Chris Clarke

Faculty of Mathematical Studies, University of Southampton

Southampton, SO17 1BJ, UK

A volume such as this is characterised by the diversity of the authors' approaches, and so it seems particularly necessary here to assist the reader at the start of this chapter by explaining ¾ only in the briefest way ¾ the general background to my own approach.

I see our aim as that of helping people, individually and as social groups, to live good lives. Of course, finding the meaning of this last phrase is a large part of the religious quest. In the pursuit of this aim we tell stories about the world we live in, because this is part of the essence of human life, and the stories are effective when they are well grounded in experience. Some stories derive from science, in which case there is a spectrum between mathematical models that are testable experimentally, and pictures in the sense of contributions to a general world-view supported by science. Other stories are drawn from the great myths of religion. There is nothing to stop us weaving together these stories to create a greater whole in the light of our deepening experience; but Kant (and, in a way more applicable to today, Merleau-Ponty) has taught us that we must guard against thinking that any part of such a story represents a reality that stands independently of ourselves as a thing in itself. Nor are all parts of such a composite story equally supported by experimental tests.

To say that a part of a story (religious or scientific) is not independent of ourselves does not imply, however, that it is arbitrary or entirely subjective. I have developed my personal position in a recent paper, of which a very rudimentary sketch can be given here. I start from the phenomenological position of Heidegger and, more especially, Merleau-Ponty, who accept that the everyday world is a construction, but that its ground is a pre-conceptual experience of Being-in-the-world, of presence. Unlike the Cartesian picture, the ground of experience is not sensations (the elements revealed by neurophysiology, though this was unknown to these authors) but this encounter with a totality. What we call the objective world arises from the overlaps between the total worlds of different beings. This overlapping is more than just a passive intersection, however, but a dynamic dialogue, characterised by Merleau-Ponty's description of perception, when "suddenly the sensible takes possession of my ear or gaze, and I surrender a part of my body, even my whole body, to this particular manner of vibrating and filling space." [2] So-called objective reality is thus primarily located neither in the subject, not the object, but in the second-person relationship between the two. The overlapping of the domains of perspectives on the universe which arises in relativistic version of the histories which I describe below is a static and physicalist version of this conception of reality. When viewed in its fully dynamic form, however, it provides an articulation of the participatory approach to reality of Heron and Reason[3}. According to them, any dividing line between the subjective self and the Other must remain indeterminate, the witness of philosophy since Kant indicating to me that the project of determining such a line is based on a misunderstanding of the human condition. My aim is not to discover what part of my view coincides with ultimate reality, but to grasp the concreteness of the world as it in fact emerges in the interplay of self, society and Other. Quantum theory for me plays an essential role in understanding this interplay, but it does not thereby impose an unnecessarily realist metaphysics on the basic openness of our experience.

Quantum theory emerged historically as a way of handling the physics of the very small. This is reflected in the role within the theory of a fundamental constant, Planck's constant h, whose value when expressed in conventional laboratory units is a very small number. But the formalism of the theory happily extends to arbitrarily large systems. If we knew nothing about the behaviour of small systems we could have used a quantum formalism to describe statistical Newtonian mechanics, from which the mechanics of single Newtonian systems can be derived, though not (as Michael Berry shows in this volume) simply by letting Planck's constant tend to zero. Indeed Hamilton's approach to mechanics was almost exactly this. From this point of view, which focuses on the formalism, quantum mechanics is not an alternative to classical mechanics, but a generalisation of classical mechanics. Should we not, therefore, resolve to use a quantum mechanical description for everything, so as to encompass all our physics in a unified way? At this point we encounter the central problem of the theory: it has appeared that, if we do this, then the theory predicts macroscopic phenomena that seem flagrantly at variance with our experience ¾ the Schrödinger's cat "experiment", involving the appearance of a superposition of a live and a dead cat, being the standard case. [4]

Several other papers in this volume (e.g. Butterfield) give careful analyses of the options that have been tried to avoid this problem. From the point of view of the histories formalism to be examined here, the slogan to indicate the nature of the problem is "how does the classical world emerge from quantum mechanics?" This can be unpacked into three interconnected sub-problems.

1. Why does the classical world not contain any states corresponding to the superpositions that appear in the quantum world?

This sub-problem can be reformulated, not quite equivalently, in terms of the logic pertaining to the classical world (that is, the rules for combining yes/no propositions about states using "or", "and", and so forth). We can ask instead:

1. Why does the classical world obey Boolean logic (the propositions form a distributive lattice) rather than quantum logic (an orthocomplemented lattice)? [5] Following from this, we can ask
2. Where the classical behaviour is indeterministic as a result of superpositions appearing at the quantum level (live + dead cat) which are not manifested at the classical level (just one alternative is seen), then:-
1. What determines the particular distributive lattice (i.e. Boolean logic) of quantum states that have corresponding classical states? (known as the "basis selection problem");
2. How is the statistical mechanics of these probabilistically chosen classical states derived from the dynamics of quantum theory?
3. On each individual occurrence, what determines which one of the classically allowed states is in fact actualised?
3. When the classical behaviour is (at least to a high accuracy) deterministic, how is the classical dynamics derived from the quantum dynamics?

I claim that there is no formalism available that solves all of (1) - (3). Berry (this volume) surveys some of the subtle issues involved in solving (2)(b) and (3). I shall be examining the histories formalism attempt to solve (1) and (2a) and offering some speculations on (2)(c).

As James Cushing carefully explains in this volume, Bohmian mechanics remains an option for understanding what is happening here in a way that in principle avoids the above problem. In the early days of quantum physics, when our underlying beliefs favoured a particle ontology for the universe, this mechanics was probably the front runner for an interpretation, and only failed to achieve that status because it arrived late on the scene. From the viewpoint of the conventional/Bohr interpretation, the main difference that adopting Bohmian mechanics makes is that, because of the particle- (or more generally the "position-space-") ontology, the answer to question (2)(a) about what Boolean logic to use is: the logic generated by pure states in which the particles have definite positions. At first sight this seems very reasonable. After all, the problem with Schrödinger's cat is precisely that it involves superpositions of particles in different positions. On the other hand, the criterion is less obvious when applied to, say, electromagnetic fields. And when it comes to, say, deciding whether or not quantum theory has anything to do with consciousness, then do we really want to give the positions of microscopic particles precedence over the global state of all the fields in the brain? For the purposes of this chapter, in any case, I will stick to orthodoxy.

The traditional way of avoiding paradoxes such as Schrödinger's cat had been through the approach of Bohr, later rigidified by Von Neumann, which rejected the idea of bringing all of the physical world within quantum theory. The Bohr/von Neumann approach is excellent in a laboratory context, but it depends on (a) a division of the world into a (microscopic, quantum) system and a (macroscopic, classical) environment; and (b) the introduction of two quite separate processes of time development in the quantum part of the world, namely Hamiltonian deterministic evolution and stochastic state-reduction. Von Neuman's demonstration[6] that it did not matter (subject to conditions that could be, and have been, stipulated) where the division in (a) was placed ensured that this scheme was consistent and workable, but I would take the position (based on a sort of principle of sufficient reason) that a basic ontological division between two categories of reality, macroscopic and microscopic, cannot be entirely arbitrary: it needs to rest on some definable criterion.

If we are looking for a story that includes divine action, and if we want to enlist some of the characteristic features of quantum theory (indeterminism, connectivity) to support such a story, then an overall picture of the world is precisely what is wanted. The story cannot be confined, moreover, to a laboratory setting, since divine action extends to the whole arena of life. The story needs to include physics within it, in such a way that the physics used applies a consistent formalism to all of the physical world, including the observer/apparatus and the observed system on the same footing.

The essential difficulty with the Von Neumann approach lies in the enigmatic nature of the process of state-reduction, and so the theories in which we are interested can be classified, following Butterfield's "Dynamics/Extra Value" dichotomy [this volume] roughly into (1) those which introduce a specific physical mechanism to carry out the process of state reduction, and (2) those which try to reformulate quantum theory in such a way that it can be claimed that the idea of state reduction is unnecessary. Many of those who hold to type 2 also hold to the "Definite Appearance" (but indefinite macroscopic reality) pole of Butterfield's second, independent, dichotomy, leading to theories which seem contrary to some of our intuitions. For example, Zurek[7] has remarked that "the only 'failure' of quantum theory is its inability to provide a natural framework for our prejudices about the workings of the universe." I shall nail my own colours firmly to the mast of recognising that it is only appearances (at best) that are definite.

Typically, a dynamics that allows state reduction (type 1) cannot be added onto quantum theory without some consequent modification of the theory itself, which might be minimal[8] or might involve the adoption of a whole new theory (or putative theory, if the hoped for one does not yet exist) as in the work of Penrose[9]. Modification in the dynamics can usually be tested experimentally in principle. Their details vary widely and in some cases, including that of Penrose, a great many essential details remain to be worked out. [10] In this paper I shall, like Butterfield, not consider these type 1 modifications in detail, but will focus mainly on one particular example of type 2 only briefly covered by Butterfield in this volume: the consistent histories approach, though it will be helpful to make comparison with the many worlds and many minds views as we progress.

A history is a time-sequence of asserted propositions (I will later generalise this somewhat to arrays that are not strictly sequences). The times of the propositions might be assigned as part of the specification of the history, or it might be that only the temporal order of the propositions is significant. A history might be a formalisation of the life-experiences of a person, or of a society; or a formalisation of the whole sequence of events in the course of the universe. It might also be a formalisation of the trajectory of a fundamental particle. In the next section I give examples of several 3-event histories in the case of a particle going through three Stern-Gerlach-type observations. For simplicity, but in contradiction with the tenets of relativity theory, in the main text I assume that it is possible to refer all events to a chosen time coordinate, and I will not discuss the question of the choice of this coordinate. This restriction is not essential: a fully relativistic version of the concept of history has been developed by Isham and co-workers[12] and is implicit in the work of Donald[13]. Also for simplicity I assume that each history involves only a finite sequence of propositions.

A quantum history is one where "proposition" is understood with the flavour that this word has in quantum theory. Typically, this will involve the idea that, at any given time, there is a much wider range of propositions available than in the classical case, because in addition to the assertions of the form "particle P is in the region of space R" there are also assertions of the form "particle P is in a superposition of being in regions R₁ and R₂." Moreover, not all such propositions will be simultaneously decidable: because of the uncertainty principle it may not, for example, be possible to determine both that the particle is in a space-region R and also that its momentum is in a mathematical region Q of momentum-space. This has consequences for the nature of the logic that must be employed if we want to handle the totality of quantum propositions available at any one time[5].

The role of a physical theory is taken to be that of assigning probabilities to histories. These can either be absolute probabilities, or probabilities conditional on a given initial state. We will work with mixed sates in the quantum case, or probability distributions in the classical case. Thus we will usually be concerned with the quantities P(ρ,H) - the probability of history H conditional on the initial state ρ.[14] There is also the possibility of including the specification of a final state, giving an interesting time-symmetric concept of causation. We choose the Heisenberg formalism for quantum theory, in which time dependence is built into the operators (including the propositions that make up histories) and the states stay fixed in time.

As an example, consider a deterministic theory of single-particle motion, using classical histories and classical mechanics. Suppose the history H consists of two propositions H = ( P₁ , P₂ ), where the first specifies the position and momentum of the particle at some initial time to be (x₁, p₁) and the second asserts that at some later time the particle is in some region R of space, and suppose that the initial sate ρ is an atomic probability distribution concentrated at (x₀, p₀). Then the probability P(ρ,H) is 0 unless we have both that (x₁, p₁) = (x₁, p₁) and also that, on the given theory, the particle arrives in the given region of space at the later time, when the probability is 1. In a classical theory probabilities can often be 0 or 1; whereas in quantum theories probabilities strictly between 0 and 1 are almost inevitable. As another example (developed in the next section), in quantum theory a history containing a succession of propositions about the results of measurements might be used to describe a particle passing in turn through a sequence of measuring devices, each one preparing the particle for the next.

The histories approach can be thought of as nothing more than the linking together of a sequence of Bohr preparation/measurement pairs, in which each measurement become the preparation for the next. The only differences from the Bohr approach is that the entire sequence gives a context for each "measurement" and this context offers the hope of avoiding having to introduce the Von Neumann "cut" between macro and micro. If we think of it in this way, however, then we must realise that the term "measurement" is hardly appropriate when we are considering an arbitrary sequence of manifestations.

The language of histories is a flexible tool covering many approaches. It can be used, for example, in the context of Albert and Loewer's[15] version of the many-minds approach, in which minds migrate along the branching pathways provided by brain-states which periodically split into numbers of different quantum possibilities. [16] The experience of one of these minds can conveniently be described in terms of a history, with one proposition corresponding to each of the splitting-points of the brain-state.

The hope is, then, that the context of the entire history can provide a unified scheme (free of notions such as "many worlds" which might be seen as unnecessary metaphysical impedimenta) for discussing the whole range of physical phenomena, quantum and classical. Such a hope is, however, soon challenged. The formalism has to overcome the problems enumerated earlier which face any non-deterministic version of quantum theory. (1) Schrödinger's cat strikes again with the realisation that, whenever a measurement takes place on a state that is a superposition of two states giving definite values for the measurement (i.e. two eigenstates) then in the histories formalism a non-zero probability would be assigned not only to the histories consisting of the preparation followed by one or other of the definite outcomes, but also to the history in which the measurement is followed by a superposition of the two outcomes of the measuring apparatus. (2) Even when we have solved 1, the formalism only gives us probabilities for the different histories, it does not tell us which one is actually instantiated. So far, it appears that nothing has been achieved by stacking together the Bohr "measurements" to obtain histories.

The consistent histories approach tries to rule out superpositions of macroscopically distinguishable states by demanding at the outset that all the histories being considered are chosen from a set which satisfies a mathematical condition (given in note [19]) which ensures that the probabilities emerging from the formalism will have the form that should have in classical logic. I will describe the motivation for this and then explain some of the formalism for handling it. The theory (also known, rather confusingly, as the decoherent histories approach because of its intimate connection with the phenomenon of decoherence) was first introduced by Griffiths in 1984 and later by Gell-Mann and Hartle in the context of cosmology. The theory received a severe blow through the work of Dowker and Kent, [11] who showed that it could deliver all that was claimed for it. Subsequent work (a recent survey is by Bassi and Ghirardi[17]) has been devoted to (largely unsuccessful) attempts to refine the consistency condition to avoid the problems highlighted by Dowker and Kent, or to shift the philosophical rules so as to make their conclusion acceptable. My approach here will be based on a replacement of the condition which removes many of the problems and enables us to make contact with the themes of human and divine action.

The problem with the development so far is that there are too many histories. Something is needed to weed out all those that we do not, in fact, observe when the history refers to the experience of an individual ¾ namely, those involving unacceptable superpositions of states that are quite different at a macroscopic level. On the face of it, the Schrödinger's cat argument seems to demonstrate that the "something" is not already contained implicitly in the dynamics, and so we require a selection criterion that needs to be imposed as an extra additional ingredient. (We have already cited the way in which Bohm's mechanics would give such an extra ingredient, on a physical basis.)

Essentially all interpretations face in some form or other of this selection problem. It is equivalent to the question: what exactly is it that constitutes a measurement / a macroscopic state / a branching of worlds / a branching of minds / an acceptable history? The attraction of the histories approach is that it highlights one particularly natural way of weeding out a lot of the unacceptable histories. I shall describe this, and then go on to examine the implications of the fact that this still does not eliminate enough of the unobserved histories.

If we consider one history in isolation then there is nothing about it that can characterise it as acceptable or unacceptable if we regard it purely from the point of view of the formalism of quantum mechanics. This is because or the unitary invariance of the rules for forming probabilities: the probability of a history remains unchanged if we apply the same unitary transformation (the transformation in Hilbert space that is analogous to a rotation in physical space) to all the propositions and to the initial state. But a unitary transformation can be chosen so that it turns any given state into any other given state, and so given any history with a non-zero probability we can always obtain from it another history that has a non-zero probability for states that are unacceptable superpositions. As a result, any criterion must either (i) refer to a particular representation of the quantum formalism in terms of, for instance, wave functions in physical space; or (ii) it needs to be applied to whole sets of histories and their inter-relations; or (iii) we need to pose the question in the sharpened form of "how do we restrict the possibilities of future histories, given a fixed (and "acceptable") history up to time t₀ ?"

The first option could be involved if, for instance, we decreed as unacceptable any wave function that represented a superposition of two states in which a given macroscopic object was in two positions widely separated in space. Unless, however, one adopts the ontological position of Bohm's approach, then ruling out things like this by fiat seems arbitrary. Where does one draw the line? How big is "macroscopic? How wide is "widely"? To my mind the only solution to such questions lies in a modification of the dynamics such as is considered by Penrose. Here I shall consider option (iii) above, looking at an unmodified dynamics, but within sets of histories. The simplest such sets to consider are those consisting of all histories of the form ( P₁ , P₂ , ... , P_n ) where each P_i is required to come from a given fixed set of possible propositions σ_i . I will concentrate on this case.

If we take this approach then a simple and strong limitation on the possible sets of histories can be derived from a particular definition of consistency. It is this that defines the standard consistent histories approach. In the next section I will examine whether of not this condition is in fact adequate to the task of narrowing the range of histories to those that are in fact observed. I will build up this restriction in three stages: first, examining the relation between the propositions in one particular σ_i , then moving to a minimal condition, and finally generalising this to the condition conventionally imposed.

The first restriction (which I shall from now on take as a standard restriction on all sets of histories, consistent or otherwise) is essentially logical. We choose the propositions {P_i⁽¹⁾, P_i⁽²⁾, ... , P_i⁽ⁿⁱ⁾ } which make up a given σ_i to be exclusive, in the sense that if P_i^(j) is found to be true, then an immediately following examination of P_i^(k) for k different from j will prove to be false. [18] In addition we require their sum to be the identity operator; which is not really a restriction, since it can always be achieved by adding in a further proposition that takes in everything that the others miss out.

Following on from this, having selected exclusive propositions, we require that the probabilities P(ρ,H) satisfy the laws of probability appropriate to this; namely, that if (for fixed index i) we consider histories

H' = (P₁ , P₂ , ... , P_i', ... , P_n )

H '' = (P₁ , P₂ , ... , P_i'', ... , P_n )

H = (P₁ , P₂ , ... , P_i' + P_i'', ... , P_n )

(where P_i' and P_i'' are exclusive members of σ_i and P_i'+ P_i'' denotes the disjunction ["or"] of the two propositions) then

P(ρ,H) = P(ρ,H') + P(ρ,H'') . (1)

At this stage we can give the standard example of where this does not hold, which illustrates the role of this condition (and hence of the consistency condition which implies it). Suppose we are considering an uncharged particle with minimum non-zero magnetic moment passed through three successive magnetic fields, having intensity gradients such that the deflection of the particle in the two fields measures first its spin (which I here take to be equivalent to magnetic moment) in the x-direction and then in the y-direction and finally in the x-direction again (the Stern-Gerlach experiment). The results of such measurements are quantised, yielding values that are either +1/2 or -1/2 in standard atomic units. We can then consider the histories:

H ' = (P₁ , P₂', P₃) where P₁ = {x-spin is +1/2}, P₂' = {y-spin is +1/2}

H'' = (P₁ , P₂'', P₃) P₂'' = {y-spin is -1/2}, P₃ = {x-spin is +1/2}

H = (P₁ , P₂' + P₂'', P₃)

with an initial state of ρ = 1/2I (no initial information). Then the quantum mechanical probabilities are

P(ρ , H ') = 1/8

P(ρ , H '') = 1/8

P(ρ , H ) = 1/2.

The failure of equality between P(ρ , H ') + P(ρ , H '') and P(ρ , H ) arises because the last of these involves P₂' + P₂'' ("spin 1/2 or spin -1/2") expressed as a single measurement, which is equivalent to no measurement at all being performed; whereas the first two involve a measurement being performed, and then the probabilities of both possible outcomes being added together. In quantum mechanics the very act of measurement has an effect, even if the result is not taken into account in the final analysis, and so the two quantities being compared represent quite different physical sequences. In logical terms, the failure arises because

P₁ and ( P₂' or P₂'' ) ≠ (P₁ and P₂' ) or (P₁ and P₂''),

(the definition of a logic that is non-distributive).

This is as far as the restrictions on histories can be taken on the basis of purely logical considerations. In practice the restriction is strengthened in a third step by generalising the expression that arises when the condition (1) is evaluated using quantum mechanical rules [19]. Several authors give a full discussion of this generalisation[11]. The strengthening enables the condition to be expressed in a more generally useful form. Sets satisfying this strengthened condition are called consistent.

The significance of this restriction for ruling out unacceptable superpositions now comes from the phenomenon of decoherence, described by Butterfield. If the system interacts with the environment in between the determination of propositions, then random phases are introduced into its quantum state. When these are taken into account, it turns out that the sets of "ordinary" histories ¾ i.e. histories which do not have propositions that select superpositions of macroscopically distinct states ¾ are consistent to a very high degree of approximation. Moreover, it can plausibly argued that close to any very-nearly-consistent set there is a consistent set which is observationally indistinguishable from it, so that nothing is lost by demanding as a principle that all sets involved are consistent.

The question now is, is the imposition of consistency enough to restrict the histories to those we actually observe, or is some further structure required? First, it helps to clarify the whole question of what structure we are talking of when we refer to quantum theory. If by "quantum theory" we follow a purist text-book and mean just the specification of an abstract Hilbert space, a Hamiltonian (the operator that gives the time evolution) and a vector giving the state at some initial or fiducial time, then it is clear that this is hopelessly inadequate. This is because such a structure is completely characterised by the spectrum of the Hamiltonian (essentially a measure on the real line) and the state by a function on this spectrum, so that there is almost no scope here for distinguishing features such as space, particles etc, as is needed for interpretation. On the other hand, the quantum theory that is used in practice involves a much richer structure. It is actually a hierarchy of theories, going down through quantum field theory, with various successive layers of fundamentality and unification, and moving out of sight of experiment in speculative attempts at quantum gravity that go beyond space and time; and then up through various phenomenal theories, solid state theories (lasers, superconductivity), and perhaps jumping to quantum cosmology. More precisely, it manifests, in a rather compressed form, the epistemological case of the hierarchy of reductionism discussed by Ellis and Murphy[20]. The links between the levels are subtle and often not formalised. Each level uses not an abstract Hilbert space, but a particular representation in terms of structures on space and time. The states are not bare vectors but already have interpretations in terms of a wealth of special operators imported from other levels (definitions of what constitutes a particle and so on). We are never starting from a blank slate, but how much of this richness is part of the ongoing development of the details of scientific theory, and how much is absolutely essential for quantum theory to make any sense at all?

So, returning to our initial question as to whether extra structure is needed in order to single out acceptable histories, we can ask more precisely: though all the acceptable histories are consistent (because of decoherence), is the converse true? The answer is, No. More strongly, Dowker and Kent[11] make a convincing case that if a history is acceptable (i.e. "classical") up to a certain point then there is nothing to stop it becoming, with a reasonably high probability, unacceptably non-classical thereafter and still remaining within a consistent family. On this approach, although in the past live cats and dead cats have never entered into superposition, yet it could be that tomorrow this may be possible.

Yet we need to be cautious about this way of speaking, because it marks a transition from the language of histories, which talks about the probabilities of different eventualities, to the language of scientific realism, which talks about what "actually happens". As a first step, let us go back to the Schrödinger's hypothetical cat. The first proposition of the history might be the specification that the apparatus has been set up appropriately, the second proposition the ascertaining of whether the cat is alive. To illustrate the idea with a cat having only two degrees of freedom (!), we can denote the corresponding quantum states for the second proposition by |liveñ and |deadñ . Then the problem seems to be that, while we can derive probabilities for each of these states from the consistent histories approach, and interpret them without problem, we can equally well derive probabilities for (|liveñ + |deadñ ) and (|liveñ - |deadñ ), or for any other of infinitely many possible superpositions. The idea obviously generalises to many degrees of freedom. The temptation is to ask, but what really happens? This well illustrates Zurek's remark (above) about our prejudices on the universe.

The key point here is that there is no way in which we can practically test for the occurrence of some particular set of superpositions of live and dead states, [21] whereas we can very easily test for the occurrence of the live, or dead states. The consistent histories approach does not set out to tell us "what really happens": rather, it gives us probabilities for those propositions that we can actually measure (or observe with our sense), within the historical context within which we measure them. It enables us to understand the past and to (probabilistically) predict the future. Prediction, however, involves two steps: first the determination of what set of propositions (Boolean logic) is being considered out of the continuously infinite range of mutually inconsistent logics allowed by the theory; and second, the determination of probabilities within that chosen logic. This prediction is reached not just from the abstract formalism of the interpretation, because , as Dowker and Kent have shown, this is inadequate for the first step in such a prediction. It requires the formalism together with our knowledge of the contingent circumstances that determine what is measurable for us. The results of the propositions so far determine the set from which the next proposition can be drawn. A consequence of this (representing a variation of the picture described so far) is that in principle the set of consistent histories within which we are operating is not defined at the outset, but progressively constructs itself; though in practice (with exceptions to be indicated later) it can be stipulated in advance for laboratory purposes.

I propose, therefore, that the "additional structure" which we were earlier seeking is to be found not in the general laws of the theory, but in the contingent circumstances within which we are operating the theory, and which are expressed through the past history up to the present moment, from which it is desired to predict the future history. [22] Within these circumstances, I have proposed[23] the introduction of a physically based condition, namely a form of stability in relation to the repetition of propositions, defined in terms of the time-scale characteristic of the present circumstances.

The view that the past history determines criteria for the subsequent propositions, has some similarities with the views of Bohr with which we started, that the present state determines the nature of any subsequent measurement ¾ bearing in mind that the past history and the present state are equivalent. Both recognise the contingent role of the actual macroscopic situation. But the proposal here differs from Bohr's in important respects. First, there is no suggestion of a division into classical and quantum domains. On this way of construing reality, the whole universe is described in the language of quantum theory, and all sets of histories satisfying the stability condition are on an equal footing in terms of prediction. Second, there is no collapse of the wave function ¾ indeed, there is no wave function! There is no world[24] (in the realist sense of an entity that is supposed to be independent of perceiving beings and yet rationally discoverable) other than the initial state (fixed - in the Heisenberg picture).

I should stress that much needs to be done in this formalism in finding the most satisfactory form for the consistency condition, which still some ad hoc features. The next aim of the work would be to reduce "stability" to a more fundamental physical basis. I am claiming, however, that the "measurement problem" is in this formalism freed from the disguise it carries in other systems, which throws into clearer relief just what more is needed. Though this formalism could be applied to the state of the universe as a whole, it seems to me much more natural, and more in keeping with the realities of our human experience, to apply it at the local level, by examining what it means for the selection of my set of histories, and then to extend this to examine how this process, operating through a variety of organisms, builds up an overall dynamics. This is in keeping with the spirit, being developed here, of not adopting a realist view based on a particular physical ground that is declared to be fundamental; but rather recognising that science is concerned with a hierarchy of explanatory levels meshed together. If, like Dowker and Kent, we ask why, given that the universe has been classical so far, it will continue to be so; then we now give an answer not in terms of the logic and dynamics of the formalism, but in terms of the stability built into a construction of histories in which the history so far determines each successive proposition.

To end this section, I want to return to the idea of decoherence in order to analyse more closely the issue of the persistence of the classical world. The core idea of Zurek, Omnès and others is that decoherence is a well analysed physical process that takes place in the world, and its result is that histories constructed out of laboratory-measurable propositions are in fact (almost) consistent. This results in probabilities (including as a special case certainties) which (almost) obey a classical "logic." [25] I have argued that this can be made a strict deduction, provided that one replaces consistency with stability, and provided that one is, given the past history, in the sort of world we in fact live in where decoherence operates as it does.

This gives a different emphasis from that placed on the subject by Dowker and Kent. For them, consistency is a regulative principle placed on the system in order to obtain a classical world, which makes its relation to coherence rather problematic. On my approach stability is a regulative principle (but one which I believe is reducible to a more fundamental one with fewer ad hoc features) and decoherence, and hence (approximate) classicality, is contingent on the past history. We find that the universe is approximately classical because we are verifying a histories approach from the vantage point of our own personal histories, which are, as a contingent fact, govern by decoherence.

Our aim is to connect quantum theory with divine action, which I am here viewing as an analogical extension of human action. To do this I need to examine what light the quantum mechanics developed so far sheds on human action; that is, on the histories of the organism that is myself. If I wanted to specify just what propositions were determined as the next step in my personal history, then I would need an analysis of the sort of system that I am. At this stage, therefore, we need to address the arguments of Ho[26] that living organisms exhibit coherence. The word is used in a rather generalised sense, but it includes the claim that living organisms can maintain the phase relations between the quantum states of their constituents over considerable distances and times. If this is the case, then (as I shall discuss shortly) this would significantly alter the picture given above, and would open the possibility that our experiences might not obey a classical logic. The evidence for there being sufficient control over quantum phases for this to be possible falls into two classes. First: there is specifically quantum-related evidence - controversial in both data and interpretation - from observations of anti-bunching in biophoton emission. [27] Anti-bunching is a strictly quantum effect indicative of second-order coherence in the light emission[28], and it is hard to see how this could be achieved without a corresponding coherence of the light-generation mechanism, which is thought to extend over the whole organism. Second there is evidence for large-scale coordination in organisms which could be explained by underlying large-scale quantum effects. Examples are the alignment of birefringent intracellular structures over large distances, and the global coordination of muscular contraction.

Possible mechanisms for such quantum effects have been discussed for some time. I have reviewed[29] the most promising of these, involving the formation of a Bose pseudo-condensation[30] arising from a generalisation of the Fröhlich mechanism. In the particular case of the application of these ideas to consciousness, Tegmark[31] has claimed that decoherence invalidates mechanisms for large scale quantum coherence. However, he is addressing almost entirely one particular model, due to Hameroff, where his calculations have been shown[32] to be in error by some 4 orders of magnitude. He fails entirely to address the Fröhlich mechanism, referring only to popularised accounts of it. Unlike other quantum effects, such as superconductivity, it has the property that it increases in strength with increasing temperature.

In the light of this we can now return to the issues raised at the end of the last section, where I argued that the choice of a set of histories was determined by the contingent situation in which it was possible to verify predictions. That was an implicitly first-person perspective: I have certain experiences that I want to explain, which are predicated on my situation. The histories appropriate to me are to be determined by my structure and its state at each stage. We now need to make the decisive transition to a perspective when recognise that other people, and other organisms, are also to be described in the same way. I observer them, and at the same time they carry out their own process as described by the histories formalism. At the philosophical level we start to engage with the "many minds" problem; at the scientific level we encounter the "Wigner's friend" variant of the Schrödingr's cat paradox": what is the quantum mechanical description of an organism that is both observed and is an observer? If observe another organism then our histories become intermeshed (in way indicated below): I will be exploring those aspects of its responses that I can interact with, which will on the whole involve gross movements rather than subtle quantum information. On the other hand these movements may arise from that organism's detection of quantum information. In particular, if this organism can respond to quantum phase information then I can potentially observe behaviour that arises from histories pertaining to that organism which are non-classical and "inconsistent." Returning to the first person perspective, if I am such an organism, then my own experience can be non-classical. We conclude, therefore, that if we accept the histories interpretation, with a stability criterion but without the arbitrary imposition of consistency, then there may be non-Boolean aspects of an organisms own history (its own consciousness, if that language is applicable) which are observationally detectable by an external observer.

The interplay, and the distinguishing, of different "persons" is crucial to these ideas. A first person subject, the world is displayed through my history, over which, as I discuss in the next section, I have some creative control. Within this I discern other entities as external objects, treating them through a third person perspective, even, at times, when there may have their own first person aspect. Mediating the two, however, is the second-person perspective, the I-thou relationship that can be established through empathy with another sentient being, in which we share a subjectivity. Quantum mechanically, we share histories, at least momentarily. It is the possibility of this that is the key to the "other minds" problem. I see this as the essential component in linking the phenomenology of Heidegger with the concept of the objective world, through the interrelation of beings-in-the-world discussed in section 1. This second person process goes beyond Heidegger into what Levinas[33] called the "face-to-face". It allows the consciousness of others to play a role in my world, even when at the classical third-person level consciousness is merely epiphenomenal (as I have discussed elsewhere[34])

Whether or not we accept the arguments of Ho, and their consequences just sketched, makes a crucial difference to the theory. If they are not accepted, then all the histories of organisms are consistent as a result of decoherence. To see a particular consequence of this, suppose that we have two organisms with histories H = (P₁ , ... , P_i , P_i+1 ... , P_n ) and K = (Q₁ , ... , Q_j , ... , Q_m ) respectively, and with Q_j chronologically between[35] P_i and P_i+1 . Then we might ask whether we should really be including Q_j in this position in the history H. But (as we discussed in detail above) the consistency condition implies that if we include Q_j but without specifying its value (that is, we include the mere fact that some proposition is in that place) then this does not make any difference to the probabilities for H. Probabilities for histories can be computed without knowing whether or not other organisms are in the neighbourhood, and without knowing what sets of histories they might be determining. I shall suggest later that this determination of histories is connected with free will, and hence in this situation the exercise of free will has no external effect: it is purely an epiphenomenon.

On the other hand, if Ho's arguments are accepted the situation changes dramatically. All the probabilities become interlinked; we cannot discount the free interactions between organisms. The universe changes from an orderly collection of histories, each of which is defined and computable in isolation from the others, to an interconnected holarchy in which any computation is conditional on the actions of other parts. There is considerable leeway available as to how we generalise the formulae for probabilities in this case, which will require considerable further exploration.

These considerations have taken us from the rather logic-dominated realm of the history formalism to a perspective that starts to take on many of the characteristics of a many-minds approach involving a theory of mind. The differences are perhaps differences of style rather than content, the style being that of avoiding metaphysical accoutrements. Probabilities are not derived from randomly migrating minds, but inserted "by hand" from the formalism of the consistent histories approach. Ontological assumptions about entities that I am not observing (other "branches" of a many minds view) are avoided.

I have proposed that the choice of sets of histories is determined by the current situation: but how is it to be determined? For I have not demonstrated that the stability condition singles out a unique set of histories, only that those that it does single out have the properties that we require of them. The arguments I have presented so far suggest that this determination requires a theory of life as much as a theory of mind ¾ perhaps the key factor is sentience. Moreover the emphasis as far as the first-person perspective is concerned would be on a theory of external awareness rather than, for instance, Donald's theory[13] of switching networks whose relation to awareness is unclear.

If we now move from the rather spartan realm of quantum mechanics to the sorts of stories that can include the core issues of our living in the world, it might be appropriate to elaborate the account just given of the interaction of first-person and third-person perspectives[36]. I necessarily attribute reality to my own being-in-the-world, a reality which is shared by those other beings that show themselves within my world[37]. This reality is, as I have described, progressively constructed through the unfolding of a probabilistic history, and meshes with the similar histories of other beings that I observe. To anticipate later discussion in this section, a part of this unfolding of histories involves activity that corresponds to the subjective experience of free will and which has a creative aspect to it. It is appropriate to describe this whole process in terms of the co-creation of the universe through a set of intermeshing histories. I suggest that it is this co-creative picture that will provide a framework both for an understanding of human action, and hence for the analogical extension of this understanding to stories about divine action. There is little point in including divine action, however, before we understand human action; and if the latter is creative then the former can shed light on the larger creative processes in the universe. This takes us to a consideration of free will.

Many of the issues surrounding free will are well examined philosophical problems that are independent of quantum considerations (see, for example, the survey in McCall[38]). Some of the problems appear to be pseudo-problems arising from the imposition of a Cartesian ontology onto experiences that are in themselves not problematic. We might be puzzled, for example, about how it is that I can exercise my will on my body in such a way that my arm lifts: what is the causative action involved here, of the mind on the body? Why should this causative action of will be restricted to the body? The apparent problems here arise entirely from a curious, though by now well established, conceptual splitting between "I" and "my body." The sentence "I raise my arm" does not, in actual experience, imply the existence of two quite distinct entities, "I" and "my arm." It is only a reflexive idiom used to describe my arm-raising in an emphatic manner suggestive of deliberation. If we avoid Cartesian and related dualities then many of these pseudo-problems are eliminated.

A second class of problems with free will involve a basic antinomy, which runs as follows. If we have sufficient grounds for making one particular choice then, it could be argued, our choice is determined by those grounds and is not free. On the other hand, if there are insufficient grounds for the choice then it would appear to be a purely random event and not a true choice or a rational act of will. Free will appears to be "free" only if it is not "will". On the histories approach there certainly are non-determined aspects to my future behaviour, whose probabilities are computed to be strictly between 0 and 1. The emergence of such an action may be a conscious process, and this may be experienced as making a "choice." It is not clear, however, either from physics or introspection, what there is about these actions which distinguishes them from the merely random. We are tempted to say that our consciousness is the agent of the choice; and this is an illegitimate deduction. The evidence of introspection[39] is that at this stage we participate passively, not actively: we observe and report a decision process whose actual origin is non-conscious. The point is discussed in some detail by Velmans[40]. His main evidence comes from the work of Libet[41] whose experiments are usually interpreted to imply that we make our decisions subconsciously[42] a short time before we are consciously aware of them. Velmans goes on to point out that consciousness does seem to be involved in the process of attention in which a certain range of propositions is assembled for the purpose of scrutiny and processing, and this act of attention and selection continues until a decision emerges. But the decision itself is not an act of consciousness. The purely logico-deductive part of decision-making (which for Descartes was the only part of consciousness that he regarded as unquestionably non-mechanical) is something that can be, and increasingly is, fully automated and so has little necessary connection with the conscious process.

There is, however, a class of decisions which are suggestive, introspectively, of a category that is neither determined nor random, and which matches well with a characteristic feature of the histories approach, namely the need to determine the choice of a set of histories out of the range of possibilities. This class is involved with changes of meaning, or changes of the "reference frame" - the whole fabric of meaningful connections. In making decisions, in practice we start off by trying a variety of (usually incompatible) rules of thumb, combined with a process of imagining possible outcomes of decisions and evaluating the nature of our emotional reactions to these outcomes. So far this is essentially the same as the construction of a chess-playing computer (except that the evaluation process is not called "emotions"). If a decision does not then emerge, we do not resort to a randomiser; rather, we alter the rules of the game. We re-construe, re-categorise the situation.

As Delal[43] has pointed out, this involves radical disruptions to classical logic, alternating between classical logic and a wider context in which there is a local application "rules of deduction" which would be destructively inconsistent if applied globally. Associated entities are identified, even (indeed, especially) to the extent of identifying propositions with their own contraries It is this, I suggest, that enables a shift from one consistent Boolean logic to another to take place, and thereby constitutes the essence of conscious human decision making[44]. From the point of view of an external description employing a fixed classical logic, it is an acausal manoeuvre which, by default, can only be described as "random". From the person's point of view, however, it is an act of creative meaning-making: the situation comes to make sense, and the change of cognition itself makes sense, but only retrospectively, from the perspective of the new frame of reference once achieved.

There is a parallel, amounting to an identity of logical structure, between this change of reference frame and the determination of one out of a range of possible sets of disjoint projections for the next such set in the histories available to an organism which utilises coherence. Each set of histories is governed by classical logic, but the collection of all available propositions requires quantum logic (as noted above), so that shifting projection-sets requires moving from classical logic to quantum logic and back again, just as suggested by Delal in the case of radical human thinking. In both cases we are dealing with a change that is neither a random choice within a given set of possibilities (as happens with the outcome of a quantum observation within a given set of exclusive possibilities) nor a determined choice (which is really a special case of a random choice with probability 1), but a third possibility, a shift of underlying meaning, in the case of free will. On the basis of these parallels, and supposing that Ho's thesis is valid, I propose that we are capable of acts of free will whose physical correlates are selections of projection sets.[45]

This sort of act of will, which I call creative will, is characterised by the simultaneous creation of a volition and of the framework of meaning within which that volition is justified. It is this simultaneity of volition and meaning that distinguishes this act from, on the one hand, an act that is determined by prior psychological and physical factors; and, on the other hand, an act that is random in that one of a number of pre-existing alternatives is selected with no perceptible rationale. (These last two cases can, or course, be combined in various proportions.) The existence of this category of creative will is the solution to the basic antinomy of free will noted above. The approach here is quite separate from that of Donald MacKay. He recognises the potential in the indeterminism of quantum theory for a separate determinative factor which could be God or human will[46], and he recognises[47] the role of meaning in the human capacity for information processing, which goes beyond the sort of information theory that was available at the time, but he fails to unite the two in the concept of change of frame of meaning. It is this connection which both gives a new understanding of the uniquely creative nature of sentient action, and which links it with the most distinctive feature of quantum theory.

It is only once we have a picture that contains human will together with the physical universe that we can usefully extend our story to include divine will, on the basis of a metaphorical extension from human will. I want to begin by drawing attention here to a particular feature of quantum theory which I shall call entrainment. It relates to the idea of "top-down causation" in the context of the action of will, whether human or, if the metaphor is to mean anything, divine.

Let us imagine a chaotic physical system such as one vessel of water draining turbulently into another, initiated in a state that is a superposition of two quantum states that are indistinguishable by macroscopic measurements. If we suppose an essentially quantum universe whose state is occasionally constrained by macroscopic information, then such superpositions will be the normal state of affairs. The evolution of this system will then, in a time depending on the Liapounov exponents of the chaotic modes, evolve to a superposition of states that are macroscopically distinguishable. If we now ascertain which one of these distinguishable states is in fact realised, and retrodict from this what was happening in the past, we will retrodict precisely that sequence of detailed microscopic fluctuations that led from the initial state to the final state now realised.

For example, suppose that we had also filmed the system from the start through a microscope. Then (following the usual quantum mechanical argument) the film would be in an entangled quantum state such that, when we ascertained the final state, if we then looked at the film it would show a sequence of events that led precisely to the final state that we had ascertained.

Expressed in terms of histories, if ρ is the initial state and the two histories terminating at the propositions that distinguish the final superposed states are H = ( P₁ , P₂ , ... , P_n ) and H' = ( P₁' , P₂' , ... , P_n') , then the probability of ( P₁ , P₂ , ... , P_n-1) conditional on P_n becomes much higher than the probability of ( P₁' , P₂' , ... , P_n-1' ). On the other hand we are supposing that the organism can have some say in the choice of which pair ( P_n , P_n' ) appears in the history, a say which is exercised at the time when these propositions are added to the history. Indeed, as indicated in note [45], this may even extend to a determination of which proposition out of ( P_n , P_n' ) is in fact realised. Thus the organism appears also to be "influencing" the previous events ( P₁ , P₂', ... , P_n-1') leading up to P_n, even though these were at an earlier time.

A more accurate account, however, would seem to be that it is the fact of these events having occurred that is realised at the time of P_n, rather than the occurrences themselves. I shall describe this situation as the entrainment of these preceding events into the final determination of P_n. The organism in question will in general have no interaction with the details of these preceding events ¾ it will merely be observing, or even influencing, the final event. But in the course of doing this the smaller scale predecessors are automatically entrained into the process.

It seems to me that this account of the action of human will and choice within a quantum mechanical setting provides the most natural metaphor for divine action. I find myself part of a nested sequence of organisms. I contain cells, organelles etc; and I am contained in a family, a community, in Gaia etc. Each of these has its proper activity which includes, in all probability, freedom of the sort that I have been discussing. We can note that the freedom of the parts is not compromised by being included in the freedom of the whole, situation that is even more true in quantum theory than in classical theory because of the greater number of real degrees of freedom that are contained in the phase relationships between the parts that make up the whole. This makes sense of the way that I experience the intermeshing of my own freedom with the choices of the organisms of which I am part. My experience of the divine is of a guidance that is immanent, in being part of the concrete flow of events around me, and transcendent in the sense of not being contained within any given contextual framework, of being always greater than my current horizons. It thus makes sense for me to think of the divine as analogous to an ideal outermost system, beyond any imaginable context, and of divine action as being the top-down action of will through entrainment that coordinates and informs all the individual acts of will that it contains. [48]

For me this top-down description of divine action accords more with experience than does the idea of a god who influences each atomic process individually so as to build up the whole. A top-down action is a story that builds on the metaphor of my own acts of will, which are also delivered at the macroscopic level. The picture that this produces goes somewhat beyond pantheism, in that the entire universe is strictly within god; in this sense it is panentheistic. That word tends, however, also to imply that god is in some sense also embodied in the world, and this is not obvious in the present system. I would regard such embodiment, the complementation of transcendence with immanence, as necessary for a fully religious perspective. This I define in terms of the second mahavakya[49] of the Upanishads: "This atman is Brahman." Religion is living one's life in the light of the experience that the transcendent ultimate within is identical to that without. Our current physical picture cannot close up the nested sequence of inclusions into the circle that this implies, in which god is both the ground of being and the context of meaning.