside)—then we may ask: at what stage does such a reduction take place? Again there are different views. Perhaps it occurs quickly and spontaneously at some level just not quite reached by experiments designed to detect violations in Schrödinger evolution. Or perhaps it is delayed until the latest allowable moment, as the results of the observation finally enter the mind of some conscious observer (Wigner1_{). As we well know}

(von Neumann13) either view, or any other viewpoint according to which the reduction takes place at some intermediate stage between these two extremes, is equally and completely compatible with all the experiments.

This frustrating and beautiful fact is, at one and the same time, among the greatest strengths and the most disturbing weaknesses of the theory. It is such a strength because it enables the theory to operate—as it does with such extraordinary accuracy and power—without our needing to have the remotest idea of what ‘actually’ takes place during the reduction process. This strength is also the theory’s weakness, since for that very reason it offers us almost no clue as to what is physically going on during reduction. Moreover, it leads us into endless arguments about ‘interpretation’ where strange (and, to my mind, highly questionable) philosophy is often invoked to lull one into thinking that no new physical theory is needed to explain the details of the actual physics of reduction.

The reader will realize that I am expressing a very personal view here, and that much more discussion and open-mindedness is required than I have allowed myself in the foregoing remarks. However, my purpose here is not to be open-minded but to make some suggestions, so I hope that the reader will continue to bear with me and, instead, take upon himself or herself this burden of open-mindedness. I wish

to make some comments concerning the place to look for a new physical theory of reduction, and then to make a speculative suggestion as to how this might ultimately relate to the phenomenon of consciousness. In bearing with me, the reader must, as a first step, be prepared to envisage that there is indeed a real physical process of reduction, where the Schrödinger equation and linearity are presumably both violated and where determinism may perhaps be violated also.

It may seem that, as I am suggesting a connection with consciousness, I propose to follow the Wigner1 _{extreme, where reduction takes place only at}

the level of conscious thought. In fact I am not at all happy with that viewpoint and my suggestions with regard to reduction will be quite different. I share the discomfort of many others that the ‘Wigner view’ seems to imply a markedly different physical behaviour in our own small corner of the universe from that which would be taking place almost everywhere else, in the absence of local conscious observers to keep things under control. Of course, von Neumann again comes to the aid of that viewpoint to ensure that no contradiction with actual observation (by conscious beings) occurs. Or so it would seem. In any case, I personally find such a lop-sided picture of a real physical universe too unattractive to be believable.

Alternatively, as quite a number of physicists now seem to argue, perhaps one should go beyond Wigner and say that reduction fails to take place, even at the level of consciousness. This leads us, instead, to Everett’s many-worlds picture. It seems that the main motivation behind this idea is the

understandable desire for economy and uniformity of description. Rather than having to have two seemingly incompatible modes of evolution for the universe—Schrödinger’s equation and reduction—is it not more economical and unified to settle just for the former and discard the latter? A faith in the elegance of Schrödinger linearity and the strong experimental support for Schrödinger evolution are among the arguments put forward for this view, but in my opinion both arguments are somewhat misconceived. Linearity seems elegant only when one has not seen an even more attractive, essentially non- linear, generalization. (Compare the analogy of Newtonian gravity and general relativity.) And the unquestionably impressive experimental support that quantum theory continues to have is not for the Schrödinger evolution of a quantum state. It is for that absurd concoction of Schrödinger evolution on the one hand and state-vector reduction on the other which defines the standard Copenhagen interpretation. That is where the support lies and it is that with which we must come to terms in our attempts toward an improved theory.