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8.5 One Global Process Time?

The above scheme for local process times is based on the causal connections which hold between events, and therefore depends on these causal relations being local. In quantum mechanics, however, we have good reason to believe that there are at least some kinds of non-locality, namely the correlations that can exist between spatially separated parts of a wave function.     The existence of these non-localities can be quantitatively confirmed by the violations of Bell's inequalities that have been experimentally confirmed (see Clauser & Shimony [1978]).

    As discussed in Redhead [1987], one consequence is that any realistic extension of quantum mechanics will have to use non-local properties and/or causal connections.   Maxwell [1985] points out that even the very notion of actualising of potentialities involves some basic kind of non-locality if the potentialities are distributed in space. This is because only one actuality can be realised, and the choice of one actualising, in order to `block off' all the incompatible possibilities, must be immediately felt in all regions of the potentiality distribution.     In quantum physics, this process is known as the `reduction of the wave packet', and has been the subject of considerable controversy ever since it was proposed by von Neumann [1932]. I am proposing a similar process here. Proposals for how it could work will be examined in chapter 12. In the meantime, however, we are concerned with constructing a notion of time which renders such a process feasible. We are not yet in any position to discuss wave functions or wave packets, but we can discuss the actual past-events, and the order (if any) in which they come to be. The point is that the presence of even restricted kinds of non-local causal connections renders inadequate the multiple `local process times' of the previous section.

Figure: Conventionality of Distant Simultaneity The time $\tau $ of a distant event E can be only partially determined by sending a light pulse at time t1 and having it reflected at E to return at time t2 . The $\tau $ must be in the range $ t_1 < \tau < t_2 $, so the parameter $\epsilon $ in $\tau = t_1 + \epsilon (t_2 - t_1 )$ must be in the range $ 0 < \epsilon < 1. $ We usually use $ \epsilon = \frac{1}{2}, $ but this choice is purely conventional (see Grübaum [1973]).

  One might have expected, independently of experiments in quantum physics, that our actual past-events should all have determinate relations between them (once they exist of course). This is because `actualities' were defined as those particular existing things which were fully determinate in every respect. On this basis alone, we may have expected that there should be definite relations between all these actualities, and if they turn out to be past-events, then we should expect that all these past events should be related in some definite order. This would indicate that there ought to be some kind of `global process time' that goes considerably beyond the `local process times' of the previous section. A global order of actual events would form a complete rather than a partial order, so that for all pairs of events, one of the pair will be definitely (and actually) before the other. The existence of such a global order can be immediately seen to be counter to the usual formulation of the special theory to relativity. Special relativity can either be formulated as expressing the relations between the observations of observers in uniform rectilinear motion,         but, as Grünbaum [1973] shows, it can equally well be expressed as the conventionality of distant simultaneity. According to special relativity, there are no intrinsic relations of simultaneity between spatially separated events -- the $\epsilon $in 8.5. may be set arbitrarily in the range $ 0 < \epsilon < 1. $ The existence of a global process time would seem to be directly counter to this arbitrariness. There are several ways in which a total ordering of all natural events could formulated. The actual ordering of events could be
according to some particular observer's `legal time',
according to the centre-of-mass frame of the universe,
according to some sequence of non-intersecting spacelike hyper-surfaces in spacetime, or
according to some contingent properties of the individual natural processes as they proceed.


1. Some Observer's Legal Time?

The first option is not tenable given special relativity, as it would make some particular observer's t = const spacelike hypersurfaces special. It would not be invariant with respect to change of velocity, as other observers moving with different velocities, who might think their t = const spaces to be just as valid, would have to wrong for no apparent reason. More seriously, Grünbaum shows that `legal time' is a purely conventional notion even for one observer, and so it would be most peculiar if some purely arbitrary choice were responsible for ordering all the actual events in the universe in some sequence.


2. Centre-of-mass of the universe?

The second option is somewhat more plausible, as the choice of a reference frame based on the centre-of-mass of the universe is not quite so arbitrary.   This would be a kind of `Mach's Principle', whereby some global property of the universe gives rise to local properties of individual things and processes. Mach originally proposed that the inertial mass of individual bodies arose because of the frame of reference provided by the `fixed stars', or the bulk of matter in the universe. The present option based on a similar principle, in that the actual order of individual events is supposed to arise also from the frame of reference provided by the overall distribution of matter in the universe. Similar kinds of Principles have been invoked to explain the direction of time, if the laws of physics themselves do not specify any direction   (see e.g. Cramer [1983]).

Problems with this scheme arise however, if the universe were expanding and unbounded, as then different averaging procedures could give different different results for the centre-of-mass velocity at any given place. This is because the calculation of centre-of-mass velocity of an expanding universe presupposes some notion of simultaneity in order to determine the velocity of distant galaxies. The option can hardly be used, therefore, to define what is meant by simultaneity and/or actual ordering of distant events.

Figure: Successive Hypersurfaces The boundaries separating actualities ($\bullet $) from emptiness.


3. General Spacelike Hypersurfaces?

The third option is the most general case of using time orders, taking note of the fact of the arbitrary nature of any t = const hypersurfaces that could be used to order actual events as in :FIGREF refid=suchyp. (We would require here that there did exist such hypersurfaces which extended globally, and were free from singularities.) There is no reason to expect that any actual ordering of events should follow the time coordinates of any observer who uses rectilinear coordinates. General Relativity is an example of a theory which allows for the essential arbitrariness of coordinate systems, and allows any curvilinear system of coordinates to be used to formulate the theory. Successive `times' in such a systems may be labelled by an index $\tau $, say. If mapped into some other rectilinear coordinates $(t,\vec{x})$, the successive `times' $\tau $ define spacelike hypersurfaces according to some function $t = \sigma ( \vec{x} ; \tau ),$ where $ \mid
\partial t / \partial x_i \mid ~ < ~ 1/c $ for speed of light c, and where t increases strictly monotonically with $\tau $.

The trouble is, that we know of no reasons to select any one such set of hypersurfaces, rather than another, as the basis for the actual ordering of all events. Furthermore, if we did know of a basis to select one such set, this would seem to be counter to the theory of relativity, which requires that all laws of nature must be invariant with respect to changes of velocity and position.   It is all very well to argue, as Maxwell [1985] does, that there must be some selection of a set of spacelike surfaces, but some basis for them must be provided.


4. Contingent Actualising?

The fourth option uses some contingent properties of the individual natural processes (as they proceed) to give rise to their actual sequence. The word contingent is most important here. We are not giving any physical law which necessarily determines the actual ordering of events, but are saying that the actual events are ordered according to, say, details of the way the potentialities for these events are in fact realised. General laws are specified for the necessary orderings of past and future relations, and for how realisations of potentialities influence the actual ordering of events. We do not however specify those actual orders in advance, and cannot, because they depend, we now postulate, on details which only come to be true once those events have definitely happened. Although the actual ordering is not determined by any law, once the events have occurred the ordering is definite, just as the precise time of the decay of a radioactive atom is not determined by quantum laws, but (if actually observed) is completely definite once it has occurred.

    In the Aspect version of the EPR experiment, for example, the detection events can be placed in a spacelike relation, so that their places alternate with each other. Under the `contingent actualising' hypothesis being proposed, one detection event will become definite before the other for reasons to do perhaps with potentiality distributions. This first event will non-locally determine the potentialities for the second event's outcomes, even though the second event is in a spatial relation with the first, because it is a basic feature of potentialities that only one outcome is possible for every event. All events have a definite order, and according to this actual order, earlier events can affect the propensities for later events with which they are correlated.

Spontaneous Symmetry Breaking

Figure: Spontaneous Symmetry Breaking Although there is no predetermined direction for the ball at the top to fall, it is in an unstable situation and will eventually fall in some definite direction. Once that has happened, the initial symmetric situation is `broken', as the final situation has no rotational symmetry.

    The laws of physics are symmetric and invariant with respect to changes of velocity and position of coordinate frames, but this symmetry is broken once actual events start occurring. Other examples of `spontaneous symmetry breaking' are well known in physics, such as the Weinberg-Salaam theory of the electromagnetic and weak interactions. After spontaneous symmetry breaking, symmetric laws are broken by the requirement that something actually happens. Then, as we shall see in the present case, what has actually happened in the past propagates its effects to influence present symmetry breaking.

Spontaneous symmetry breaking in physics is often illustrated by the example from elementary classical mechanics of a ball on the top of the `Mexican Hat' shape shown in 8.5. The initial state of the system is clearly symmetric under rotations about the vertical axis, and there is no predetermined direction for the ball to move, because gravity acts only in the vertical direction. Yet, the initial state is unstable, and the ball will eventually fall in one direction or another. As it loses energy through friction, it will eventually come to rest somewhere in the rim of the `hat'. Its final situation there is stable, but no longer has rotational symmetry. We say that the rotational symmetry in the underlying forces is `broken' by what actually happens. The actual position in the rim is purely random, and cannot be predicted by the theory. The actual outcome has no deep significance, yet it influences what happens from then on, for all times.  

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Next: 8.6 Necessary and Contingent Up: 8. A Theory of Previous: 8.4 Process Time
Prof Ian Thompson


Author: I.J. Thompson (except as stated)