E-mail : passante@iaif.pa.cnr.it

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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 "

Newton's mechanics does not distinguish among different inertial frames, but it does make a sharp distinction between inertial and non-inertial frames : A

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.

Einstein's theory of relativity was the first physical theory to reject the absolute character of space. His

- all inertial frames are equivalent for

- the speed of light

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

Einstein's

In quantum field theory, a physical system is described in terms of two different mathematical objects:

- the field operators

- the quantum states.

A

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

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

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

A well-known example of an uncertainty relation involves the position

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

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.

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.

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,

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

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.

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.

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

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.

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

(b) In this experiment described in Newton's

(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.

Gompagno G, Passante R, Persico F.

Milonni PW.

Saunders S, Brown HR (eds) The philosophy of vacuum, Clarendon Press, Oxford, 1991.