Somehow we are always surprised when we have our first snow in mid-December, as if we had never seen the cold white stuff so early before. The snowfall last night was unusually heavy – 27 centimeters. It was still snowing at nine this morning. Jack, our neighbor, has a 4×4 pickup, so he was able to back out of his driveway and get to work. Kay had to be at the dentist’s at 11:30, so I went out to clear our driveway and give her a good start. The snow was still crisp and fluffy. After a while though I was glad to lean on the shovel.
Kay sneaked up behind me and heaved a snowball past my ear. I whirled and doused her with a shovelful. When she closed in and grabbed my arm, the threat of war ended in laughter.
“I hope the snowplow comes this way soon,” she said. “I hate driving in snow.”
“But you love snow as long as you don’t have to drive in it or shovel it,” I reminded her. “Last February out at the lake, I remember you watching snowflakes falling on the dark wood of the picnic table. You were fascinated by the variety of patterns in those flakes.”
It was still snowing lightly. Kay caught some white flakes on her glove and inspected them closely. She chuckled a little. “I wonder if there’s a special prize somewhere for anyone who finds two snowflakes alike.”
I shrugged, and kept on shoveling.
“Oh, here comes the plow,” Kay exclaimed.
Rounding the corner, a heavy truck with a wide blade mounted up front was turning into our street. It was scraping the snow out of its path but leaving it piled up like a mini-mountain range along the curb on our side of the street. Soon that oncoming avalanche completely blocked the entrance to our driveway. Kay waved gratefully to the driver. My feelings were somewhat more mixed as I began to attack the newly heaved-up ridge with my shovel.
There isn’t much about moving snow shovelful by shovelful to occupy one’s mind. My mind was still in my study digging into the possibility that time transforms the world Now-state after Now-state. As that truck moved steadily and powerfully on down the street, it also made its way into my reflections. The plow’s smooth advance was being powered by successive individual strokes of pistons in the cylinders of its engine. The ignition is timed to fire vapor-filled cylinders one after another at short intervals. Time also seems to come along smoothly without any obvious abrupt starts or stops – but just maybe it actually proceeds by consecutive steps that are too tiny and swift for the eye to see or the mind to grasp. Breaks in time, or continuity? How could one tell? What would serve as evidence pointing one way or the other?
A scatteration of ideas showered into my mind like snowflakes. I just had to put them on paper. With that motivation, the shoveling was finished in record time.
When I had hung up my jacket and was heading for the study, Kay handed me a mug of hot chocolate. I explained that I was about to set down some thoughts on a sort-of-atomic theory of time – that moments may perhaps come along “blip-blip-blip.”
She remarked, “And while you are doing that, you will of course be drinking your chocolate sip by sip.” She understood.
Time: a continuum or a succession of discrete moments?
From my reading, I know that a considerable number of world-class thinkers have believed that any length of time is the sum of many distinct and separate “little times,” “moments” or “instants.” Processes and activities have long been understood in terms of stages, phases or moves. Only in the last few centuries has time been conventionally conceived as an uninterrupted, uniform continuity, an undifferentiated unity, a continuum completely without intrinsic parts.
In ancient times, thinkers speculated as to whether the world might be composed of hordes of things so tiny that they could not be divided further into smaller portions. In Greek the word “atom” referred to “what cannot be divided or cut.” In his letter to the Corinthians, the apostle Paul used the word “atom” when he wrote of a “moment of time, the twinkling of an eye.” Today the adjective “discrete” is used to describe entities which, like atoms, are individually distinct and separate from one another.
Many famous thinkers have taken seriously the notion of the discreteness of time and have espoused it publicly. They include Zeno of Elea (fl. 475 BCE), Martianus Capella of Carthage (fl. 470 CE), Isadore of Seville (d. 636), the Venerable Bede (d. 735), al-Khazali (d. 1111), William of Ockham (d. 1349), Evangelista Torricelli (d. 1647), René Descartes (d. 1650), Arnold Geulincx (d. 1669), Nicolas de Malebranche (d. 1715) and David Hume (d. 1776).
In the eighteenth century, mathematicians came to think of the number sequence in spatial terms. Since space had long been accepted as an unbroken continuum, by analogy mathematicians decided and agreed that between any two rational numbers (integers or fractions) there are always smaller fractional numbers. Since the numbers which appear in natural order around the spatial perimeter of a clock-face were accepted as markers of time, it easily came to be assumed that time, like space and the number sequence, is also a continuum.
In the nineteenth century therefore any suggestion that time might be discrete or “granular” would have been deemed ridiculous. Everyone then “knew” that time flows continuously.
Arguments in favor of the hypothesis of temporal discreteness
From quantum mechanics
Space, time and numbering having been assumed to be infinitely divisible continua. It was also taken for granted that any given amount of energy could be divided again and again into ever smaller fractional portions. There would be no limit to how infinitesimally small an amount of energy could be. Energy, it was therefore believed, could be radiated, absorbed or transported in any amount that could be mathematically specified.
In 1900, however, Max Planck proved that action (the product of energy and time) is not endlessly divisible into ever tinier fractional portions. Once a certain tiniest measure of action – which Planck dubbed a “quantum” – has been reached by a process of division, further fractionalization of energy becomes impossible.
Planck thus showed that action comes, as it were, in discrete packets, each of which has a standard, equal, and lowest possible value – much like the coin which has the lowest value in a monetary system. Higher denominations of currency must always be multiples of the value of some least coin. Each item offered for purchase must be priced in terms of that coin or it cannot be paid for precisely.
Similarly in the physical world no action involving less than one quantum of energy can ever take place. The value of each energy transaction will always be a multiple of the basic quantum unit, which is 6.626 x 10-34 joule-seconds. Unless and until the energy input into some situation has reached the magic measure of at least one quantum, no new action can take place.
The shocking news of Planck’s discovery that energy comes in units which are actually discrete and indivisible erupted among natural scientists like a volcano. No doubt many of them fervently hoped that somehow this unsettling revelation would go away.
In 1905, however, Albert Einstein used the quantum principle to explain a physical conundrum known as the “photoelectric effect.” Physicist Philip Lenard had noticed that when light rays hit a metal surface, electrons are ejected, and that the speed at which they are ejected never exceeds a certain maximum limit. According to classical energy theory, the more intense the light, the more energy should hit the surface. With increasing intensity of light, the speed of the ejected electrons should become greater and greater. For years the question of why the exit speed of light-dislodged electrons is constant had been a head-scratching puzzler.
Einstein’s proposal was that light – indeed, every kind of electromagnetic radiation – is emitted in quantized units. The energy in a light ray consists of such quantized units, which came to be called “photons.” The amount of energy emitted depends on the vibrational frequency of the light’s source. Einstein maintained that, no matter how intense the illumination may be, each photon which hits the metal possesses only a single quantum of energy. When applied to an electron in the metallic surface, that single quantum of energy can eject it only so fast and no faster. Hence the mysterious maximum speed limit on light-ejected electrons. Einstein’s quantized photoelectric theory was soon experimentally verified by Robert Millikan.
The Danish physicist Niels Bohr developed a model of the hydrogen atom which resembled the solar system. In his conception, the atom had a central nucleus around which a lighter particle called an “electron” could move in little closed orbits, much like the paths in which planets swing around the sun. Bohr noticed that the atomic electrons travel in only a strictly limited number of orbits, which are spaced out one after the other at increasing but specifiable distances from the nucleus.
In the normal energy state of the atom – the “ground state” – the electron moves in the orbital track closest to the nucleus. If however the atom becomes “excited” by a certain input of energy, the electron will then leap into an orbit farther out from the nucleus. Curiously, however, while the electron is transferring from one orbit to another, it can never be found between the orbits.
Bohr explained this behavior and the spacing of the orbits by quantum theory. If a single quantum of energy is imparted to the atom, the excited electron will speed up and leap over into the orbit which is next farther out from the nucleus. Two quanta will send the electron two orbits farther out, three will move it three, and so on. When no further input of energy is forthcoming, sooner or later the excited electron will begin to radiate off the quanta of energy which it had earlier absorbed. When it loses one quantum, the electron will move back into the orbit which is next closer to the nucleus. Losing two quanta, it will move two orbits inward, and so on, until the atom returns to its unexcited ground state at which it radiates no more.
In making these discrete “quantum leaps,” the electron seems to jump the full distance from one orbit to another instantly. It cannot be detected somewhere in between the prescribed orbits. The particular spacing of orbits appears to be due to the fact that no fractions of a quantum of energy exist which might move the electron only part of the distance over to a new orbit. Without the input of at least one quantum of energy there will be no leaping.
Bohr’s speculation that energy may be transferred to an atom only in quantum units was proved correct by the experiments of James Franck and Gustav Hertz. It is now known that, while a quantum of energy may not only power an electron to jump between orbitals or energy-states, it can also abruptly change a particle’s linear momentum, its angular momentum, its magnetic polarity, or its charge. A quantum can even change the inclination of an orbital by altering the direction of the axis of a rotating particle (its precession).
The quantum unit – known as Planck’s constant, symbolized as h – turns up in every investigated aspect of microphysics. Quantized micro-discontinuities in transfers of energy and quantic discreteness are accepted in today’s physics as unreservedly as are natural numbers.
In order to give a scientific description of a certain movement in a physical system, the masses of the components involved and the distances through which they moved during a certain interval of time must be measured. These measurements can be inserted into well-known equations to derive values for the dynamic aspects of a motion, such as its speed, acceleration, momentum, work, energy, action and frequency.
In all of the formulae for these aspects of motion, units of time are essential. Without a measure of the time during which a motion or process goes on, the equations are utterly useless. Consider the following. Frequency – a cyclic phenomenon’s rate of repeating – is obviously a direct function of time. The amount of work done is described in terms of an amount of mass moved over an amount of distance during a certain time. The power of a certain mover tells how much work it can do in a given time. The amount of energy which was expended to accomplish a given result is determined by the product of the power used and the length of time during which it was operating. Without time electric charges would never move, and if electric charges were never in motion, there would never be a magnetic field to power electric motors.
The values of conventional units of mass, distance and time were set by human officials according to different cultural customs or arbitrary decisions. There have been hundreds of these units: units of weight, such as grain, ounce and gram; units of length such as inch, meter, fathom and furlong; units of time such as second, hour, week, month, shift and watch (aboard ship). The value of the quantum of action, however, was not set by any human decision. In what was conceived to be the great “continuum of cosmic energy flow” no one had ever imagined that energy possessed inherently discrete units, the size of which is given, definite and universal – that minimal amount of energy which is denominated by Planck’s constant.
Why does this “natural” unit exist? Why is this unit just so large and no larger – or no smaller? Why is h a constant?
Being a unit of applied energy, the numerical value of Planck’s constant is a constant joint product of three factors: a mass x a distance moved x an elapsed time. The unexplained universal constancy of this minimal product cries out for an inquiry into the source of the quantum’s bounding limits. Do its limits arise from its mass factor, its distance factor or its time factor?
The boundaries of a given mass may vary widely with temperature. If a sealed glass jar full of water is cooled below –2°C, the water will expand and shatter its container. If a similar jar is heated beyond 100° C, it will explode and its contents will turn into a boundless cloud of steam. Masses do not have fixed bounding limits.
Also, as far as we know, the space in which a distance can be measured is boundless, not at all self-limiting. Beyond any markers which we may set up, and beyond any edges or surfaces which we encounter, spatial distance goes on and on.
Neither mass nor distance, then, can account for the bounding limits which give discreteness to the quantum of energy.
What about the time factor? In dealing with time are there any boundaries which are universally encountered? Does time involve limits?
Most definitely yes! As Christmas draws near you may see a boy peering through a shop window at the display inside. Though dearly desiring to get his hands on those computer games, the transparent glass stands between him and them. Pressed against the clear glass, his nose cannot smell the hot roasted nuts inside the store. Time is like that situation.
If I ever yearned to change the past or to discover what the future holds for me, I would always find myself in much the same predicament as that boy – only doubly so. Like a bug which has somehow managed to get in between the panes of a double-glazed window, I seem to exist in a narrow “time crevasse” between the sheer walls of “no longer” and “not yet.” Confined between these invisible, impenetrable time-barriers, I can neither reenter times past nor force my way into the future. One barrier prevents me from going back and changing or erasing what I once did; another barrier makes me wait in anticipation or dread of the future.
If a moment of time is always bounded by at least the two aforementioned limits, and if no inherent boundaries have ever been found for mass or space in general, it seems reasonable to maintain that what gives distinctive discreteness to the quantum is the intrinsic limitation of its time factor.
In my opinion, therefore, the universal existence of quanta in matter and motion bears testimony to an inherent and universal discreteness of time.
From quantic matter-waves
In 1905 physicists were shocked when Einstein showed that light consists of particles (photons), even though under certain experimental conditions light obviously behaves like waves. Waves and particles, however, are incompatible. Physicists sustained another shock in 1926 when Davisson and Germer at the Bell Laboratories verified a hypothesis which had been proposed by De Broglie two years earlier – that particles have wave-like characteristics.
In 1927 the most notable explanation of matter-waves came from Erwin Schrodinger, who suggested that electrons are not hard little balls circling about the nucleus but patterns of “standing waves.”
You can approach this idea by jerking a rope. Tie one end of a rope to an unmoving support and stretch the rope fairly tightly out to a convenient length, then quickly jerk the free end up and down. A hump-like wave followed by a trough will travel along the rope. When they come to the fixed end, they will reverse and start back along the rope to your hand. If you jerk your hand up and down, a series of waves will travel over and back in such a way that some returning humps will coincide with out-going humps, making the hump larger, and some returning troughs will coincide with out-going troughs, making the trough deeper. Some out-going humps, however, may be entirely cancelled out by returning troughs. Along the rope these coincidings will set up an “interference pattern” consisting of large, loopy “standing waves” separated at regular intervals by stationary “nodes.”
If you increase the frequency and vigor of your regular jerkings, the number of standing waves and nodes will also increase. That number will always be a whole number – one single loop, two loops, three loops, etc., all separated by nodes. Any additional minor fractional humpies or troughies which you introduce into the rope will quickly disappear. A given length of rope will sustain only a whole number of standing waves – that whole number which will divide the rope’s length evenly. This means that the number of standing waves in a given length of rope cannot be changed gradually but only discontinuously.
The general principle governing the production of standing waves is called constructive and destructive interference. This principle is easily seen when water waves which encounter and pass through one another also develop standing waves. Coinciding crests of waves rise much higher and coinciding troughs sink much lower. If the waves are regularly spaced and of similar amplitude and length, their constructive interference can produce standing waves. A crest and a trough can cancel each other out between standing waves and become, as it were, a nodal patch – neither high nor low.
In a flash of insight, Erwin Schrodinger related the idea of interference patterns, with their standing waves and nodes, to the difficulty which particle physicists have trying to locate an electron in orbit around an atomic nucleus. The location of the electron at a certain distance from the nucleus could be specified only in terms of the statistical probability of finding it in a given location. Schrodinger developed a time-dependent equation which quickly became the key to wave mechanics. That equation implied that an electron could be considered as a train of standing waves. The frequency and wavelength of an orbiting electron determine the radius and overall length of its circular orbital – just as frequency of jerks determines the number of standing waves which fit evenly into the length of a vibrated rope. The electron therefore could be found most easily in its “loops” or times of maximal amplitude, but not at all in its nodes of zero amplitude.
This conception helped physicists to understand the elusiveness of electrons, but made those entities extremely difficult to visualize. If particles aren’t really little round balls whizzing around larger balls, what are they? If they are really waves, exactly what is it that is vibrating? Physicists have proposed a variety of explanations.
A railway track constrains the route of trains, although a train appears on the track only at certain times. Similarly, perhaps there is an invisible but constraining path shaped like standing waves over which an electron – conceived as a periodically expanding and condensing cloud of energy – must travel. Or perhaps the electron is a wavetrain-shaped belt of energy which surrounds the atomic nucleus like an orbital but never moves around it at all. Perhaps a matter wave is really just a “wave of probability” which, on detecting an electron during an experiment, suddenly turns into a definite actuality.
No one knows what an electron or any other fundamental particle is made of. The size and shape of the electron’s position or path, however, does depend on the frequency and wavelength of its quantized vibrational energy. Truly it can be detected here but it can’t be detected there. Its “trajectory” is certainly characterized by baffling discontinuities.
In my view, the standing-wave hypothesis could be completed by postulating the alternating existence and subsequent annihilation of an electron, whatever it actually is. In Schrodinger’s wave equation, the train of waves around a quantically prescribed size of electronic orbital is obviously completely dependent upon time. The wave-theory of matter thus fits perfectly with the hypothesis that the creation of things in successive Now-states automatically creates time. This statement is easily transposed into the statement that the discreteness of time is the successive creation and annihilation of Now-states of the universe.
From psychology
The process by which we receive information from the world around us and within us is obviously discontinuous. Take a look at the scene in front of you. There are various objects of different sizes, shapes and colors, all in different places and directions from each other. Your attention may hop from one object to another, or you may take in at a glance several aspects of the scene. At the same time you are supported by floor or chair, maintaining your balance, breathing easily and comfortable in your clothing. At each moment a flock of messages from all those different directions is converging on your brain. At an incredible speed this multifarious information is being transmitted by discrete “off-and-on-and-off-and-on” firings of nerve cells. While journeying toward your brain, these neural impulses must leap across the microscopic gaps which separate one neuron from another. Though you never experience the world as flickering in and out of your consciousness, the information which your sensory nerves send you is by no means an uninterrupted, smoothly flowing stream.
The discrete incoming impulses are processed at several different locations in your brain, each of which registers a different characteristic such as color, edge or pattern. How all this discretely distributed information is eventually integrated to give you a meaningful, unified experience moment by moment is a profound mystery.
The pictures on a movie screen don’t actually move. A series of still pictures, each slightly different from the one before, is projected onto the screen. Frame follows frame – flick-flick-flick – at just the right rate to superimpose a somewhat differing image upon your eyes before their brief visual impression of the previous picture disappears. Due to lagging retinal chemistry and the brain’s ability to retain and compare successive impressions, the resulting experience of “motion” seems smooth and natural. Unless the movie projector slows down, you will not become aware of the “breaks” between successive frames.
Similarly when you are watching television, you are never conscious of the swiftly moving, pulsating electron beam which lights up a selection of the discrete pixels (picture elements) which are spread in gridlines over the face of the picture tube. You see only one whole, integrated picture which seems to change in much the same way as things change in real life.
Although time seems to come on smoothly and continuously, it may actually be coming in discrete pulsations. If our psychological makeup enables us to fuse a succession of discrete states of unmoving pictorial elements into what we perceive as an uninterrupted, continuously moving transition, it is conceivable that our mental constitution performs the same remarkable feat with discrete Now-states of time.
The notion of discreteness implies the existence of separation, differences, limits, boundaries and edges. These implications could also apply to discrete Now-states of time.
Edges of all kinds are enormously important for our consciousness. When driving at night we need to know where the edge of the pavement and the lane markings are. Normally we assume that the road conditions ahead are going to be continuously the same. When an animal, a child or a stalled vehicle suddenly appears in the road directly ahead, the situation is now alarmingly different. We have come to a discontinuity – the edge of our security.
When we are walking it is important to look out for curbs, holes, fallen branches and stones over which we might trip. Edges warn us that currently familiar circumstances are about to end and that we should prepare to meet new and possibly dangerous conditions. When approaching steps “down,” setting the table or rolling over in bed, awareness of edges is a serious concern. The retinas of our eyes are admirably set up to recognize the contrasts which mark the edges of things – a feature which obviously has great survival value for mobile beings with roving eyes and feet.
If our conscious awareness of the world were constantly being turned on and off in a drastically stroboscopic fashion, life would be synonymous with surprise and alarm. Any scene that ever came up would always be meaningless, and our existence would be perpetually fearful and utterly hopeless.
If the time in which we live actually comes as Now-state after Now-state, edge after edge, how is it that we normally do not live in unrelieved confusion and panic?
Although our world is always changing with time, our consciousness somehow retains memories of the immediate and distant past, including the conceptual knowledge which we have gained from experience. Being thus able to anticipate much of the oncoming future, our fear of it is largely dissipated. Our mentality somehow succeeds in blurring the hard edges of new Now- states. Barring the occurrence of unanticipated catastrophic events, most of the time we manage to live fairly comfortable lives without drastic forgetfulness or increasing fearfulness.
We should actually be thankful that time does bring a series of changes. Without these changes there would be no need for consciousness at all. Within a situation which is quite edgeless and homogeneous, such as a pitch-black room or a dense fog, there is nothing specific upon which to focus our attention. Without the occurrence of significant distinctions, things would all meld together. If all sound were as continuous as unbroken silence, there would be no aural communication or music. Any phenomenon which is always the same, such as the weight of our wristwatch, is utterly ignored by our consciousness. Difference, discontinuity and the ability to discriminate are absolutely essential for conscious human existence. Without broken continuity, without change, without time, our consciousness – if any – would be only a useless blank.
From the intellect
Our minds function most confidently and efficiently when our intellects possess clear and distinct ideas. The world is full of things that people can talk about. In order to be certain about which of all those things one is presently talking about, its extent and distinctiveness must be specifiable. The subject should possess a definite boundary or edge which marks its limit. It must not be confused with anything else whose characteristics are quite different. The rules of logical thinking rightly require definiteness. A thing is what it is and is not what it is not. A thing may be either this or that, but not both at the same time or in the same respect. Vagueness, ambiguity and uncertainty are sure to spawn misunderstanding.
Dictionaries list definitions which help us to keep the meanings of our words as clear as possible. Distinctions are drawn between the various kinds of things, actions, qualities, quantities and relations which we are likely to encounter. The number system in mathematics combined with clearly specified standard units of magnitude enables us to be precise about measurements. Clocks are useful because they assign numbers to successive “time-intervals” and thus mark off definite portions of elapsed mundane time.
It is so important to know exactly what we are dealing with that, where there are no obvious natural distinctions or dividing lines, we often draw them ourselves. Into what size of pieces shall I cut this cake? In drawing or painting a certain scene, where on the canvas should I place the horizon line? Where exactly on the ocean is the international dateline or the offshore boundary of a maritime country?
Distinctions, boundaries, divisions, breaks, interruptions, edges, voids, beginnings, endings and betweens – all of these are implied in the notion of discreteness. We cannot say that we intellectually understand anything which does not in some way incorporate discreteness. If we can analyze complex matters – that is, take them apart – intellectual satisfaction becomes attainable.
If our intellects are ever to comprehend the nature of time, discreteness must enter the picture.
From the nature of clocks
It is commonly believed that time is a continuum and not a series of discrete intervals. If this is so, no clock can register time with precise accuracy. Mechanical clocks jerk ahead at intervals of accelerating speed, each of which is abruptly terminated by a controlling escapement. If real time goes on at a constant rate while clocks are moving on through discrete intervals of varying speed, none of those clocks can be completely accurate at all times. Digital and atomic clocks must also count repetitions of regular motion: the vibrations of crystals. Any back-and-forth movement must slow to a stop, pause, then begin to accelerate in the reverse direction. The revolutions of a regular circular movement can be counted, but time goes on unregistered while an incomplete revolution is yet in progress.
Clocks must display successive numbers. No series of displayed finite numbers can register completely the infinity of numbers which compose a continuum. The numbers which a clock counts off will always fail to register exactly that portion of the time-continuum which elapsed between numbered registration times.
Ultimately it must be realized that clocks are composed of molecules and atoms. The necessary discreteness of the atomic structure of matter cannot provide an infrastructure which will sustain an absolutely continuous time-measuring procedure.
From the sequence of events
In the world around us we know that events in any given place occur one after another. Wave after wave rises and breaks upon the shore. We are used to the discreteness of our heartbeats and the throb of our pulse. Events always have a “before” and an “after.” They begin and end. While something is happening here, in other places other quite separate events are also happening. The world’s story is ordinarily understood as chains of “parallel” successive and discrete macro-events and, in telling that story, historians write or utter one word at a time.
Most cosmologists say that, at the beginning, the whole mass of the universe was concentrated within a very small volume. This unimaginable primeval speck is believed to have exploded in a “Big Bang.” For billions of years since then the universe has been expanding. Some scientists speculate that, since the energy which expands the universe is continually dissipating, gravity will eventually slow down this expansion until the movement reverses. The universe will then begin to contract and will ultimately produce the grand collision known as the “Big Crunch.”
If these savants are willing to accept an abrupt beginning and ending for the macro-universe, they ought to be able at least to consider seriously the possibility of momentaneous discretenesses occurring throughout the time between that beginning and that end. So it appears that a sizeable list of intimations for time’s discreteness can be compiled. Will these be sufficient to convince anyone in our traditional time-culture that time actually does come discretely? Time will tell!
