Timely Confessions

But how should I know whether what he said was true?  And if I did know it, would it be from him that I knew it?  No it would not; it would be from inside me, from that inner house of my thought, that Truth, which is neither Hebrew nor Greek nor Latin nor Barbarian, and which speaks without the aid of mouth or tongue, without any sound of syllables, and at once I would be sure...

                                             Saint Augustine, 400 AD

The eleventh book of Augustine's "Confessions" begins as a contemplation on the opening lines in the biblical book of Genesis, but soon digresses into a lengthy and rather abstract consideration of time. It's clear that the bishop of Hippo was deeply perplexed by questions about the nature and the meaning of time. He commented that in ordinary conversation we use the word "time" (and related words, such as when, now, before, soon, etc.) more than any other, and yet if we're asked to explain time, we find that we cannot. (At one point he raises the interesting possibility that perhaps we do understand time, we simply don't know how to express this understanding.)

Augustine says he was taught as a boy that time consists of the past, present and future, and yet (he reasons) the past no longer exists and the future does not yet exist, so this leaves only the present, but the present has no actual extension, so he is tempted to think that time does not exist at all. However, he rejects this conclusion on the grounds that, since the concept of time is so unambiguously useful, it must represent something meaningful, even though we may have difficulty describing it. He goes on the consider the common idea that time is nothing other than a measure of the motions of bodies, particularly the motions of the sun, moon, and stars in the heavens, but this, too, he rejects, arguing that the passage of time is not dependent on the movements of any objects. In support of this assertion he cites a biblical story, observing that

When, at a man's prayer, the sun stood still so that he might fight and win a battle, the sun did indeed stand still, but time went on; it went on for the space necessary for that battle to be fought and brought to its conclusion...  That no body can move except in time is something I understand, but that the motion of a body actually is time is something which I do not understand.... A body may sometimes be in motion, at varying speeds, and may sometimes be standing still, but by means of time we measure not only its motion but its rest...  Time therefore is not the motion of a body.

What exactly are we measuring when we measure an interval of time? What do we mean when we say two intervals of time have the same length, or one interval is twice as long as another? Augustine's conclusion is that intervals of time actually exist only in the mind, in the form of memories of the past and anticipations of the future, and that the measures of time are actually measures of these psychological formations. It may seem odd that he found this to be satisfactory, considering that it predicates time on the existence of a human consciousness, whereas measures of time are invoked in Genesis prior to the creation of Adam. However, we can interpret his view more broadly, so that when we says (11;27)

As things pass by, they leave an impression in [my mind]; this impression remains after the things have gone into the past, and it is this impression, which I measure in the present, not the things which, in their passage, caused the impression. It is this impression which I measure when I measure time.

we could extend this to "impressions" created in other media, besides just our minds. Under this interpretation there is a fascinating parallel between Augustine's conclusion and the modern understanding of measurements in quantum mechanics. An event is considered to have definitely occurred only when a thermodynamically irreversible impression has been made, and it is, in fact, these irreversible impressions that we measure when we measure the occurrence of events and the passage of time.

Another intriguing parallel between Augustine's discussion and modern ideas is the concept that only things with beginnings and endings can be measured. He says

Let us consider the case of a bodily voice.  The voice begins to sound, it sounds, it continues to sound, and then it stops sounding... Now consider another voice.  This voice begins to sound; it still goes on sounding; it sounds at the same pitch continuously with no variation...  we must measure it, and say how long it is... but measurement is only possible from the beginning, when it started to sound, to the end, when it stopped sounding.  What we measure is the space between a beginning and an end.  Therefore, a voice that has never ceased to sound cannot be measured.

Here we see Augustine explicitly considering a purely monochromatic wave, which, if such a thing existed, could have no beginning and no ending (technically because any finite starting or stopping point would introduce other frequencies into the Fourier spectrum). Such a wave carries no information at all, and hence, just as Augustine suggests, there is nothing to measure. Indeed we could go further and claim that such a "voice" (if it existed) would be undetectable, just like a free photon that was never emitted and will never be absorbed.

It's also interesting to see that Augustine did not automatically rule out the possibility of temporal symmetry with some kind of "advanced action". He wrote (11;18)

When I recollect the image of my boyhood, and tell others about it, I am looking at this image in time present, because it still exists in my memory.  Whether a similar cause operates with regard to predictions of the future - namely, that images of things which do not yet exist are felt in advance as already existing - this, I confess, My God, I do not know.

We can see in this passage some hint of things like the Wheeler-Feynman absorber theory of electrodynamics, in which radiation reaction is "felt in advance" of "things which do not yet exist", and the basic laws are regarded as perfectly symmetrical with respect to the direction of time.

Incidentally, there are frequent references to "syllables" in Augustine's discussion of time. For example, he writes

What, then, is it that I measure?  Where is that short syllable by which I measure?  Where is that long syllable which I measure?  Both have sounded; have fled away; have gone into the past, and no longer exist.

Surely the Confessions would have been read by Shakespeare, so it's probably not mere coincidence that we find in Macbeth's famous soliloquy (5;5)

Tomorrow and tomorrow and tomorrow creeps in this petty pace from day to day to the last syllable of recorded time.

Fifteen centuries after Augustine wrote his Confessions, the very same questions about the nature and measurability of time were discussed by Henri Poincare in an essay called "The Measure of Time". The similarities between these two discussions are remarkable. Poincare pointed out that we have no direct intuition of the equality of two intervals of time:

People who believe they possess such an intuition are dupes of an illusion.  When I say the same time passes from noon to one as from two to three, what meaning has this affirmation?  The least reflection shows that, by itself, it has none at all.

He goes on to consider the idea that the motions of the heavenly bodies constitute time, but, like Augustine, he rejects this idea, pointing out that tidal interactions between the earth and moon cause the earth's rotation to slow in a predictable and measurable way, whereas this physical fact could have no meaning if time were simply defined in terms of rotations of the earth. Also, like Augustine, Poincare notes that we could define time in terms of our psychological impressions, but he seeks a more objective definition of time, one more suitable for the practice of physics.

Poincare proposes a definition based on the principle of sufficient cause, i.e., the notion that equal causes lead to equal effects. If we are given a physical system that repeatedly returns to the same intrinsic state, and if the extrinsic conditions are unchanging, then we can assert by the principle of sufficient cause that the "periods of time" between returns to this state are equal. He stresses, however, that this is merely a convention. Furthermore, this approach relies on exact equality between the periodic conditions, which is never achieved in practice. For example, after a spring-powered clock has gone through one cycle of its motion, the spring has uncoiled slightly, so the state of the clock at the beginning of the second cycle is not exactly identical to the state at the beginning of the first cycle. In order to argue that the durations of the two cycles are equal we must claim that this difference in the state of the spring does not significantly affect the duration of a cycle. Likewise the descending mass which drives a pendulum clock is slightly lower on each cycle; and the angular momentum of the earth is slightly less from one rotation to the next. Indeed the laws of thermodynamics imply that the states of any (classical) mechanical clock on successive cycles cannot be identical. Hence the applicability of the principle of sufficient cause is not as straightforward as we might wish.

The same difficulty arises with respect to the extrinsic (or external) conditions. Consider, for example, two identical clocks in two different locations, and ask whether the principle of sufficient cause still applies. Does the mere difference in locations affect the durations of the clock cycles? To some extent this is (again) a matter of convention, since we have no direct means of comparing clocks at different locations. We might separate two clocks and then bring them back together, but even if we find that they have recorded the same total lapse of time we cannot prove that they ran "at the same rate" while separated. It can always be argued that their readings were affected by the act of separating and re-uniting them, in just such a way as to give equal elapsed times overall. On the other hand, if the clocks are at fixed positions in relation to each other, we can send signals (assumed to be of constant speed) back and forth between them, and show that the results are consistent with the idea that position does not affect their rates of operation. (Of course, this does not prove that the intervals are uniform, since we have no a priori reference with which to compare them, but it does imply that they are mutually co-linear. In effect, then, we identify time with the largest equivalence class of mutually co-linear physical processes that we can find.) Hence the homogeneity of space with respect to the workings of clocks is a viable convention to impose on the interpretation of experience.

This is fortunate from a practical standpoint, because the location of a clock cannot be ascertained on the basis of purely local measurements. In other words, if we enclose the workings of a clock inside a sealed container, there is no way to determine the clock's absolute position based on measurements inside the container, without reference to outside entities. By the same token, we cannot distinguish between different states of uniform motion based on purely local measurements. On the other hand, it is possible to locally sense acceleration, and we know that accelerations can affect the workings of a dynamical mechanism. This leads us to the concept of an ideal clock, which is a clock that operates on the principle of sufficient local cause. To construct an ideal clock we begin with any ordinary "sufficient cause" clock that has been constructed to work under specific reference conditions, and then we determine the sensitivity of the clock to each locally sensible effect, and we compensate for each effect, either by re-designing the clock so that it is inherently insensitive to the effect, or by including a sensor (such as an accelerometer, a thermometer, etc.) to sense the condition and apply a suitable correction. Once we have done this for every locally sensible effect, we have an ideal clock.

Needless to say, in practice we can only approximate a truly ideal clock, so there is always some tolerance on the accuracy of any real clock. Nevertheless, these tolerances are defined relative to our notion of a perfectly ideal clock, i.e., a clock that is perfectly compensated for every locally sensible effect.

Why do we stress "local" sensibility? This is certainly not the only possible way of defining an ideal clock. We could, in principle, construct clocks that take into account the absolute position of the clock relative to the distant stars, and the velocity of the clock relative to those stars (or to the cosmic background radiation). Indeed this would enable us to establish an absolute frame of reference at each location. However, in order to do this, we would need to inform our clocks with global information, because the locally sensible variations in incoming radiation (for example) have no appreciable effect on the workings of a clock (as can be shown by two identical clocks, side by side, one shielded from all external radiation and the other not). Therefore, the conventional notion of an ideal clock (and, implicitly, of intervals of time) is defined in terms of local physics, i.e., periodic physical processes with compensation for local effects only, not global effects.

As a result, it's conceivable that two ideal clocks, periodically intersecting with each other so that their readings may be directly compared (for example, two clocks in differing freefall orbits, one circular and the other highly elliptical, around a gravitating body), may show different elapsed times between meetings. This is actually the case according to the general theory of relativity. The intervals measured by an ideal clock represent proper time, but the proper time along different paths between the same two events can differ, due to how those paths are embedded globally in the spacetime manifold, even though the locally "felt" physical conditions along those paths are identical. This raises the question "what is proper time?" If we allow the elapsed times along the individual worldlines of physical objects to be independent, why then can we not claim that the elapsed proper time along the path of an unchanging object is zero? In other words, if no physical change occurs along a certain span of a worldline, and if the duration of the span is not constrained by any extrinsic requirements, then what gives the span a definite duration? In general, what exactly are we measuring when we measure the elapsed proper time along a certain worldline?

The best answer today seems to be that proper time corresponds to the phase of the quantum wave function. This explains why the elapsed time is not necessarily zero for a stationary physical system, because even such a system has an advancing quantum phase. Recall that the Schrodinger equation for the complex wave function Ψ(x,t) for a particle of mass m in one dimension is

and the probability density at any given x,t is d(x,t) = Ψ*Ψ. If the potential V is static, meaning that it depends only on x, then the form of this partial differential equation is such that the general solution can be factored into two parts Ψ(x,t) = α(x)β(t). Substituting this into the above equation and dividing through by Ψ gives

Since the left side is purely a function of t and the right side is purely a function of x, and these two variables are independent, it's clear that each side must be a constant, which we will call E (because it turns out to represent the energy of the system). Thus we have two separate equations

From the left hand equation we immediately have

independent of V. Then, given the potential function V(x) and the energy E of the system, the right hand equation enables us to determine the function α(x), and thus we have

The factor β(t) has unit magnitude, so it has no effect on the probability density, other than to advance the phase at a frequency proportional to the energy E. Thus the probability density is stationary, independent of time, i.e.,

A stationary state is one for which the probability density is strictly a function of the position x, independent of the time t. This might lead us to overlook the fact that the actual quantum wave function itself is still a function of time. The phase of the wave function is always advancing in proportion to the coordinate time t, or, more accurately, in proportion to the proper time τ, even for a physically stationary system. This answers the ancient riddle about what it means for time to progress for a physical object that is not changing. It becomes clear only after we realize that physical systems have a phase as well as an amplitude. Of course, the quantum phase of a system, by itself, is unobservable, but the effects of the phase become apparent in the form of interference when two or more wave functions are superimposed.

It was mentioned previously that time is identified with the largest equivalence class of mutually co-linear physical processes, and now we see that this is fundamentally based on the frequencies of the quantum wave functions, because their phases are always linear functions of time.

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