The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe - John D. Barrow (2002)
Chapter 7. The Box That Can Never Be Empty
“There is an element of tragedy in the life-story of the ether. First we had luminiferous ether. Its freely given services as midwife and nurse to the wave theory of light and to the concept of field were of incalculable value to science. But after its charges had grown to man’s estate it was ruthlessly, even joyfully, cast aside, its faith betrayed and its last days embittered by ridicule and ignominy. Now that it is gone it still remains unsung. Let us here give it a decent burial, and on its tombstone let us inscribe a few appropriate lines:
Then we had the electromagnetic ether.
And now we haven’t e(i)ther.”
IT’S A SMALL WORLD AFTER ALL
“This [quantum] theory reminds me a little of the system of delusions of an exceedingly intelligent paranoiac, concocted of incoherent elements of thoughts.”
One of the greatest truths about the character of the physical universe, which has come increasingly into the spotlight during the past twenty-five years, is the unity of its laws and patterns of change. Once upon a time it would have been suspected that the nature of the most elementary particles of matter had little to do with the shapes and sizes of the greatest clusters of galaxies in the astronomical realm. Likewise, few would have believed that a study of the largest structures in the Universe would be able to shed light upon the smallest. Yet, today, the study of the smallest particles of matter is inextricably linked to the quest for a cosmological understanding of the Universe and the organisation of matter within it. The reason is simple. The discovery that the Universe is expanding means that its past was hotter and denser than its present. As we retrace its history back to the first minutes, we encounter a cosmic environment of ever-increasing energy and temperature which ultimately reduces all the familiar forms of matter – atoms, ions and molecules – to their simplest and smallest ingredients. The number and nature of the most elementary particles of matter will thus play a key role in determining the quantities and qualities of the different forms of matter that survive the childhood of the Universe.
This cosmic link between the large and the small also features in the fate of the vacuum. We have just seen how the theory of gravity that Einstein created can be used to describe the overall evolution of the physical universe. In practice we choose a mathematically simple universe that is a very good approximation to the structure of the real one that we see through our telescopes. At first, we have seen how it was that Einstein’s theory reinforced his expulsion of the ether from the vocabulary of physics by providing a natural mathematical description of universes which are completely devoid of mass and energy – ‘vacuum’ universes. No ether was necessary even if electrical and magnetic fields were introduced to curve space. Yet there was to be a sting in the tail of this new theory. It permitted a new force field to exist in Nature, counteracting or reinforcing the effects of gravity in a completely unsuspected way, increasing with distance so that it could be negligible in its terrestrial effects yet overwhelming on the cosmic scale of the Universe’s expansion. This ubiquitous ‘lambda’ force allows itself to be interpreted as a new cosmic energy field: one that is omnipresent, preventing the realisation of nothing. But, if such a vacuum-buster exists, where does it come from and how is it linked to the properties of ordinary matter? Astronomers like Lemaître and McCrea posed these questions but did not answer them. They hoped that the world of subatomic physics would enable a link to be forged with the vacuum energy of Einstein.
Einstein’s development of the theories of special and general relativity was one half of the story of the development of modern physics. The other half is the story of quantum physics, pioneered by Einstein, Max Planck, Erwin Schrödinger, Werner Heisenberg, Niels Bohr and Paul Dirac. Whereas the new theory of gravity was a single-handed creation by Einstein, needing no revision or interpretation, the quantum theory of the microworld was the work of many hands which had a tortuous path to clarity and utility. The task of unravelling what it meant combined challenging problems of mathematics with subtleties of interpretation and meaning, some of which are far from resolved even today. Each year several popular science books will appear which seek to explain the mysteries of quantum mechanics in a manner that readers who are not physicists will be able to understand.3 Each of these authors is motivated to try yet another explanation of how it works by some of the unnerving words of warning from the founding fathers. From Niels Bohr, its principal architect,
“Anyone who is not shocked by quantum theory has not understood it”;4
or from Einstein,
“The quantum theory gives me a feeling very much like yours. One really ought to be ashamed of its success, because it has been obtained in accordance with the Jesuit maxim: ‘Let not thy left hand know what thy right hand doeth’”;5
or Richard Feynman,
“I think I can safely say that nobody understands quantum mechanics”;6
or Werner Heisenberg,
“Quantum theory provides us with a striking illustration of the fact that we can fully understand a connection though we can only speak of it in images and parables”;7
or Hendrick Kramers,
“The theory of quanta is similar to other victories in science; for some months you smile at it, and then for years you weep.”8
Yet for all this ambivalence, the quantum theory is fabulously accurate in all its predictions about the workings of the atomic and subatomic worlds. Our computers and labour-saving electronic devices are built upon the things it has revealed to us about the workings of the microworld. Even the light-detectors that enable astronomers to see supernovae near the edge of the visible universe rely upon its strange properties.
The quantum picture of the world grew out of the conflicting pieces of evidence for the wavelike and particlelike behaviour of light. In some experiments it behaved as if it were composed of ‘particles’ possessing momentum and energy; in others it displayed some of the known properties of waves, like interference and diffraction. These schizophrenic behaviours were only explicable if energy possessed some revolutionary properties. First, energy is quantised: in atoms it does not take on all possible values but only a ladder of specific values whose separation is fixed by the value of a new constant of Nature, dubbed Planck’s constant and represented by the letter h. An intuitive picture of how the wavelike character of the orbital behaviour leads to quantisation can be seen in Figure 7.1, where we can see how only a whole number of wave cycles can fit into an orbit.
Second, all particles possess a wavelike aspect. They behave as waves with a wavelength that is inversely proportional to their mass and velocity. When that quantum wavelength is much smaller than the physical size of the particle it will behave like a simple particle, but when its quantum wavelength becomes at least as large as the particle’s size then wavelike quantum aspects will start to be significant and dominate the particle’s behaviour, producing novel behaviour. Typically, as objects increase in mass, their quantum wavelengths shrink to become far smaller than their physical size, and they behave in a non-quantum or ‘classical’ way, like simple particles.
Figure 7.1 Only a whole number of wavelengths will fit around a circular orbit, as in (a) but not in (b).
The wavelike aspect of particles turned out to be extremely subtle. The Austrian physicist Erwin Schrödinger proposed a simple equation to predict how a wavelike attribute of any particle changes in time and over space when subjected to forces or other influences. But Schrödinger did not have a clear idea of what this attribute was that his equation could so accurately calculate. Max Born was the physicist who saw what it must be. Curiously, Schrödinger’s equation describes the change in the probability that we will obtain a particular result if we conduct an experiment. It is telling us something about what we can know about the world. Thus, when we say that a particle is behaving like a wave, we should not think of this wave as if it were a water wave or a sound wave. It is more appropriate to regard it as a wave of information or probability, like a crime wave or a wave of hysteria. For, if a wave of hysteria passes through a population, it means that we are more likely to find hysterical behaviour there. Likewise, if an electron wave passes through your laboratory it means that you are more likely to detect an electron there. There is complete determinism in quantum theory, but not at the level of appearances or the things that are measured. Schrödinger’s amazing equation gives a completely deterministic description of the change of the quantity (called the ‘wave function’) which captures the wavelike aspect of a given situation. But the wave function is not observable. It allows you only to calculate the result of a measurement in terms of the probabilities of different outcomes. It might tell you that fifty per cent of the time you will find the atom to have one state, and fifty per cent of the time, another. And, remarkably, in the microscopic realm, this is exactly what the results of successive measurements tell you: not the same result every time but a pattern of outcomes in which some are more likely than others.
These simple ideas laid the foundations for a precise understanding of the behaviour of heat radiation and of all atoms and molecules. At first, they seem far removed from the definite picture of particle motion that Newton prescribed; but remarkably, if we consider the limiting situation where the particles are much larger than their quantum wavelengths, the quantum theory just reduces to the conclusion that the average values of the things we measure obey Newton’s laws. Again, we see this important feature of effective scientific progress, that when a successful theory is superseded it is generally replaced by a new theory with an enlarged domain of applicability which reduces to the old theory in an appropriate limiting situation.
At first, the quantum theory seems to usher in a picture of a world that is founded upon chance and indeterminism, and indeed it was Einstein’s belief that this was so, and it led him to spurn the theory he had helped to create as something that could not be part of the ultimate account of how things were. He could not believe that ‘God plays dice’. Yet, on reflection, something like the quantum theory is needed for the stability of the world. If atoms were like little solar systems in which a single electron could orbit around a single proton with any possible energy, then that electron could reside at any radius at all. The slightest buffeting of the electron by light or distant magnetic fields would cause tiny shifts in its energy and its orbit to new values because all possible values are permitted. The result of this democratic state of affairs would be that every hydrogen atom (made of a proton and an electron) would be different: there would be no regularity and stability of matter. Even if all atoms of the same element started off identical, each atom in Nature would undergo its own succession of external influences which would cause a random drift in its size and energy. All would be different.
The quantum saves us from this. The electron can only occupy particular orbits around the proton, with fixed energies: hydrogen atoms can only have a small number of particular energies. In order to change the structure of the atom it must be hit by a whole quantum of energy. It cannot just drift into a new energy state that is arbitrarily close to the old one. Thus we see that the quantisation of atomic energies into a ladder of separate values, rather than allowing them to take on the entire continuum of possible values, lies at the heart of the life-supporting stability and uniformity of the world around us.
One of the most dramatic consequences of the wavelike character of all mass and energy is what it does for our idea of a vacuum. If matter is ultimately composed of tiny particles, like bullets, then we can say unambiguously whether the particle is in one half of a box or the other. In the case of a wave, the answer to the question ‘Where is it?’ is not so clear. The wave spans the whole box.
The first application of the quantum idea was made in 1900 by the great German physicist Max Planck, who sought to understand the way in which energy is distributed amongst photons of different wavelengths in a box of heat radiation – what is sometimes called ‘black-body’ radiation.9 Observations showed that the heat energy apportioned itself over different wavelengths in a characteristic way. Our heat and daylight are provided by the Sun. Its surface behaves like a black-body radiator with a temperature of about 6000 degrees Kelvin.10 There is little energy at short wavelengths. The peak is in the green part of the spectrum of visible light but most of the energy is emitted in the infrared region which we feel as heat (see Figure 7.2). The shapes of the curves change as the temperature increases in the manner shown in Figure 7.3. As the temperature increases, so more energy is radiated at every wavelength, but the peak of the emission shifts to shorter wavelengths.
Figure 7.2 The spectrum of a ‘black-body’ radiation source with a temperature of 6000 degrees Kelvin, similar to that of the Sun.
Tantalisingly, before Planck’s work, it was not possible to explain the overall shape of this curve. The long-wavelength region where the energy steadily falls could be explained, as could the location of the peak, but not the fall towards short wavelengths. Planck was first able to ‘explain’ the curves by proposing a formula of a particular type. But this was not really explaining what was going on, merely describing it succinctly. Planck wanted a theory which predicted a formula like the one that fitted so well. He was impressed by the fact that the black-body energy distribution had a universal character. It did not matter what the emitter was made of; whether it was a flame, or a star, or a piece of hot iron, the same rule applied. Only the temperature mattered. It was a bit like Newton’s law of gravity: the material out of which things are made does not seem to matter to gravity, they can be cabbages or kings; it is just their mass that determines their gravitational pull.
Figure 7.3 The changing shape of the Planck curves as temperature, in degrees Kelvin, increases.
Planck wanted to describe the behaviour of black-body radiation by the action of a collection of tiny oscillators, gaining energy by collisions with each other as heat is added, and losing energy by sending out electromagnetic waves at a frequency determined by that of the oscillations. Here, Planck had his most brilliant insight. In the past it had always been assumed that the oscillators in a system like this could emit any fraction of their energy, no matter how small. Planck proposed instead that the energy emission can only occur in particular quotas, or quanta, proportional to the frequency, f. Thus the energies emitted can only take the values 0, hf, 2hf, 3hf, and so on, where h is a new constant of Nature,11 which we now call Planck’s constant. Planck modelled the whole glowing body as a collection of many of these quantised oscillators, each emitting light of the same frequency as it vibrates. Their energies can only change by a whole quantum step. At any moment it is more likely that the hot body contains more oscillators with low energies than those with high energies because the former are easier to excite. From these simple assumptions, Planck was able to show that the radiation emitted at each wavelength was given by a formula that precisely followed the experimental curves. The ‘temperature’ is a measure of the average value of the energy. Better still, the energy that his formula predicted would be emitted at wavelengths not yet observed subsequently proved to be correct.
Ever since these successful predictions, Planck’s black-body law has been one of the cornerstones of physics. Most dramatically, in the last twelve years, astronomers have managed to measure the heat radiation left over from the hot early stages of the expanding Universe with unprecedented precision using satellite-based receivers, observing far above the interfering effects of the Earth’s atmosphere. What they found was spectacular: the most perfect black-body heat spectrum ever observed in Nature, with a temperature of 2.73 degrees Kelvin.12 This famous image is shown in Figure 7.4.
Figure 7.4 The spectrum of the heat radiation left over from the early stages of the Universe and measured by NASA’s Cosmic Background Explorer satellite (COBE). No deviations from a perfect Planck curve have been found.
Planck’s deductions about the nature of thermal equilibrium between matter and radiation at a given temperature were widely explored and ultimately led to the creation of a full quantum theory of all atomic interactions. This picture turned out to have one mysterious aspect to it. It described the intuitive idea of an equilibrium of radiation in a container. If the radiation started hotter than the walls of the container then the walls would absorb heat until they attained the same temperature as the radiation. Conversely, if the walls were initially hotter than the radiation they enclosed then they would emit energy that would be absorbed by the radiation until the temperatures were equalised. If you tried to set up an empty box whose walls possessed a finite temperature then the walls of the box would radiate particles to fill the vacuum.
As the implications of the quantum picture of matter were explored more fully, a further radically new consequence appeared that was to impinge upon the concept of the vacuum. Werner Heisenberg showed that there were complementary pairs of attributes of things which could not be measured simultaneously with arbitrary precision, even with perfect instruments. This restriction on measurement became known as the Uncertainty Principle. One pair of complementary attributes limited by the Uncertainty Principle is the combination of position and momentum. Thus we cannot know at once where something is and how it is moving with arbitrary precision. The uncertainty involved is only significant for very small things with a size comparable to their quantum wavelength. One way of seeing why such an uncertainty arises is to recognise that the act of making a measurement always disturbs the thing being measured in some way. This was always ignored in pre-quantum physics. Instead, the experimenter was treated like a bird-watcher in a perfect hide. In reality, the observer is part of the system as a whole and the perturbation created by an act of measurement (say light bouncing off a molecule and then being registered by a light detector) will change the system in some way. Another, more sophisticated and more accurate, way to view the Uncertainty Principle is as a limit on the applicability of classical notions like position and momentum in the description of a quantum state. It is not that the state has a definite position and momentum which we are prevented from ascertaining because we change its situation when we measure it. Rather, it is that classical concepts like position and velocity cannot coexist when one enters the quantum regime. In some ways this is not entirely surprising. It would be a very simple world if all the quantities that describe the behaviour of very big things were exactly those that were needed to describe very small things. The world would need to be the same all the way down to nothing.
THE NEW VACUUM
“The vacuum is that which is left in a vessel after we have removed everything which we can remove from it.”
James Clerk Maxwell13
The Uncertainty Principle and the quantum theory revolutionised our conception of the vacuum. We can no longer sustain the simple idea that a vacuum is just an empty box. If we could say that there were no particles in a box, that it was completely empty of all mass and energy, then we would have to violate the Uncertainty Principle because we would require perfect information about motion at every point and about the energy of the system at a given instant of time. As physicists investigated more and more physical systems in the light of quantum theory, they found that the last stand mounted by the Uncertainty Principle manifested itself in the form of what became known as the zero-point energy. If one looked at the impact of quantisation on systems like the oscillators that lay at the heart of Planck’s description of heat radiation equilibrium, it emerged that there was always a basic irreducible energy present that could never be removed. The system would not permit all its energy to be extracted by any possible cooling process governed by the known laws of physics. In the case of the oscillator, the zero level was equal to one-half of hf, the quantum of energy.14 This limit respects and reflects the reality of the Uncertainty Principle in that if we know the location of a particle oscillator then its motion, and hence its energy, will be uncertain, and the amount is the zero-point motion.
This discovery at the heart of the quantum description of matter means that the concept of the vacuum must be somewhat realigned. It is no longer to be associated with the idea of the void and of nothingness or empty space. Rather, it is merely the emptiest possible state in the sense of the state that possesses the lowest possible energy: the state from which no further energy can be removed. We call this the ground state or the vacuum state.
As an illustration, consider a rather corrugated terrain of valleys and hills of different depths and heights, like that in Figure 7.5. The valley bottoms are the different minima of the system. They have different heights and are characterised locally by the simple fact that if you move slightly away from them in any direction you must travel uphill. One of these minima is lower than the others and is called the global minimum. The others are merely local minima. In the study of energies of systems of elementary particles of matter, such minima are called vacua to emphasise the characterisation of the vacuum by a minimum energy state. This example also illustrates something that will prove to have enormous importance for our understanding of the Universe and the structures within it: it is possible for there to be many different minimum energy states, and hence different vacua, in a given system of matter.
Indirect evidence for the physical reality of the zero-point energy appears every time a successful prediction emerges from quantum theories of the behaviour of radiation and matter. However, it is important to have a direct probe of its existence. The simplest way to do this was suggested by the Dutch physicist Hendrik Casimir in 1948 and has been known ever since as the Casimir Effect.
Figure 7.5 An undulating terrain displaying several local peaks and valleys.
Casimir wanted to instigate a way for the sea of zero-point fluctuations to manifest themselves in an experiment. He came up with several ideas to achieve this, of which the simplest was to place two parallel, electrically conducting metal plates in the quantum vacuum. Ideally, the experiment should be performed at absolute zero temperature (or at least as close to it as it is possible to achieve). The plates are set up to reflect any black-body radiation that falls on them.
Before the plates are added to it we can think of the vacuum as a sea of zero-point waves of all wavelengths. The addition of the plates to the vacuum has an unusual effect upon the distribution of the zero-point waves. Only rather particular waves can exist between the two plates. These are waves which can fit in a whole number of undulations between the plates. The wave has to begin with a zero amplitude at the plate and end in the same way on the other plate. It is like attaching an elastic band between the two plates and setting it vibrating. It will be fixed at each end and the vibrations will undergo one, or two, or three, or four, or more, complete vibrations before the other plate is reached (see Figure 7.6).
The simple consequence of this is that those zero-point waves which do not fit an exact number of wavelengths between the plates cannot reside there, but there is nothing stopping them from inhabiting the region of space outside the plates. This means that there must be more zero-point fluctuations outside the plates than between them. Therefore the plates get hit by more waves on their outside than they do on the inside-facing surfaces. The plates will therefore be pushed towards one another. The magnitude of the pressure (force per unit area) pushing the plates together is πhc/480d4, where d is the distance between the plates, c is the speed of light, and h is Planck’s constant. This is called the Casimir Effect.15 As you might expect, it is very small. The closer the plates can be placed (the smaller d) so the bigger will be the pressure pushing them together. This is to be expected since the effect arises because some wavelengths have been excluded from the collection between the plates as they don’t fit in. If we separate the plates a little further then more waves will be able to fit in and the disparity between the number of waves present between and outside the plates will get smaller. If the plates are separated by one half of one thousandth of a millimetre then the attractive pressure will be the same as that created by the weight of a fifth of a milligram16 sitting on your finger tip, similar to that of a fly’s wing.
Figure 7.6 In the presence of a pair of parallel plates those vacuum energy waves that can fit a whole number of wavelengths between the plate will be present there. All possible wavelengths can still exist outside the plates.
Casimir had hoped that a spherical version of this model might provide a viable picture of the electron but unfortunately it was not possible to balance the repulsive electrostatic force against an attractive Casimir force as he expected. In fact, when one replaces the parallel plates in the zero-point sea by a spherical shell, or by other shapes, the calculations become very different (and very difficult) and the overall effect need not even have an attractive effect. The shape of the region placed in the vacuum is critically important in determining the magnitude and sense of the resulting vacuum effect.17
Casimir’s beautifully simple idea has been observed in experiments. The first claim to see the effect was made by Marcus Sparnaay18 in 1958, using two plates one centimetre square made of steel and chromium. However, the uncertainties in the final results were large enough to be consistent with no attractive effect being present, and it was not until 1996 that a completely unambiguous detection of the effect was made by Steve Lamoreaux19 in Seattle with the help of his student Dev Sen. One of the greatest difficulties in performing these experiments is ensuring that the two plates are very accurately aligned parallel. In order to see an attractive effect between the plates which is as small as Casimir predicts, one must be able to control their separations to an accuracy of 1 micron over a distance of 1 centimetre. This job can be made easier by replacing one of the plates by a spherical surface so that it does not matter how it is orientated with respect to the flat plate – it always sees the same curvature. So long as the spherical surface is almost flat (or, at least, is not significantly curved over a distance equal to the distance between its surface and the flat plate – in Lamoreaux’s experiment the separation was varied between 0.6 and 11 microns, whilst the radius of curvature of the curved surface was two metres) the expected attractive force can be recalculated to high accuracy. In the experiment, the force is measured by attaching one of the surfaces to the end of one arm of a torsion pendulum. Both surfaces are made of gold-coated quartz to maximise conductivity and robustness. The other end of the pendulum arm is placed between two conducting plates across which there is a voltage difference. A precise measurement of this voltage difference enables one to determine the electric force needed to overcome the attractive Casimir force between the plates and keep them at a fixed separation. The separation is measured with a laser interferometer20 which is able to detect twisting of the pendulum to an accuracy of 0.01 of a micron (see Figure 7.7). The measured attraction of about 100 microdynes agrees with Casimir’s prediction to an accuracy of five per cent.
What these beautiful experiments show is that there really is a base level of electromagnetic oscillation in space after everything removable has been removed. Moreover, this base level changes as the plate separations are changed and it exists between the plates and outside the plates at different levels. The energy in a given volume of the space between the plates is greater when the plates are closer than when they are far apart. This is understandable. If the plates attract one another you need to expend energy to separate them, after which the vacuum energy between them will be lower than before.
Figure 7.7 The experimental set-up used to measure the Casimir force of attraction between two plates in a quantum vacuum.
Even more ingenious experiments have been devised to probe the quantum fluctuations between the Casimir plates.21 Atoms can be perturbed so that their electrons will change from one quantum orbital to another. When this happens they will emit light with a particular wavelength determined by the quantum of energy equal to the difference between the two energy levels. Allow this process to occur between a pair of Casimir plates and the normal decay will not be able to occur if the emitted light has a wavelength that does not fit between the plates. The atom will not decay as expected. Instead, it will remain in its perturbed state. If the wavelength of the emitted radiation fits nicely into the distance between the plates then the atom will decay more rapidly than it otherwise would in a space without the plates present.
There are many other experimentally observed effects of the zero-point energy. One of the earliest to be discovered was by Paul Debeye in 1914, who found that significant scattering of X-rays still occurred from the lattice of atoms making up a chunk of solid material even when the temperature started to approach absolute zero. This scattering is produced by the zero-point energy of the vibrations in the solid.
In the last few years a public controversy has arisen as to whether it is possible to extract and utilise the zero-point vacuum energy as a source of energy. A small group of physicists, led by American physicist Harold Puthoff,22have claimed that we can tap into the infinite sea of zero-point fluctuations. They have so far failed to convince others that the zero-point energy is available to us in any sense. This is a modern version of the old quest for a perpetual motion machine: a source of potentially unlimited, clean energy, at no cost.
While this more speculative programme was being argued about, wider interest in the vacuum was aroused by a phenomenon called ‘sonoluminescence’, which displays the spectacular conversion of sound-wave energy into light. If water is bombarded with intense sound waves, under the right conditions, then air bubbles can form which quickly contract and then suddenly disappear in a flash of light. The conventional explanation of what is being seen here is that a shock wave, a little sonic boom, is created inside the bubble, which dumps its energy, causing the interior to be quickly heated to flash point. But a more dramatic possibility, first suggested by the Nobel prize-winner Julian Schwinger,23 has been entertained. Suppose the surface of the bubble is acting like a Casimir plate so that, as the bubble shrinks, it excludes more and more wavelengths of the zero-point fluctuations from existing within it. They can’t simply disappear into nothing; energy must be conserved, so they deposit their energy into light. At present, experimenters are still unconvinced that this is what is really happening,24 but it is remarkable that so fundamental a question about a highly visible phenomenon is still unresolved.
Puthoff (see note 22) has claimed far more speculative uses for vacuum energy, arguing that by manipulating zero-point energies we could reduce the inertia of masses in quantum experiments and open the way for huge improvements in rocket performance. The consensus is that things are far less spectacular. It is hard to see how we could usefully extract zero-point energy. It defines the minimum energy that an atom could possess. If we were able to extract some of it the atom would need to end up in an even lower energy state, which is simply not available.
ALL AT SEA IN THE VACUUM
“I must go down to the sea again, to the lonely sea and the sky,
And all I ask is a tall ship and a star to steer her by,
And the wheel’s kick and the wind’s song and the white sail’s shaking,
And a grey mist on the sea’s face and a grey dawn breaking.”
During the first half of the nineteenth century, an illustrated nautical book appeared in France26 containing advice to mariners on how to deal with a host of dangerous situations encountered at sea. Some involved coping with adverse weather conditions and natural hazards, whilst others dealt with close encounters with other vessels. The Dutch physicist Sipko Boersma noticed that this handbook contained a peculiar warning to sailors of something that is reminiscent of the Casimir effect that we have just described.27
Sailors are warned that when there is no wind and a strong swell building, then two large sailing ships will start to roll. If they come close together and lie parallel to one another then they are at risk. An attractive force (‘une certaine force attractive’) will draw the two ships together and there will be a disaster if their riggings collide and become entangled. The sailors are advised to put a small boat in the water and tow one of the ships out of the range of the attractive influence of the other. This sounds like a strange warning. Is there any truth to it? Remarkably, it turns out that there is. The attractive force between the ships arises in an analogous way to the force of attraction between the Casimir plates although there is no quantum physics or zero-point fluctuations of the vacuum involved – ships are too large for those effects to be big enough to worry about. Instead of waves of zero-point energy, the ships feel the pressure of the water waves.
The analogy is quite clear. Although we were dealing with radiation pressure between Casimir’s plates, the same ideas apply to other waves as well, including water waves. In Figure 7.8, we see the situation of two ships, oscillating from side to side in the swell. The rolling ship absorbs energy from the waves and then re-emits this by creating a train of outgoing water waves. If the principal wavelength of these waves is much bigger than the distance between the two ships then they will rock together in time like a pair of copy-cat dancers. However, the waves that they radiate towards each other will be exactly out of phase. The peaks of one ship’s waves will coincide with the troughs of the other ship’s waves. The net result is that they will cancel each other out. As a result, there is virtually no radiated waterwave energy in between the two ships, and the pushing together of the ships, caused by the outgoing waves from the other sides of the ships, is not balanced. Thus, rolling ships will approach one another, just like atoms in a sea of vacuum fluctuations.
Figure 7.8 Two nearby ships rolling in a swell of ocean waves. Some waves are excluded from the region between the ships and the ships are forced together by the higher wave pressure on their outer sides.
The calculations28 show that two 700-ton clipper ships should attract one another with a force equal to the weight of a 2000-kilogram mass. This is a reasonable answer. It is a force that a large boat of rowers could overcome by concerted effort. If the force were ten times bigger then all such efforts would be hopeless, whereas if it were ten times smaller the attraction would be negligible and no action would be needed to avert a collision. Boersma also discovered that the attractive force between the boats is proportional to the square of the maximum angle that they swing back and forth in the swell. In breezy conditions these oscillations will die out fairly quickly as the sails take up their energy. Thus we see the reason for the warning about the naufragous effects of coming too close to another ship in fairly calm conditions.
THE LAMB SHIFT
“I used to be Snow White … but I drifted.”
One of the greatest successes of the quantum theory was to explain in exquisite detail the structure of atoms and the characteristic frequencies of the light waves that they emit when their electrons change from one quantum energy level to another. The first calculations of these levels got very accurate answers, in line with all observations, without realising that the vacuum energy might have an effect on the levels. Fortunately, the effect is very small and requires very sensitive measurements to detect it. It was not until 1947 that instruments were sensitive enough to detect these tiny changes. The electrons near the atomic nucleus feel tiny fluctuations created by the zero-point motions around them. These slight jigglings should result in a slight change in the path of the electron’s orbit and a tiny shift of the energy level of the electron compared to its expected value if we ignore the vacuum fluctuations. In particular, in the hydrogen atom, two energy levels which would otherwise have the same level are split by a tiny amount, four millionths of an electron volt – more than three million times smaller than the energy needed to remove an electron from the atom. This tiny energy difference, now called the ‘Lamb Shift’, was first measured by the Americans Willis Lamb and Robert Retherford,30 in 1947, using some of the techniques developed for the use of radar during the Second World War. Lamb received the Nobel prize for physics for this discovery in 1955.
FORCES OF THE WORLD UNITE
“God is in the atoms… A superposition, if you like. Or whether you don’t like actually, that’s what it’s called. A superposition is like God in that the quantum object occupying a number of different states simultaneously can be everywhere at once. A superposition is a kind of immanence. Without these superpositions, quantum objects would simply crash into each other and solid matter could not possibly exist.”
The quantum vacuum with its seething mass of activity has ultimately proved to be the foundation of all our detailed understanding of the most elementary particles of matter. We have found only four distinct forces of Nature acting in the relatively low-energy world in which we live. Their properties are summarised in Figure 7.9. The action of each of these forces is sufficient to understand almost all the things that we see around us.32 The quartet of forces includes gravity and electromagnetism, which are both familiar to us in everyday life, but they are joined by two microscopic forces which have only been explicitly isolated during the twentieth century. The ‘weak’ force lies at the root of radioactivity whilst the ‘strong’ force is responsible for nuclear reactions and the binding together of atomic nuclei. Each of these forces is described by the exchange of a ‘carrier’ particle which conveys the force. The quantum wavelength of this particle determines the range of influence of the force. The force of gravity is carried by the exchange of a massless particle, the graviton, and so has an infinite range.33 Gravity is unique in that it acts on every particle. The force of electromagnetism also has an infinite range because it is carried by the exchange of another massless particle, the photon of light. It acts on every particle that possesses electric charge. The weak interaction is different. It acts upon a class of elementary particles called leptons (Greek for light ones), like electrons, muons, tauons, and their associated neutrinos, and is carried by three very massive particles, the so-called intermediate vector bosons (W+, W− and Z0). These particles are about 90 times heavier than the proton and the weak force they mediate has a finite range 100 times less than the radius of an atomic nucleus.
Figure 7.9 The four known fundamental forces of Nature.
The strong force is more complicated. Originally, it was regarded as acting between particles like protons which undergo nuclear reactions. However, experiments in which these particles were collided at high energies revealed that they did not behave as if they were elementary indivisible pointlike particles at all. Rather, the proton deflected incoming particles as if it contained three internal pointlike scatterers. These internal constituents are known as quarks and they possess an analogue of electric charge that is called colour charge. This has nothing to do with the usual meaning of colour, as a hue determined by the wavelength of light absorbed when we observe it. It is just a particular attribute (like electric charge) which is conserved in all the processes that we have ever observed. The strong force acts on every particle that carries the colour charge and for this reason is sometimes called the ‘colour force’. The colour force is mediated by the exchange of particles called gluons which have masses about 90 times less than the W and Z bosons and so the strong force has a range about 90 times greater. It is equal to the size of the largest atomic nucleus, a reflection of the fact that it is this force that binds it together.
Quarks possess both colour charge and electric charge. Gluons also possess colour charge and are therefore very different to photons. Photons mediate the electromagnetic interactions between electrically charged particles but do not themselves possess that electric charge – you can’t have electromagnetic interactions of photons alone, they need charged particles like electrons to participate as well. Gluons, by contrast, carry the colour charge and mediate interactions between particles that possess colour charge – you could have strong interactions of gluons alone without any quarks. In this respect the gluons are more akin to the gravitons which mediate the gravitational force. Since gravity acts on everything that has mass or energy it also acts on the gravitons which convey it.
The most elementary particles of matter are believed to be the families of identical quarks and leptons listed in Figure 7.10. ‘Elementary’ means that they display no evidence of possessing internal structure or constituents.
The story of how this picture was established and the feats of engineering performed to establish the identities of the particles involved and the roles they play in Nature’s great particle play is one about which whole books have been written.34 Our interest is in a particular chapter of the story which reveals the reality and crucial properties of the quantum vacuum.
Figure 7.10 The three known ‘families’ of quarks and leptons. Each pair of quarks is related to a charged lepton (either the electron, muon or tau) and an uncharged neutrino.
This theory appears succinct and appealing. It enables us to explain just about everything that is seen and has enabled a succession of successful predictions to be made. However, there is something unattractively in-complete about it all. Physicists believe deeply in the unity of Nature. A universe that rests upon four fundamental laws governing different populations of particles appears to them like a house divided against itself. The unity of Nature reveals itself in a host of different places and provokes us to show that these forces are not really different. If only we could find the right way of looking at them they would fall into place as different pieces of a single big picture, different parts of just one basic force of Nature from which everything derives. An analogy might be found in the behaviour of water. We see it in three very different forms: liquid water, ice and steam. Their properties are different yet they are all manifestations of a single underlying molecular structure for a combination of two hydrogen atoms and one oxygen atom. Despite appearances there is an underlying unity.
Any attempt to unify the quartet of basic forces seems doomed from the start. They look too different. They act upon different classes of elementary particles and they have very different strengths. The relative strengths are shown in Figure 7.9. We see that gravity is by far the weakest. The gravitational force between two protons is about 1038 times weaker than the electromagnetic force.35 At laboratory energies, the weak force is about a hundred million times weaker than the electromagnetic force and the strong force is ten times stronger than electromagnetism.
The fact that the four separate forces have such different strengths and act upon separate sub-populations of elementary particles is deeply perplexing for anyone seeking a hidden unity behind the scenes that would unite them into a single superforce described by one all-encompassing ‘theory of everything’. How can they be united when they are so different? The answer that has emerged reveals the vacuum to be the key player.
“Thirty spokes share the wheel’s hub
It is the centre hole that makes it useful.
Shape clay into a vessel;
It is the space within that makes it useful.
Cut doors and windows for a room;
It is the holes which make it useful.
Therefore profit comes from what is there;
Usefulness from what is not there.”
We used to think of the strength of a force of Nature like electro-magnetism as a fixed constant of Nature, one of the defining features of the Universe. It could be described by combining the basic unit of electric charge carried by a single electron, the speed of light in a vacuum, and Planck’s constant, h. These can be organised into a combination that possesses no units of mass, length, time or temperature. Thus, it provides us with a universal measure of the strength of electromagnetic forces of Nature irrespective of the units of measurement that we employ for the pieces that go into it (so long as we use the same units for all of them). The value obtained37 by experiments of great accuracy and ingenuity for this pure number, called the fine structure constant and denoted by the Greek letter alpha, is equal to
α = 1/137.035989561…
Usually, it is regarded as being approximately equal to 1/137 and physicists would love to explain why it has the precise numerical value that it does. We say that it is a fundamental constant of Nature. Accordingly, the number 137 is instantly recognised by physicists as significant and I have no doubt that the key codes of the locks on the briefcases of a significant number of physicists around the world involve the number 137. For an example of the type of numerological flights of fancy that this quest can inspire see Figure 7.11.
The fine structure constant tells the strength of the interaction that occurs when we fire two electrons towards each other. They have the same (negative) electric charge and so they repel one another like two magnetic North poles (see Figure 7.12).
Figure 7.11 Some numerological flights of fancy involving the number 137, compiled by Gary Adamson.38
In a world without quantum mechanics this interaction should produce the same degree of deflection regardless of the temperature or energy of the environment. All that counts is the number 1/137. In a nineteenth-century vacuum composed of empty space there would be nothing more to be said.
The quantum vacuum changes all that. Our two electrons are no longer situated in a completely empty space – the Uncertainty Principle forbids us from entertaining any such notion. They are moving in the quantum vacuum and that is far from empty. It is a hive of activity. You recall that the Uncertainty Principle reveals that there are complementary pairs of properties that we cannot measure at once with unlimited precision. The energy and lifetime of a particle or a collection of particles is one of these so-called ‘complementary’ pairs. If you want to know everything about the energy of a particle you have to sacrifice all knowledge about its lifetime. Heisenberg’s Uncertainty Principle tells us that the product of these uncertainties always exceeds Planck’s constant divide by twice the number pi:
(uncertainty in energy) × (uncertainty in lifetime) > h/2π (*).
Figure 7.12 Two electrons deflecting in a world with an empty ‘classical’ vacuum.
Any observed particle or physical state must obey this inequality. Observability requires that it be satisfied.
The quantum vacuum can be viewed as a sea composed of all the elementary particles and their antiparticles continually appearing and disappearing. For example, let us focus attention upon the electromagnetic interactions only for the moment. There will be a ferment of electrons and positrons.39 Pairs of electrons and positrons will appear out of the quantum vacuum and then quickly annihilate each other and disappear. If the electron and the positron each have mass m, then Einstein’s famous formula (E = mc2) tells us their ‘creation’ requires an energy equal to 2mc2 to be borrowed from the vacuum. If the time they exist before annihilating back into the vacuum is so short that the Uncertainty Principle (*) is not obeyed, so
(uncertainty in energy) × (uncertainty in lifetime) <h/2π, (**)
then these electron-positron pairs will be unobservable. Hence, they are called virtual pairs. If they live long enough for (*) to be satisfied before they annihilate each other and disappear, then they will become observable and are called real pairs. The creation of virtual pairs seems like a violation of the conservation of energy. Nature allows you to violate this principle so long as no one can see you doing it and this is guaranteed so long as you repay the energy quickly enough. It is useful to think of the virtual condition (**) rather like an ‘energy-loan’ arrangement. The more energy you borrow from the energy bank the quicker you have to pay it back before it is noticed.
The upshot of this is that we can think of our quantum vacuum as containing a collection of continually appearing and disappearing virtual pairs of electrons and positrons. This sounds a little mystical, for if they are unobservable why not just ignore them and opt for a simpler life? But let us reintroduce our two electrons that are all set to interact. Their presence creates an important change in the quantum vacuum. Opposite electric charges attract and so if we put an electron down in the vacuum of virtual pairs the positively charged virtual positrons will be drawn towards the electron, as shown in Figure 7.13(a).
The electron has created a segregation of the virtual pairs and the electron finds itself surrounded by a cloud of positive charges. This process is called vacuum polarisation. Its effect is to create a positively charged shield around the bare negative charge of the electron. An approaching electron will not feel the full negative electric charge of the electron sitting in the vacuum. Rather, it will feel the weaker effect of the shielded charge and be scattered away more feebly than if the vacuum polarisation was absent.
This effect changes if we alter the energy of the environment and the incoming electron. If it comes in rather slowly, then it will not penetrate very far into the shielding cloud of positive charges and will be deflected weakly. But, if it comes in with a higher energy, it will penetrate further through the shield and feel the effect of more of the full negative electron charge within. It will be deflected more strongly than the less energetic particle. Thus we see that the effective strength of the electromagnetic force of repulsion between the two electrons depends upon the energy at which it takes place, as shown in Figure 7.13(b). As the energy increases so the interaction appears to get stronger. It is a little like covering two hard billiard balls with a soft woolly padding. If the balls collide very gently then they will deflect only slightly because the hard surfaces will not hit and rebound. Only the woolly shields will gently rebound. But if they are made to collide at high speed the shields will have little effect and the balls will rebound very strongly. The trend is clear: as the energy of the environment increases so the stronger does the effectiveelectromagnetic interaction become. As the energy rises, the incoming particle gets a closer ‘look’ at the bare point electron charge beneath the cloud of virtual positrons and is deflected more.
Figure 7.13 (a) An incoming electron (B) with low energy scatters weakly due to the outer shield of virtual positive charges around the central negative charge of the electron (A); (b) an incoming electron with high energy penetrates the cloud of positive virtual charges and feels a strong repulsion from the central negative charge of the second electron.
The same study can be made of the strong interaction that affects particles, like quarks and gluons, which carry the colour charge. The situation is a little more complicated than that of the electromagnetic interaction. When we considered the effects of the repelling charges of virtual electrons and positrons we could ignore the photons mediating their electromagnetic inter-action because they have no electric charge. However, if we put a quark of fixed colour charge down in the vacuum and fire another coloured quark towards it, there are two vacuum polarisation effects to consider. Just as before, there will be a cloud of quark-antiquark pairs which will tend to surround any quark with a screening cloud of opposite colour charge. As with the electrons, the overall effect will be to make the strong interaction effectively stronger at higher energies. However, the presence of the gluons also affects the pattern of colour charge. Virtual gluons have the opposite effect and tend to smear out the central colour charge. When scattering occurs from a more extended, less pointlike object it tends to be weaker. The winner between these two opposed tendencies depends on how many varieties of quark there are to pop up in virtual pairs. If the number is as low as the six that we observe in Nature, then it is the gluon smearing that wins out and the strong interaction is predicted to get effectively weaker as we go to higher energies.
This property, called ‘asymptotic freedom’ because it implies that if one continues to extrapolate to indefinitely increasing energies there would be no apparent interaction at all – the particles would be free – was predicted in 1973 and was quite unexpected. It is now confirmed by observations of the interaction strength at different energies. It revolutionised the study of elementary particles and high-energy physics and opened the door to making serious studies of the first moments of the expanding Universe when temperatures would have been high enough for these effects to be very significant. Before 1973, there had been widespread belief that the strong interaction was going to be hopelessly complicated and there was not much chance of understanding interactions at very high energies. It was assumed that they got stronger and stronger at higher and higher energies and so became increasingly intractable. Asymptotic freedom meant that in many ways things got simpler and simpler and it was possible to make real progress.
These important effects of the quantum vacuum enable us to see how the puzzling obstacle to unification of the forces of Nature posed by their different apparent strengths might be overcome. The force strengths do indeed differ significantly in the low-energy world where life like ours is possible, but if we follow the changes expected in those forces as we go to higher and higher energies, they can become closer and closer in strength until a particular energy is reached where the strengths are the same (Figure 7.14). Unification exists only in the ultra-high-energy environment that would have existed in the early stages of the Universe. Today, things have cooled off, and we are left searching for the remnants of a symmetrical past, disguised by billions of years of history. At the energies of our life-supporting environment these forces look very different and the unity of the forces of Nature is hidden. The deep symmetry of the forces that should be found at high energies is possible only because of the contributions of the quantum vacuum. This sea of virtual particles is really there. Its effects can be observed, as predicted, by the change in strength of natural forces as energies increase. The vacuum is far from empty. Nor is it inert. Its presence can be felt and measured in the elementary-particle world, and without its powerful contribution, the unity of Nature could not be sustained.
Figure 7.14 Asymptotic freedom. The weakening of the strong force between quarks as the energy of interaction increases predicts that it can have the same strength as the electromagnetic force at very high temperatures.
“Confinement to the Black Hole … to be reserved for cases of Drunkenness, Riot, Violence, or Insolence to Superiors.”
British Army Regulation, 1844
One of the most recurrently fascinating concepts in the whole of science has proved to be that of the ‘black hole’.40 This cosmic cookie monster, relentlessly devouring everything that strays too close, has captured the popular imagination like no other scientific concept, starred in Hollywood movies, and inspired a host of science-fiction stories. Black holes are regions where the gravitational field of matter is too strong for anything, even light, to escape from its grasp. In Einstein’s picture of curved space, the concentration of mass within a small region can grow so great that the geometry curves dramatically and pinches off the region surrounding it, preventing any signals getting out. The mass concentration is surrounded by a surface of no-return, called the event horizon, through which material and light can flow in but not out.
Despite their popular image, black holes are not necessarily solid objects possessing enormous densities. The huge black holes that seem to be lurking at the centres of many large galaxies have masses about a billion times larger than that of the Sun, but their average density is only about that of air.41 We could be passing through the event horizon of one of these vast black holes at this very moment and nothing would seem strange. No alarms bells would ring as we crossed the horizon and we wouldn’t be torn apart.42 Later on, it would: gradually we would find ourselves drawn inexorably towards conditions of increasing density at the centre. If we ever tried to reverse our path, we would find that there was a definite limit to how far back we could get and none of the signals beamed back to base outside the hole would ever be received.
Black holes are predicted to form whenever a star that is more than about three times the mass of our Sun exhausts its nuclear fuel. It will then cease to have any means of supporting itself against the inward pull of gravity exerted by its constituents. No known force of Nature is strong enough to resist this catastrophic implosion and it will continue to compress the material of the star into a smaller and smaller region until a horizon surface is created. From the inside, the compression just carries on going but from the outside it ceases to be visible. A distant observer looking at the black hole would see light from just outside the horizon becoming redder and redder as it loses energy climbing out of the very strong gravity field.43 The only trace that remains is its gravitational pull.
Evidence has steadily mounted to such an extent that the existence of black holes is regarded as established beyond all reasonable doubt by astronomers. The trick is to catch a black hole in orbit around another luminous star.44The orbit of the visible star will betray the presence of an unseen companion and the star will have material steadily pulled from its outer regions by the companion’s gravitational pull. This material will be heated to millions of degrees Kelvin as it swirls down into the plughole produced by the black hole. At these temperatures there will be a profuse emission of X-rays from the heated material, colliding with other particles, on its inspiralling trajectory. When it nears the horizon surface the wavelength of the flickering of these X-rays will tell us the size of surface they are disappearing into. Black holes have a very particular relationship between their mass and the size of their event horizon. The information obtained from the motion of the visible star and the flickering of the X-rays enables this relationship to be checked. A number of such ‘X-ray binary systems’ are now known and they provide very strong evidence that black holes result when very massive stars end their careers and collapse in upon themselves.
Up until 1975 this picture of black holes was regarded as the full story. Things went into black holes. They never came out. But then the picture changed in a dramatic way. Stephen Hawking45 asked what would happen if a black hole was placed in a quantum vacuum. Remember what we have just seen when the Casimir plates are placed in a quantum vacuum. The sea of vacuum fluctuations of all wavelengths is affected. Now imagine what would happen if a black hole were introduced. If a virtual particle-antiparticle pair appeared very close to the horizon then one of the particles could fall inside the horizon surface while the other stayed outside. The virtual particles would become real; the outgoing particle would be detected by a distant observer and the black hole would appear to be radiating particles from all over its horizon surface.46 This process should be happening continuously and the net result is that all black holes will slowly evaporate away. Black holes are not truly ‘black’ when the quantum vacuum is taken into account. Further investigation revealed that the radiation of vacuum particles followed the laws of black-body thermodynamics originally discovered by Planck. Black holes were black bodies. Sadly, the rate at which particles are expected to be radiated is very slow when black holes are as large as those seen in X-ray binary star systems. In order for Hawking’s radiation process to be visible,47 we would have to encounter black holes which are only about the mass of a large mountain or asteroid. Their horizon size is equal to that of a single proton! These ‘mini’ black holes cannot form today when stars die. But they can be formed in the dense environment of the Big Bang if it is irregular enough. If they were, then these mountain-sized black holes would be in the final stages of evaporation today. The climax of the process will be a dramatic explosion that would show up as a burst of high-energy gamma rays accompanied by radio waves arising from the fast-moving electrons emerging from the explosion at speeds close to that of light. They would radiate 10 gigawatts of gamma-ray power for a period of more than forty billion years and could be seen many light years away. Radio telescopes could see the radio waves from one of these atomic-sized explosions occurring two million light years away in the Andromeda galaxy.
Observers have searched for evidence of black-hole explosions but none has yet been found. All we can say is that if exploding mini black holes do exist then they are few and far between, with no more than one occurring per year in every sphere of space, one light year in diameter.
The Hawking radiation process is of great significance for our understanding of the way in which the great laws underlying Nature are interwoven. It is a unique example of a process which is both relativistic, quantum, gravitational, and thermodynamic. Again, we see that its existence is a direct consequence of the reality of the vacuum and the sea of fluctuations within it. The steep gradient in the gravitational force field near the horizon of the black hole pulls the virtual pairs apart and prevents them annihilating back into the vacuum. They become real particles at the expense of the energy of the gravitational field of the black hole.48
In this chapter we have seen the vacuum move to centre stage in our story. Its existence and universality turn out to underlie the workings of all the forces of Nature. It influences the strengths of the electromagnetic, weak and strong forces of Nature, and links the force of gravity to the quantum character of energy. Each of these influences provides us with observational evidence for the reality of the quantum vacuum and the fluctuations that support it. These successes have flowed from a new conception of the vacuum that gives up the ancient picture of the vacuum as completely empty space. In its place is the more modest view that the vacuum is what is left when everything is removed from space that can be removed. What is left is the lowest energy state available. Remarkably, this means that the vacuum might change, steadily or suddenly. If it does then it can alter the complexion of the entire Universe. In the next chapter we see how.