In physics, a virtual particle is a particle which exists for such a short time and space that its energy and momentum do not have to obey the usual relationship. Virtual particles exhibit some of the phenomena that real particles do such as obedience to the conservation laws. A single virtual particle cannot be detected because that would result in it becoming real. The concept of virtual particles necessarily arises in the perturbation theory of quantum field theory where interactions between real particles are described in terms of exchanges of virtual particles. Any process involving virtual particles admits a schematic representation known as a Feynman diagram which facilitates understanding of calculations.

A virtual particle is one that does not obey the ${\displaystyle m^{2}c^{4}=E^{2}-p^{2}c^{2))$ relationship. In other words, their kinetic energy may not have the usual relationship to velocity, indeed, it can be negative. The probability amplitude for them to exist tends to be canceled out by destructive interference over longer distances and times. They can be considered a manifestation of quantum tunneling.

The term is rather loose and vaguely defined, in the sense that it clings to a rather incorrect view that the world is somehow made up of "real particles": it is not; rather, "real particles" are more accurately understood to be excitations of the underlying quantum fields. As such, virtual particles are also excitations of the underlying fields, but are "temporary" in the sense that they appear in calculations, but never as indexes to the scattering matrix (i.e., they never appear as the observable inputs and outputs of the physical process being modeled). In this sense, virtual particles are an artifact of perturbation theory, and do not appear in a nonperturbative treatment. As such, their existence is questionable; however, the term is useful in informal, casual conversation, or in rendering concepts into layman's terms.

There are two principal ways in which the notion of virtual particles appear in modern physics. They appear as intermediate terms in Feynman diagrams; that is, as terms in a perturbative calculation. They also appear as an infinite set of states to be summed or integrated over in the calculation of a semi-non-perturbative effect. In the latter case, it is sometimes said that virtual particles cause the effect, or that the effect occurs because of the existence of virtual particles.

There are plenty of phenomena resulting from interactions involving virtual particles:

• The Coulomb force between electric charges. It is caused by exchange of virtual photons. In symmetric 3-dimensional space this exchange results in inverse square law for force.
• The strong nuclear force between quarks - it is the result of interaction of virtual gluons. The residual of this force outside of quark triplets (neutron and proton) holds neutrons and protons together in nuclei.
• The weak nuclear force - it is the result of exchange by virtual W bosons.
• The spontaneous emission of a photon during the decay of an excited atom or excited nucleus; such a decay is prohibited by ordinary quantum mechanics and requires the quantization of the electromagnetic field for its explanation.
• The Casimir effect, where the ground state of the quantized electromagnetic field causes attraction between a pair of electrically neutral metal plates.
• The van der Waals force, which is partly due to the Casimir effect between two atoms,
• Vacuum polarization, which involves pair production or the decay of the vacuum, which is the spontaneous production of particle-antiparticle pairs (such as electron-positron).
• Lamb shift of positions of atomic levels.
• Hawking radiation, where the gravitational field is so strong that it causes the spontaneous production of photon pairs (with black body energy distribution) and even of particle pairs.

Most of these have analogous effects in solid-state physics; indeed, one can often gain a better intuitive understanding by examining these cases. In semiconductors, the roles of electrons, positrons and photons in field theory are replaced by electrons in the conduction band, holes in the valence band, and phonons or vibrations of the crystal lattice.

Antiparticles have been proven to exist and should not be confused with virtual particles or virtual antiparticles.

Virtual particles in Feynman diagrams

The calculation of scattering amplitudes in theoretical particle physics requires the use of some rather large and complicated integrals over a large number of variables. These integrals do, however, have a regular structure, and may be represented as Feynman diagrams. The appeal of the Feynman diagrams is strong, as it allows for a simple visual presentation of what would otherwise be a rather arcane and abstract formula. In particular, part of the appeal is that the outgoing legs of a Feynman diagram can be associated with real, on-shell particles. Thus, it is natural to associate the other lines in the diagram with particles as well, called the "virtual particles". Mathematically, they correspond to the propagators appearing in the diagram.

In the image above and to the right, the solid lines correspond to real particles (of momentum ${\displaystyle p_{1))$ and so on), while the dotted line corresponds to a virtual particle carrying momentum k. For example, if the solid lines were to correspond to electrons interacting by means of the electromagnetic interaction, the dotted line would correspond to the exchange of a virtual photon. In the case of interacting nucleons, the dotted line would be a virtual pion. In the case of quarks interacting by means of the strong force, the dotted line would be a virtual gluon, and so on.

It is sometimes said that all photons are virtual photons. This is because the world-lines of photons always resemble the dotted line in the above Feynman diagram: the photon was emitted somewhere (say, a distant star), and then is absorbed somewhere else (say a photoreceptor cell in the eyeball). Furthermore, in the photon's frame of reference, no time elapses between emission and absorption. This statement illustrates the difficulty of trying to distinguish between "real" and "virtual" particles as mathematically they are the same objects and it is only our definition of "reality" which is weak here. In practice, a clear distinction can be made: real photons are detected as individual particles in particle detectors, whereas virtual photons are not directly detected; only their average or side-effects may be noticed.

Virtual particles need not be mesons or bosons, as in the example above; they may also be fermions. However, in order to preserve quantum numbers, most simple diagrams involving fermion exchange are prohibited. The image to the right shows an allowed diagram, a one-loop diagram. The solid lines correspond to a fermion propagator, the wavy lines to bosons.

Virtual particles in the vacuum

Formally, a particle is considered to be an eigenstate of the particle number operator ${\displaystyle a^{\dagger }a}$ where ${\displaystyle \!a}$ is the particle annihilation operator and ${\displaystyle a^{\dagger ))$ the particle creation operator (sometimes collectively called ladder operators). In many cases, the particle number operator does not commute with the Hamiltonian for the system. This implies the number of particles in an area of space is not a well-defined quantity, but like other quantum observables is represented by a probability distribution. Since these particles do not have a permanent existence, they are called virtual particles or vacuum fluctuations of vacuum energy. In a certain sense, they can be understood to be a manifestation of the time-energy uncertainty principle in the vacuum.

An important example of the "presence" of virtual particles in the vacuum is the Casimir effect. Here, the explanation of the effect requires that the total energy of all of the virtual particles in the vacuum be added together. Thus, although the virtual particles themselves are not directly observable in the laboratory, they do leave an observable effect: their zero-point energy results in forces acting on suitably arranged metal plates or dielectrics.

Pair production

In order to conserve the total fermion number of the universe, a fermion cannot be created without also creating its antiparticle; thus many physical processes lead to pair creation. The need for the normal ordering of particle fields in the vacuum can be interpreted by the idea that a pair of virtual particles may briefly "pop into existence", and then annihilate each other a short while later.

Thus, virtual particles are often popularly described as coming in pairs, a particle and antiparticle, which can be of any kind. These pairs exist for an extremely short time, and mutually annihilate in short order. In some cases, however, it is possible to boost the pair apart using external energy so that they avoid annihilation and become real particles.

This may occur in one of two ways. In an accelerating frame of reference, the virtual particles may appear to be real to the accelerating observer; this is known as the Unruh effect. In short, the vacuum of a stationary frame appears, to the accelerated observer, to be a warm gas of real particles in thermodynamic equilibrium. The Unruh effect is a toy model for understanding Hawking radiation, the process by which black holes evaporate.

Another example is pair production in very strong electric fields, sometimes called vacuum decay. If, for example, a pair of atomic nuclei are merged together to very briefly form a nucleus with a charge greater than about 140, (that is, larger than about the inverse of the fine structure constant), the strength of the electric field will be such that it will be energetically favorable to create positron-electron pairs out of the vacuum or Dirac sea, with the electron attracted to the nucleus to annihilate the positive charge. This pair-creation amplitude was first calculated by Julian Schwinger in 1951.

The restriction to particle-antiparticle pairs is actually only necessary if the particles in question carry a conserved quantity, such as electric charge, which is not present in the initial or final state. Otherwise, other situations can arise. For instance, the beta decay of a neutron can happen through the emission of a single virtual, negatively charged W particle that almost immediately decays into a real electron and antineutrino; the neutron turns into a proton when it emits the W particle. The evaporation of a black hole is a process dominated by photons, which are their own antiparticles and are uncharged.

It is sometimes suggested that pair production can be used to explain the origin of matter in the universe. In models of the Big Bang, it is suggested that vacuum fluctuations, or virtual particles, briefly appear. Then, due to effects such as CP-violation, an imbalance between the number of virtual particles and antiparticles is created, leaving a surfeit of particles, thus accounting for the visible matter in the universe.

History

Paul Dirac was the first to propose that empty space (the vacuum) can be visualized as consisting of a sea of virtual electron-positron pairs, known as the Dirac sea. The Dirac sea has a direct analog to the structure of electronic bands in crystalline solids as described in solid state physics. Here, particles correspond to conduction electrons, and antiparticles to holes. A variety of interesting phenomena can be attributed to this structure.

A summary of observational arguments

"Are virtual particles really constantly popping in and out of existence? Or are they merely a mathematical bookkeeping device for quantum mechanics?" Gordon Kane, director of the Michigan Center for Theoretical Physics at the University of Michigan at Ann Arbor, provides this answer at the Scientific American website.