Science Tribune - Article - March 1998
Vacuum in physics
Istituto per le Applicazioni Interdisciplinari della Fisica, Consiglio Nazionale delle Ricerche, Via Archirafi 36, I-90123 Palermo, Italy
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The concept of vacuum has played an essential role in our understanding of the physical world since the beginnings of natural philosophy and of science. This article (a) briefly traces the history of the concept of vacuum and then focusses on several of its significant aspects in quantum field theory, in particular in quantum electrodynamics.
Quantum vacuum clearly has dynamic properties that are at variance with those of vacuum in classical physics. Its properties can be perturbed and these perturbations are observable. Moreover, the structure of quantum vacuum can determine the physical properties of matter. But do infinite vacuum field fluctuations really exist ? Are the effects observed (e.g. the Lamb shift, Casimir-Polder forces and the Casimir effect) the result of vacuum fluctuations or are they explained in other ways, for instance by the source-reaction field ?
The article highlights the multifaceted, controversial nature of the various interpretations of the dynamic structure of quantum vacuum and fundamentally asks whether there is a complementarity in the physical origin of all radiative processes. Of course, it cannot provide all the answers but attempts to show how quantum vacuum is a fascinating research subject where there is still much to be learnt.
The notion of space and vacuum in ancient times
In ancient times, vacuum was a synonym of space and often denoted by the same Greek word (tò kevóv) although it was not viewed in the same way by all philosophers. For example, for Parmenides (6th-5th century BC, from Elea in Southern Italy) "vacuum is nothingness, and nothingness does not exist" whereas, for Democritus (5th century BC, from Abdera in Northern Greece), vacuum was something as real as atoms and synonymous with space in the sense of "free space". However, it was Parmenides' pupil, Zeno, who first made a clear distinction between the two concepts - vacuum and space - by attributing the notion of location to space. Such a view was also upheld by Epicurus (341-270 BC, from Samos in Northern Greece) and Lucretius (98-55 BC) for whom the world was made up of physical bodies and of the vacuum in which these bodies are located and through which they move. On the other hand, Aristotle (384-322 BC), unlike Democritus, denied the existence of space because of mistaken views on movement and velocity. He argued that, in vacuo, any object should move with infinite velocity. This being clearly absurd, he concluded that vacuum does not exist.
Vacuum and space as frames of reference for moving objects : The views of Galileo and Newton
The birth of modern science witnessed extensive discussions of the concepts of space and vacuum. Galileo and Newton considered them as frames to be used to describe the motion of matter, a view that led to the famous concept of absolute space. According to Newton, absolute space exists independently of matter and cannot be modified (this is at variance with today's description as we shall see below). It is a universal frame of reference for describing the motion of any object.
A universal frame of reference is essential for the consistency of Newton's mechanics because Newton's laws are frame dependent. The second law of mechanics, for example, involves acceleration, and the reference frame used to calculate acceleration must be specified. Newton's laws are valid only if the acceleration is relative to absolute space or to any other frame moving with a uniform velocity with respect to absolute space (the so-called "inertial frames of reference").
Newton's mechanics does not distinguish among different inertial frames, but it does make a sharp distinction between inertial and non-inertial frames : A non-inertial frame is a frame with non-vanishing acceleration with respect to an inertial frame. Newton's rotating bucket experiment (b) and Foucault's pendulum (c) are experimental manifestations of acceleration relative to absolute space, thereby confering it absolute meaning. However, velocity relative to absolute space cannot be measured by any mechanical experiment; it has no absolute character because all inertial frames are equivalent. For Newton, all these considerations substantiated the "reality" of absolute space.
The 19th century concept of ether
In the 19th century, with Maxwell's theory of electromagnetism, measuring velocities relative to absolute space seemed to be possible. At the time, the concept of ether was very popular. Space was thought to be permeated by a substance, electromagnetic ether, which, according to Maxwell, Hertz and Lorentz, supported all electric and magnetic processes including light propagation very much as air supports sound waves. Moreover, it was considered immobile with respect to Newton's absolute space.
If light and sound waves were indeed similar, some optical or electrodynamical experiment might plausibly reveal a uniform velocity relative to the ether/absolute space. However, unlike for sound, the speed of light was always the same irrespective of the motion of source and observer. In Michelson-Morley's experiment, light propagation was not influenced by the motion of the light source, thus failing to show whether this source does or does not move with respect to the ether/absolute space.
Concepts that do not lead to observable manifestations cannot be incorporated into a physical theory and the concept of ether thus lost its physical meaning.
A dynamic role for vacuum in the 20th century as given by Einstein's theory of relativity
This century, the theories of relativity and quantum mechanics - and their merger into quantum field theory - have radically changed our ideas about vacuum, crediting it with a dynamic role. Vacuum has become an entity whose physical properties can be modified and that can influence the properties of physical bodies.
Einstein's theory of relativity was the first physical theory to reject the absolute character of space. His special theory of relativity has two main postulates :
- all inertial frames are equivalent for all physical phenomena (the principle of relativity);
- the speed of light in vacuo is equal in all inertial frames.
The space separating two points and, therefore, the length of physical objects thus depends on the velocity of the observer. Two observers moving with respect to each other measure different lengths (the so-called Lorentz contraction of distances). Here, the very concept of space separation is relative to the motion of the observer.
Einstein's general theory of relativity extends the principle of relativity to accelerated frames and incorporates a theory of gravitational forces consistent with an upper limit for velocities, thus reinforcing the relative character of space. The geometrical properties of the vacuum space depend on the presence of massive bodies or, correspondingly, on the acceleration of the observer. Gravitational forces are the manifestation of the curvature of space - or rather of space-time - around masses. Matter deforms the properties of vacuum and the deformed space affects the motion of matter. This view is at total variance with Newton's idea of an absolute rigid space.
A dynamic role for vacuum in quantum field theory
Modern quantum field theory even more forcibly endows vacuum with a dynamics of its own. I shall focus on quantum electrodynamics, i.e., the quantum theory of the interaction between matter and electromagnetic fields.
In quantum field theory, a physical system is described in terms of two different mathematical objects:
- the field operators
- the quantum states.
A field operator is related to the general structural properties of the field. It is independent of the particular configuration of the system considered which is described by a quantum state.
Let us consider the interaction between electrons and photons (the photons are the "particles" of the electromagnetic field). In this case, there are electron field operators and photon field operators, whose mathematical expressions are related to the physical properties of electrons and photons, such as their masses and spins. All information on a particular realization of the system (number of particles, their position and energy, etc) is contained in the quantum state which specifies how many photons and electrons are present in any admissible state (for example, with given momentum and polarization).
Let us now consider just the electromagnetic field, assuming for the moment that the charged particles with which it may interact are absent. This is an example of a free field, i.e., a field not interacting with other fields. This system is described by two field operators : the electric and the magnetic field operators.
A generic quantum state for this system is given by listing the numbers of photons present for any possible wavevector and polarization. Its energy is proportional to the space integral of the sum of the squares of the electric and magnetic fields created by the photons. The vacuum state is the state with the minimum possible energy compatible with boundary conditions as those of a cavity. (A cavity imposes conditions on the field operators at its surface). In other words, a vacuum state is a quantum state with no photons. Vacuum is thus defined as the state containing the least possible amount of anything.
But does this imply that it is a state with no electromagnetic fields at all and no energy ? This would indeed be so in classical field theory where nothing prevents all fields and all energy from vanishing to yield a state of nothingness - no field, no particles. But is this also true in quantum physics? The answer is NO. In quantum physics, some activity - in the form of field fluctuations (called vacuum fluctuations) - remains even in a vacuum. These fluctuations - a direct manifestation of the quantum nature of our world - cannot be eliminated. In a way, they are like thermal fluctuations but they survive even at a temperature of zero Kelvin where all thermal activity is frozen.
Why are there quantum field fluctuations in vacuum ?
The key reason is that a configuration where electric and magnetic fields vanish simultaneously violates the basic uncertainty principles of quantum theory. To understand this, we must briefly consider uncertainty relations. One of the postulates of quantum mechanics is that any physical observable is described by a Hermitian operator defined in a Hilbert space (d). Two physical quantities associated with the operators A and B can be measured simultaneously with arbitrary precision only if the commutator AB-BA vanishes. Otherwise, there is an upper limit to the precision with which the two quantities can be measured. Mathematically, the product of their uncertainties must be larger than some nonvanishing quantity, usually proportional to the Planck constant h. Such a mathematical uncertainty relation indicates that the two quantities considered cannot simulatneously possess well-defined values in the system.
A well-known example of an uncertainty relation involves the position x and the momentum p of a particle in one-dimensional motion. The uncertainty is expressed by the inequality Dx Dp > h, where Dx and Dp are the uncertainties in position and momentum, respectively. The more precisely the particle's position is measured, the less precisely its momentum is known, and vice-versa.
The same situation occurs when viewing an electromagnetic field from the standpoint of quantum theory. Because electric and magnetic fields are associated with non-commuting operators, they cannot both simultaneously display well-defined values. In particular, a state where they vanish together cannot exist. Consequently, even in the lowest energy state of a vacuum, the electric and magnetic fields can only be zero on average. At any particular instant, they must have finite values in order to satisfy the uncertainty relations. Electric and magnetic field fluctuations thus cannot be eradicated.
Properties of quantum vacuum fluctuations : 1°. Infinite energy ?
Vacuum fluctuations, also called zero-point fluctuations, as well as any observable effect they have on physical objects, are a manifestation of the quantum nature of the electromagnetic field. Their physical properties are rather peculiar, in particular their infinite energy because we do not expect to find divergent quantities in any meaningful physical theory. When zero-point fluctuations are decomposed in normal modes, their energy can be expressed in the form of an integration over the frequency of the field modes. The divergence, because it arises from the high-frequency part of the integral, is called an ultraviolet divergence.
This divergence in vacuum energy of course brings the concept into question. For example, according to the general theory of relativity, the infinite energy should act as an infinite source of gravitational field, which is, however, contrary to experience. Changing the zero of the energy to eliminate zero-point energy (because energy is always defined up to an additive constant), as has been proposed, is an unsatisfactory solution because vacuum fluctuations produce observable effects, can be changed, and finite changes in the zero-point energy can be observed experimentally.
Properties of quantum vacuum fluctuations : 2°. The Lamb shift
A well-known physical phenomenon that can be interpreted as an effect of vacuum fluctuations is the Lamb shift, i.e., the energy shift of atomic levels due to the interaction of the electrons with the zero-point quantum radiation field.
In a hydrogen atom, for example, the electron orbiting around the nucleus interacts with vacuum fluctuations, resulting in a fluctuating motion that is superimposed on the orbital motion. This determines a change in the electron-nucleus electrostatic interaction, yielding a shift in the atomic energy - the Lamb shift - which can be observed experimentally with very high precision.
Properties of quantum vacuum fluctuations : 3°. Casimir-Polder forces
Casimir-Polder forces - long-range interactions between neutral atoms - are another consequence of vacuum fluctuations. These are very tiny forces but that have been measured. The presence of a first atom, acting as a polarizable body, changes the energy distribution of the vacuum field modes, producing a change in vacuum fluctuations and, in turn, a change in the Lamb shift of the second atom. The first atom thus changes the energy levels of the second atom in a way that depends on their separation, yielding an interatomic potential.
This interatomic potential highlights an unexpected aspect of a quantum electrodynamic vacuum : its physical properties can be changed. The presence of the first atom perturbs the intensity of vacuum fluctuations in all of space and, in particular, there where the second atom is located.
Other properties of vacuum (spatial correlations of the fields, for example) can be similarly "deformed". In other words, quantum electrodynamic vacuum displays dynamic properties.
Do the Lamb shift and Casimir-Polder forces prove that vacuum fluctuations really exist and can be changed ?
Probably NOT, because the same results can be obtained without invoking vacuum fluctuations but by using the so-called radiation reaction field (or source field).
In the case of the Lamb shift, it is possible to show that quantum mechanics predicts the existence of fluctuations of the atom's electric dipole moment, yielding a fluctuating electric field emitted by the atom. This field can then interact with the emitting atom - it is the reason why it is called a reaction field - yielding an energy shift of the atomic energy levels that coincides exactly with that obtained when using the vacuum fluctuations argument. A similar reasoning can be applied to Casimir-Polder forces.
The Lamb shift and Casimir-Polder forces can thus be deduced with equal forcefulness by two completely different methods : as a consequence of vacuum fluctuations or of a radiation reaction field.
Yet again, we are confronted with a deep-rooted example of complementarity in the quantum world where phenomena can be explained, on the one hand, by assuming that vacuum fluctuations really exist and intervene and, on the other, by assuming they play no role.
Properties of quantum vacuum fluctuations : 4°. The Casimir effect
The Casimir effect is a force between two neutral parallel conducting plates placed in a vacuum. There is no classical counterpart to this force which stems from the quantum properties of the electromagnetic field as do the Lamb shift and Casimir-Polder forces. It is usually considered to be a manifestation of vacuum fluctuations although, here again, some caution is necessary.
Let us consider two ideal, perfectly conducting, metallic plates and compare the energies of vacuum fluctuations when the separation between the plates is increased. Only the energy between the plates needs to be taken into account, that in the external unbounded space playing no part here.
The metallic plates limit the number of admissible field modes because the electric field must vanish at their surface and not all free space field modes satisfy this condition. When the distance between the plates changes, the structure of the field modes and the vacuum energy change too. If we calculate the difference between the zero-point energies for the narrow and wide spacing of the plates, we obtain a finite change in vacuum energy which depends on plate separation, yielding a potential energy and a force between the neutral plates. This force can be evaluated explicitly and, in the case of infinite parallel plates, is attractive.
A few recent experiments have measured the Casimir force and its dependence on plate separation, revealing good agreement between experiment and theory. The Casimir force has also been studied for dielectric bodies and for different geometries. In the case of a conducting spherical shell, it is repulsive.
Does the Casimir effect prove that vacuum fluctuations really exist ?
The Casimir effect seems to result from a change in the energy of vacuum fluctuations due to boundary conditions as if the structure of vacuum were "polarized" by the external perturbations represented by these boundary conditions.
However, it is, in principle, possible to obtain the Casimir force without assuming the existence of vacuum fluctuations. For this purpose, let us consider the Casimir effect for two parallel dielectric slabs (the case of metallic plates would be the limiting case of dielectrics with infinite conductivity). There are Casimir-Polder forces among the atoms constituting the two slabs and these forces, as we have seen above, can be obtained by the use of the reaction or source field. Clearly, the sum of the Casimir-Polder forces between the slabs' individual atoms should result in a nonvanishing force between the slabs. Unfortunately, this sum has not been calculated yet. The calculation is highly complicated because these long-range forces are non-additive. In the case of more than two atoms, not only forces between pairs of atoms, but three-, four-, ... body contributions must be considered. For macroscopic objects such as dielectric slabs, the many-body components, usually negligible for few-body systems, may become comparable with the two-body component.
A quantitative comparison of the Casimir force between the dielectric slabs as obtained by simple vacuum fluctuations arguments and by summation of non-additive Casimir-Polder forces between individual atoms is thus very difficult. However, conceptually, the two approaches (vacuum fluctuations and source fields) should give the same result, once the equivalence of the two methods has been proved for interactions between individual atoms.
Properties of quantum vacuum : 5°. Vacuum polarization
I shall now address one more property of vacuum in quantum electrodynamics - vacuum polarization - although there are several others that are at variance with those of vacuum in classical physics.
In relativistic quantum electrodynamics, virtual electron-positron pairs can be spontaneously created in vacuum. (The positron is the electron's antiparticle, same mass but opposite electric charge.) These electron-positron pairs are called virtual because their existence is fleeting. In vacuum, they yield electric charge fluctuations whose physical origin is analogous to the electric and magnetic field fluctuations discussed above.
If an electric charge - let us say positive charge - is placed in the vacuum space, it can polarize vacuum charge fluctuations because the positive charge attracts the electrons of the virtual electron-positron pairs, but repels the positrons. Thus, on average, the barycentre of vacuum negative charges is closer to the charge than is the barycentre of vacuum positive charges. This results in a charge screening - as occurs for the charge in a dielectric - and the "effective" electric charge thus becomes distance-dependent. The shorter the distance, the greater the charge. Most experiments measure electric charge at a large distance, and thus measure the screened "dressed" charge and not the unscreened "bare" charge.
Vacuum polarization shows that the quantum electromagnetic vacuum acts as a polarizable medium in line with modern physics' general view of vacuum as a system endowed with intrinsic dynamics.
Properties of vacuum in quantum chromodynamics
Quantum chromodynamics is the field theory of strong nuclear forces between quarks. The basic elements of chromodynamics are quarks (matter particles) and gluons. Quarks are the constituents of protons and neutrons, as well as of other elementary particles called hadrons. Gluons are the particles of the strong nuclear interaction just as photons are the particles of the electromagnetic field.
Quantum chromodynamics is much more complicated than quantum electrodynamics The most important differences are:
i) There are 8 different kinds of gluons;
ii) Unlike photons which are electrically neutral, gluons carry a nuclear charge and can thus interact directly with each other;
iii) This nuclear charge is not small (unlike the electric charge of elementary particles), thus complicating calculations.
Vacuum fluctuations and vacuum polarization effects exist not only in the framework of quantum electrodynamics but also of quantum chromodynamics. However, the more complicated structure of quantum chromodynamics makes their detailed investigation well-nigh impossible, and we must resort to qualitative and phenomenological models in order to guess the physical properties of vacuum in quantum chromodynamics.
One of the most striking properties of this vacuum is antiscreening of nuclear charges (to be contrasted with screening of electric charges in quantum electrodynamics). Antiscreening means that the effective nuclear charge decreases to zero when the separation between quarks is reduced. Thus, quarks separated by a short distance do not interact via nuclear forces but behave as if they were free. This is called asymptotic freedom. On the other hand, the nuclear charge grows at large distances and prevents the quarks from leaving the hadrons of which they are constituents (j). The quarks are confined inside the hadrons and no isolated quarks can be observed. These effects - asymptotic freedom and confinement - of the quantum chromodynamic vacuum can hardly be visualized in physical terms but are further examples of how the properties of vacuum can determine the basic behaviour of matter. The key factor differentiating the quantum chromodynamic antiscreening vacuum from the quantum electrodynamic screening vacuum is gluon self-interaction.
All the above data concur to show that the concept of vacuum in modern physics is radically different from that in classical physics. Vacuum has dynamic properties, and the properties of vacuum and of matter are linked.
(a) The author wishes to express his thanks to G. Compagno, G. Oliveri, G.M. Palma and F. Persico for helpful discussions on vacuum physics and for reading this article with a critical eye.
(b) In this experiment described in Newton's Principia, a bucket filled with water and hung from a long cord is made to rotate around its axis. At first, the water is at rest and the relative motion between water and bucket is maximum but, little by little, the bucket transmits its motion to the water which begins to rotate as the bucket does, and their relative motion decreases. As the water starts to rotate, the centrifugal force makes it move away from the middle of the bucket and up the sides. According to Newton, this tendency of the water to recede from its axis of rotation is a manifestation of its "absolute" circular motion. The effect is maximum when there is no relative motion between bucket and water. It is not related to the motion of the water relative to nearby objects such as the bucket but to its absolute motion.
(c) In 1851, the physicist L. Foucault made an experiment with a pendulum free to rotate in any direction that showed, for the first time, the (absolute) rotation of the Earth. For simplicity, let's assume that the pendulum is located over the North Pole. As the pendulum oscillates, its oscillation plane is fixed in space while the Earth rotates beneath. An observer on Earth sees that the pendulum plane rotates over 24 hours but without any real cause. This indicates that the Earth is not an inertial frame of reference (i.e. it rotates relative to absolute space) and that fictitious inertial forces must be introduced to restore the validity of Newton's laws in the non-inertial frame.
(d) In quantum mechanics, quantum states are vectors in a Hilbert space. A Hilbert space is a particular kind of vector space with infinite dimensions. A vector space is a set of elements, called vectors, that obey specific rules as regards their addition and multiplication by numbers as do vectors in the ordinary 3-dimensional space. Operators are mathematical objects that, when acting on a state, yield another state in the vector space. Hermitian operators are a class of operators with well-defined mathematical properties.
Aitchinson IJR. Nothing's plenty. The vacuum in modern quantum field theory. Contemporary Physics 26, 331-391, 1985. (This is a very clear non-technical review on vacuum in quantum field theory).
Gompagno G, Passante R, Persico F. Atom-field interactions and dressed atoms, Cambridge University Press, Cambridge, 1995
Milonni PW. The Quantum Vacuum, Academic Press, San Diego, l994
Saunders S, Brown HR (eds) The philosophy of vacuum, Clarendon Press, Oxford, 1991.