QED vacuum

Another field-theoretic vacuum is the QCD vacuum of the Standard Model.

Fluctuations

The QED vacuum is subject to fluctuations about a dormant zero average-field condition:

A description of the quantum vacuum is given by Joseph Silk in On the Shores of the Unknown (pg. 62):

The quantum theory asserts that a vacuum, even the most perfect vacuum devoid of any matter, is not really empty. Rather the quantum vacuum can be depicted as a sea of continuously appearing and disappearing [pairs of] particles that manifest themselves in the apparent jostling of particles that is quite distinct from their thermal motions. These particles are ‘virtual’, as opposed to real, particles. ...At any given instant, the vacuum is full of such virtual pairs, which leave their signature behind, by affecting the energy levels of atoms.

Virtual particles

An intuitive picture of virtual particles can be attempted based upon the Heisenberg energy-time uncertainty principle $${\displaystyle \mathrm{\Delta }E\mathrm{\Delta }t\ge \frac{\hslash }{2},}$$ where ΔE and Δt are the uncertainties in energy and time, respectively, with ħ the Planck constant divided by 2π. Using this starting point, it could be argued that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum, thus permitting particle generation over short time intervals.

This interpretation of the energy-time uncertainty relation is not universally accepted, however. 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] = iħ). Various schemes have been advanced to construct an observable that has some kind of time interpretation while also satisfying a canonical commutation relation with energy. The many approaches to the energy-time uncertainty principle are a continuing subject of study.

Quantization of the fields

The Heisenberg uncertainty principle does not allow a particle to exist in a state in which the particle is simultaneously at a fixed location and has also zero momentum. Instead, the particle has a range of momenta, the distribution of which is attributable to quantum fluctuations; if confined, it has a zero-point energy.

An uncertainty principle applies to all quantum mechanical operators that do not commute. In particular, it applies also to the electromagnetic field; in order to understand this, it is necessary to elucidate the role of commutators for this field.

The standard approach to the quantization of the electromagnetic field begins by introducing a vector potential A and a scalar potential V to represent the electric field E and magnetic field B using the relations $${\displaystyle \begin{array}{cc}\mathbf{B}& =\mathrm{\nabla }\times \mathbf{A};\\ \mathbf{E}& =-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{A}-\mathrm{\nabla }V.\end{array}}$$ The vector potential is not completely determined by these relations, leaving open a so-called gauge freedom. Resolving this ambiguity using the Coulomb gauge leads to a description of the electromagnetic fields in the absence of charges in terms of the vector potential and the momentum field Π, given by $${\displaystyle \mathbf{\Pi }={\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{A},}$$ where ε₀ is the electric constant in SI units. Quantization is achieved by insisting that the momentum field and the vector potential do not commute. That is, the equal-time commutator is $${\displaystyle [{\mathrm{\Pi }}_i(\mathbf{r},t),A_j({\mathbf{r}}^{\prime },t)]=-i\hslash {\delta }_{ij}\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),}$$ where r, r′ are spatial locations, ħ is the reduced Planck constant, δ_ij is the Kronecker delta, and δ(r − r′) is the Dirac delta function. The notation [ , ] represents the commutator.

Quantization can be achieved without introducing the vector potential in terms of the underlying fields themselves: $${\displaystyle \left[{\hat{E}}_k(\mathit{r}),{\hat{B}}_{k^{\prime }}({\mathit{r}}^{\prime })\right]=-{\epsilon }_{k{k}^{\prime }m}\frac{i\hslash }{{\epsilon }_0}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_m}\delta (\mathit{r}\mathbf{-}{\mathit{r}}^{\mathbf{\prime }}),}$$ where the circumflex denotes a Schrödinger time-independent field operator, and ε_ijk is the antisymmetric Levi-Civita tensor.

Because of the non-commutation of field variables, the variances of the fields cannot be zero, although their averages are zero. So, the electromagnetic field has a zero-point energy, and therefore a lowest quantum state. The interaction of an excited atom with this lowest quantum state of the electromagnetic field is what leads to spontaneous emission: The transition of an excited atom to a state of lower energy by emission of a photon even when no external perturbation of the atom is present.