Less than meets the eye

If even some of the ideas presented here are correct, the world is different from what it seems. The major theses of the book, on time's arrows and on quantum measurement theory, are unified by the notion of cryptic constraints. We see, sense, specify, macroscopic states, but what we predict about these states depends on an important assumption concerning their microscopic situation. The assumption is that the actual microstate is equally likely to be any of those consistent with the macrostate. And I say, not so. For various reasons, both classical and quantum, many otherwise-possible microstates are eliminated. As presented in detail in previous chapters, such elimination impacts many areas of physics, from the cosmos to the atom. But we also make the point that this elimination can be difficult to notice and in particular there is no experimental evidence that confirms the usual assumption. By an explicit example on a model (Fig. 4.3.2), we show that a future constraint eliminating 98% of the microstates can go completely unnoticed.

My expectation is that this fundamental change in the foundations of statistical mechanics is needed. Whether or not it takes the forms I've proposed will be determined by future investigations.

In the next section I will review open problems for the program implicit in this book. The tone will be that used in speaking with colleagues: an attempt to be frank about difficulties and a willingness to be wildly speculative.

The existence of ‘special’ states can be established with ordinary quantum mechanics. We seek particular microscopic states of large systems that have the property that they evolve to only one or another macroscopic outcome, when other microstates of a more common sort (having the same initial macrostate) would have given grotesque states. Justifying the hypothesis that Nature chooses these special states as initial conditions is another matter. In this chapter we stick to the narrower issue of whether there exist states that can do the job, irrespective of whether they occur in practice.

In Section 7.1 we give several explicit examples. In Section 6.2 we exhibited an apparatus model and its special states. Here we look at the decay of an unstable quantum state, not as a single degree of freedom in a potential, but with elements of the environment taken into account as well. We also study another popular many-body system, the spin boson model. This has extensive physical applications and especially with respect to Josephson junctions has been used to address quantum measurement questions. A single degree of freedom in a potential, by the way, does not generally lend itself to ‘specializing,’ and an example is shown below.

In recent years, exotic non-local effects of quantum mechanics have been exhibited experimentally. Behind many of these lies entanglement, the property of a wave function of several variables that it does not factor into a product of functions of these variables separately.

Quantum measurement theory addresses several problems. All of them arise from applying a microscopically valid theory to the macroscopic domain. The most famous is the Schrödinger cat example in which the ordinary use of quantum rules suggests a superposition of macroscopically different states, something we do not seem to experience. Another problem is the Einstein-Podolsky-Rosen (EPR) paradox in which a fundamental quantum concept, entanglement, creates subtle and non-classical correlations among remote sets of measurements. Although it seems mere word play to observe that such an apparent micro-macro conflict ought to be viewed as a problem in statistical mechanics—by virtue of the way that discipline is defined—until recently the importance of this observation was seldom recognized.

The founders, Bohr, Schrödinger, Heisenberg, Einstein did not emphasize this direction, but over the years the realization that measurement necessarily involves macroscopic objects—objects with potentially mischievous degrees of freedom of their own—began to be felt. For me this was brought home by the now classic fourth chapter of Gottfried's text on quantum mechanics. He takes up the following problem. The density matrix ρ0 for a normalized pure state is ρ0 = |ψ〉 〈ψ|. It follows that Tr ρ0 = Tr ρ02 = 1. After a non-trivial measurement, the density matrix ρ is supposed to be diagonal in the basis denned by the measured observable, with Tr ρ = 1 but Tr ρ2 < 1.

In our experience, time is not symmetric. From cradle to grave things happen that cannot be undone. We remember the past, predict the future. These arrows, while not always—or even now—deemed suitable for scientific investigation, have been recognized since the dawn of thought. Technology and statistical mechanics give us a precise characterization of the thermodynamic arrow. That's what the previous chapter was about. But the biological arrow (memory, etc.) is elusive. Then we come to arrows that only recently have been recognized. The greatest of these is the fact that the universe is expanding, not contracting. This is the cosmological arrow. Related, perhaps a consequence, is the radiative arrow. Roughly, this is the fact that one uses outgoing wave boundary conditions for electromagnetic radiation, that retarded Green's functions should be used for ordinary calculations, that radiation reaction has a certain sign, that more radiation escapes to the cosmos than comes in. Yet more recently, the phenomenon of CP violation was discovered in the decay of K mesons. As a consequence of CPT invariance, and some say by independent deduction, there is violation of T, time reversal invariance. This CP arrow could be called the strange arrow of time, not only because it was discovered by means of ‘strange’ particles, but because its rationale and consequences remain obscure. There is another phenomenon often associated with an arrow, the change in a quantum system resulting from a measurement.

Why should special states occur as initial conditions in every physical situation in which they are needed? Half this book has been devoted to making the points that initial conditions may not be as controllable as they seem; that there may be constraints on microscopic initial conditions; that this would not have been noticed; that such constraints can arise from two-time or future conditioning; that in our universe such future conditioning may well be present, although, as remarked, cryptic. In this chapter I will take up more detailed questions: what future conditions could give rise to the need for our ‘special’ states, and why should those particular future conditions be imposed.

Before going into this there is a point that needs to be made. Everything in the present chapter could be wrong and the thesis of Chapter 6 nevertheless correct. It is one thing to avoid grotesque states (and solve the quantum measurement problem) by means of special states and it is another provide a rationale for their occurrence. I say this not only to highlight the conceptual dependencies of the theses in this book, but also because there is a good deal of hand waving in the coming chapter and I don't want it to reflect unfavorably on the basic proposal. As pointed out earlier, the usual thermodynamic arrow of time can be phrased as follows: initial states are arbitrary, final states special.

Pure quantum evolution is deterministic, ψ → exp(—iHt/ħ)ψ, but as for classical mechanics probability enters because a given macroscopic initial condition contains microscopic states that lead to different outcomes; the relative probability of those outcomes equals the relative abundance of the microscopic states for each outcome. This is the postulated basis for the recovery of the usual quantum probabilities, as discussed in Chapter 6. In this chapter we take up the question of whether the allowable microstates (the ‘special’ states) do indeed come with the correct abundance. To recap: ‘special’ states are microstates not leading to superpositions of macroscopically different states (‘grotesque’ states). For a given experiment and for each macroscopically distinct outcome of that experiment these states form a subspace. We wish to show that the dimension of that subspace is the relative probability of that outcome.

This is an ambitious goal, especially considering the effort needed to establish that there are any special states—the subject of Chapter 7. As remarked there, the special states exhibited are likely to be only a small and atypical fraction of all special states in the physical apparatus being modeled (e.g., the cloud chamber). In one example (the decay model) there is a remarkable matching of dimension and conventional probability, but I would not make too much of that. What is especially challenging about the present task is that we seek a universal distribution.

In Chapter 6 I presented a proposal for how and why grotesque states do not occur in Nature. In subsequent chapters I explored consequences and found subsidiary requirements, such as Cauchy distributed kicks. To find out whether all or part of our scheme is the way Nature works, we turn to experiment. How to turn to experiment is not so obvious, since the basic dynamical law, ψ → exp(−iHt/ħ)ψ, is the same as for most other theories. Our basic assertion concerns not the dynamical law but the selection of states. Therefore it is that assertion that must be tested. For example, one way is to set up a situation where the states we demand, the ‘special’ states, cannot occur. Then what happens? Another part of our theory is the probability postulate, and this deals not only with the existence of special states but with their abundance. It enters in the recovery of standard probabilities but has far reaching consequences that may well lead to the best tests of the theory. Such tests arise in the context of EPR situations.

The experimental tests fall into the following categories.

Precluding a class of special states. This should prevent a class of outcomes. If the changes in the system (due to precluding the class of special states) do not change the predictions of the Copenhagen interpretation, then this provides a test. In particular, with a class of special states precluded, our theory forbids the associated outcome.

Although the variational principles of classical mechanics lead to two-time boundary value problems, when dealing with the real world everyone knows you should use initial conditions. Not surprisingly, the eighteenth-century statements of classical variational principles were invested with religious notions of teleology: guidance from future paths not taken could only be divine. The Feynman path integral formulation of quantum mechanics makes it less preposterous that a particle can explore non-extremal paths; moreover, it is most naturally formulated using information at two times. In the previous chapter, use of the two-time boundary value problem was proposed as a logical prerequisite for considering arrow-of-time questions. Perhaps this is no less teleological than Maupertuis's principle, except that we remain neutral on how and why those conditions are imposed.

In this chapter we deal with the technical aspects of solving two-time boundary value problems. In classical mechanics you get into rich existence questions—sometimes there is no solution, sometimes many. For stochastic dynamics the formulation is perhaps easiest, which is odd considering that this is the language most suited to irreversible behavior. Our ultimate interest is the quantum two-time problem and this is nearly intractable.

Later in the book I will propose that the universe is most simply described as the solution of a two-time boundary value problem. A natural reaction is to wonder whether this is too constraining. Given that our (lower entropy) past already cuts down the number of possible microstates, are there sufficient microstates to meet a future condition as well?

Things happen. That may seem obvious, but it has also been maintained that all the ‘happening’ does not signify change and that the way of the world is periodic and repetitious:

I won't discuss the profound aspects of this passage, but I will do post-industrial age nitpicking. Only in the past century has humanity understood a distinction that exists among these cyclic behaviors. For the 'rising and setting of the sun, there is indeed little that is happening. To a good approximation this is non-dissipative. But as to the coming and going of generations, with the benefit of wisdom gained in building steam engines, we recognize that birth and death can occur only so long as there is a source of negative entropy.

I could continue in this vein and discuss how the failure to distinguish between free and frictional motion confused humanity's greatest minds as they grappled with elementary mechanics. But I wish to begin a technical discussion of irreversibility and only want to draw a lesson of humility from the historical perspective. Until the past few centuries, humanity failed to appreciate the most manifest of time's arrows, the second law of thermodynamics. Unless one realizes that Nature's dynamics are mostly time symmetric one does not know that there is a problem.

Introduction

The main purpose of this book is to investigate in detail long-term variations in the length of the day, or equivalently, changes in the Earth's rate of rotation, using pre-telescopic observations of eclipses. Such variations are mainly produced by lunar and solar tides, but non-tidal mechanisms are also significant. Despite its historical bias, this subject has become an important topic in modern geophysics.

In studying changes in the length of the day which have occurred in recent centuries (since the invention of the telescope), more accurate data than eclipses are available. These include occultations of stars by the Moon up to 1955 and systematic monitoring of the Earth's rotation relative to the atomic time-scale since that date. However, observations of eclipses provide by far the most consistent and reliable source of information on variations in the terrestrial rate of rotation during the pre-telescopic period.

Although numerous early accounts of eclipses are to be found in astronomical works, many others are scattered in a variety of writings – for example the Greek and Latin classics, imperial annals of China and monastic chronicles of Europe. Observations recorded in these works are often extremely crude by modern standards, but the lengthy time-scale covered is highly suited to the detection of long-term trends which are not apparent from more recent data. The investigation of this diverse material adds a whole new dimension to what might otherwise have been a somewhat narrow scientific discipline.

Introduction

In this chapter, all accessible reports of solar eclipses preserved on the Late Babylonian astronomical texts which are of value in studying the Earth's past rotation will be investigated. Compared with the substantial number of observations which must have originally been made, very few Babylonian records of solar eclipses are now extant. Reasons for this are briefly discussed in chapter 4. Reliable observations only range in date from 369 to 136 BC – less than half the period covered by lunar eclipse sightings.

The solar eclipses which form the subject of the present chapter will be divided into four categories: (i) the only known example of a total obscuration of the Sun observed at Babylon (dating from 136 BC); (ii) measurements of the local times of a variety of eclipse contacts; (iii) estimates of solar eclipse magnitude at maximum phase; and (iv) instances when the Sun rose or set whilst obscured. Several of these observations have already been discussed in chapter 3, where they were used to illustrate various analytical techniques. However, they will now be considered in greater detail. At the close of this chapter, a possible allusion to a total solar eclipse recorded in the Religious Chronicle of Babylon will be considered. This event probably took place around 1000 BC – much earlier than the period covered by the Late Babylonian texts.

All eclipse records on the Late Babylonian texts where the date is doubtful will be rejected, as will questionable readings of damaged signs.

Introduction

In chapters 4–13, more than 400 timed and untimed observations of both solar and lunar eclipses from the pre-telescopic period (date range –700 to +1600) have been investigated in detail. ΔT values or limits (depending on whether the observations were timed or untimed) have been derived in almost every case. The fundamental objective of the present chapter is to use these results to obtain the best-fitting ΔT curve to the data and hence to determine changes in the length of the day (LOD) over the historical period. In addition, the various geophysical mechanisms responsible for the observed variations will also be discussed. I am grateful to Dr L. V. Morrison of the Royal Greenwich Observatory for undertaking the data analysis which forms the basis of much of this chapter and producing several of the diagrams. This chapter is essentially an enlargement of section 5 of the paper by Stephenson and Morrison (1995). Although several tens of further historical data have been added (notably medieval Chinese timing – see chapter 9), the basic conclusions obtained in that paper remain unchanged.

All of the ΔT results derived in chapters 4–13 are summarised in tabular form in the Appendices for ready reference. For each individual observation, only the year (– or +) and appropriate ΔT value or limits are tabulated. Appendix A contains timed data and Appendix B untimed material, each being divided into sub-groups depending on the source and type of observation it contains.

Introduction

For the purpose of this chapter, the term ‘medieval’ will be loosely interpreted to mean the period between the close of the classical age in Europe (c. AD 500) and the beginning of the telescopic era. It thus includes the Renaissance. Numerous observations of both solar and lunar eclipses were recorded in Europe during this period, especially after AD 1000. In most cases, the date is accurately reported and the precise place of observation is known. However, with only a few exceptions, times were only crudely estimated to the nearest hour or so. Thus most observations may effectively be regarded as untimed.

As discussed in chapter 3, untimed lunar eclipses are of negligible value for the determination of ΔT (unless it is clearly implied that the Moon rose or set eclipsed). Most of this chapter will be devoted to the investigation of solar obscurations in which the Sun was either totally or very largely covered. In section 11.9, a few careful timings of solar eclipses from the early fourteenth century will also be analysed. Finally, in section 11.10, an eighth century report of an occultation of the planet Jupiter by the eclipsed Moon – a very rare event – will be discussed.

Historical sources

European reports of solar eclipses (as well as lunar obscurations) from the Middle Ages and Renaissance are mainly found in historical works, notably chronicles; only a few observations are reported in astronomical treatises.

Introduction

Arab observations of solar and lunar eclipses which are of value in the study of the Earth's past rotation originate exclusively from the medieval period. These records are found in two main sources: (i) chronicles, and (ii) compendia on astronomy. As might be expected, the observations reported in chronicles are essentially qualitative; measurements of any kind are fairly rare. Eclipses and other celestial phenomena (such as bright comets and meteor showers) were mainly noted on account of their spectacular nature. Yet chronicles contain some important astronomical records. The present chapter will be devoted to eclipses described in these works. Arab compendia on astronomy contain many careful measurements of the times of the various phases for both solar and lunar eclipses. These observations will be discussed in chapter 13.

The Arabic names for the Moon and Sun are respectively al-Qamar and al-Shams. In general, medieval Arab chronicles use the term khusuf al-Qamar for an eclipse of the Moon and kusuf al-Shams for an eclipse of the Sun. These designations are still in use today, both among astronomers and the general public. However, medieval Muslim astronomers mostly used kusuf for both types of eclipse, adding the appropriate term for the Moon or Sun. The word khusuf means ‘sinking’, but in describing a lunar eclipse it came to mean a failing of the Moon's light. By contrast, kusuf means a ‘cut’ – i.e. in the solar (or lunar) limb.