1. Non-zero Expectation Value

File:Vacuum fluctuations revealed through spontaneous parametric down-conversion.ogv

If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a measurement problem. In this case the vacuum expectation value (VEV) of any field operator vanishes. For quantum field theories in which perturbation theory breaks down at low energies (for example, Quantum chromodynamics or the BCS theory of superconductivity) field operators may have non-vanishing vacuum expectation values called condensates. In the Standard Model, the non-zero vacuum expectation value of the Higgs field, arising from spontaneous symmetry breaking, is the mechanism by which the other fields in the theory acquire mass.

2. Energy

In many situations, the vacuum state can be defined to have zero energy, although the actual situation is considerably more subtle. The vacuum state is associated with a zero-point energy, and this zero-point energy has measurable effects. In the laboratory, it may be detected as the Casimir effect. In physical cosmology, the energy of the cosmological vacuum appears as the cosmological constant. In fact, the energy of a cubic centimeter of empty space has been calculated figuratively to be one trillionth of an erg (or 0.6 eV).^[1] An outstanding requirement imposed on a potential Theory of Everything is that the energy of the quantum vacuum state must explain the physically observed cosmological constant.

3. Symmetry

For a relativistic field theory, the vacuum is Poincaré invariant, which follows from Wightman axioms but can be also proved directly without these axioms.^[2] Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEV's. The VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case the vacuum has less symmetry than the theory allows, and one says that spontaneous symmetry breaking has occurred. See Higgs mechanism, standard model.

4. Non-linear Permittivity

Quantum corrections to Maxwell's equations are expected to result in a tiny nonlinear electric polarization term in the vacuum, resulting in a field-dependent electrical permittivity ε deviating from the nominal value ε₀ of vacuum permittivity.^[3] These theoretical developments are described, for example, in Dittrich and Gies.^[4] The theory of quantum electrodynamics predicts that the QED vacuum should exhibit a slight nonlinearity so that in the presence of a very strong electric field, the permitivity is increased by a tiny amount with respect to ε₀. What's more, and what would be easier to observe (but still very difficult!), is that a strong electric field would modify the effective permeability of free space, becoming anisotropic with a value slightly below μ₀ in the direction of the electric field and slightly exceeding μ₀ in the perpendicular direction, thereby exhibiting birefringence for an electromagnetic wave travelling in a direction other than that of the electric field. The effect is similar to the Kerr effect but without matter being present.^[5] This tiny nonlinearity can be interpreted in terms of virtual pair production^[6] The required electric field is predicted to be enormous, about [math]\displaystyle{ 1.32 \times 10^{18} }[/math]V/m, known as the Schwinger limit; the equivalent Kerr constant has been estimated, being about 10²⁰ times smaller than the Kerr constant of water. Explanations for dichroism from particle physics, outside quantum electrodynamics, also have been proposed.^[7] Experimentally measuring such an effect is very difficult,^[8] and has not yet been successful.

5. Virtual Particles

The presence of virtual particles can be rigorously based upon the non-commutation of the quantized electromagnetic fields. Non-commutation means that although the average values of the fields vanish in a quantum vacuum, their variances do not.^[9] The term "vacuum fluctuations" refers to the variance of the field strength in the minimal energy state,^[10] and is described picturesquely as evidence of "virtual particles".^[11] It is sometimes attempted to provide an intuitive picture of virtual particles, or variances, based upon the Heisenberg energy-time uncertainty principle:

[math]\displaystyle{ \Delta E \Delta t \ge \hbar \ , }[/math]

(with ΔE and Δt being the energy and time variations respectively; ΔE is the accuracy in the measurement of energy and Δt is the time taken in the measurement, and ħ is the Reduced Planck constant) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times.^[12] Although the phenomenon of virtual particles is accepted, this interpretation of the energy-time uncertainty relation is not universal.^[13][14] One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty Δt determines a "budget" for borrowing energy ΔE. Another issue is the meaning of "time" in this relation, because energy and time (unlike position q and momentum p, for example) do not satisfy a canonical commutation relation (such as [q, p] = i ħ).^[15] Various schemes have been advanced to construct an observable that has some kind of time interpretation, and yet does satisfy a canonical commutation relation with energy.^[16][17] The very many approaches to the energy-time uncertainty principle are a long and continuing subject.^[17]

6. Physical Nature of the Quantum Vacuum

According to Astrid Lambrecht (2002): "When one empties out a space of all matter and lowers the temperature to absolute zero, one produces in a Gedankenexperiment [thought experiment] the quantum vacuum state."^[18] According to Fowler & Guggenheim (1939/1965), the third law of thermodynamics may be precisely enunciated as follows:

It is impossible by any procedure, no matter how idealized, to reduce any assembly to the absolute zero in a finite number of operations.^[19] (See also.^[20][21][22])

Photon-photon interaction can occur only through interaction with the vacuum state of some other field, for example through the Dirac electron-positron vacuum field; this is associated with the concept of vacuum polarization.^[23] According to Milonni (1994): "... all quantum fields have zero-point energies and vacuum fluctuations."^[24] This means that there is a component of the quantum vacuum respectively for each component field (considered in the conceptual absence of the other fields), such as the electromagnetic field, the Dirac electron-positron field, and so on. According to Milonni (1994), some of the effects attributed to the vacuum electromagnetic field can have several physical interpretations, some more conventional than others. The Casimir attraction between uncharged conductive plates is often proposed as an example of an effect of the vacuum electromagnetic field. Schwinger, DeRaad, and Milton (1978) are cited by Milonni (1994) as validly, though unconventionally, explaining the Casimir effect with a model in which "the vacuum is regarded as truly a state with all physical properties equal to zero."^[25][26] In this model, the observed phenomena are explained as the effects of the electron motions on the electromagnetic field, called the source field effect. Milonni writes:

The basic idea here will be that the Casimir force may be derived from the source fields alone even in completely conventional QED, ... Milonni provides detailed argument that the measurable physical effects usually attributed to the vacuum electromagnetic field cannot be explained by that field alone, but require in addition a contribution from the self-energy of the electrons, or their radiation reaction. He writes: "The radiation reaction and the vacuum fields are two aspects of the same thing when it comes to physical interpretations of various QED processes including the Lamb shift, van der Waals forces, and Casimir effects.^[27]

This point of view is also stated by Jaffe (2005): "The Casimir force can be calculated without reference to vacuum fluctuations, and like all other observable effects in QED, it vanishes as the fine structure constant, α, goes to zero."^[28]

7. Notations

The vacuum state is written as [math]\displaystyle{ |0\rangle }[/math] or [math]\displaystyle{ |\rangle }[/math]. The vacuum expectation value (see also Expectation value) of any field [math]\displaystyle{ \phi }[/math] should be written as [math]\displaystyle{ \langle0|\phi|0\rangle }[/math].