----------------- Virtual particles ----------------- Virtual particles are part of the imagery of quantum field theory. They are figurative language for abstract mathematics, used by experts and laymen as imagery for giving abstract recipes for calculating scattering amplitudes an appearance of intuitive meaning. However, any attempt to take this language literally gives a very misleading and unscientific view of the microscopic world. The virtual particle imagery stems from the 1940s and 1950s when people tried to understand how quantum electrodynamics and its generalizations can make sense. For the experts of today, the term is fully exchangable with ''internal lines in a Feynman diagram'', without any intended meaning beyond that. The collection of Feynman diagrams without loops describes _exactly_ the scattering of classical fields in a perturbation theoretic treatment; the diagrams with k loops describe quantum corrections of order O(hbar^k). If virtual particles had a meaning, then they would already exist in classical field theory, since tree diagrams have internal lines. But nobody ever claimed that predictions of classical field theories are caused by virtual particles. Popularizations often take the imagery for real since on the surface it seems far more understandable that the formal stuff. But these popularization have to pay for it (and get paid for it by the public, in terms of sold copies of their books) by having to ascribe to the virtual particles very strange properties far from both ordinary experience and measurable facts - solely to be able to maintain a pseudo-realistic view of virtual particles. Nowhere in science is such a way of proceeding deemed respectable. Indeed, quantum mechanics is much more rational and intelligible if one avoids such spurious imagery. So it is best to unlearn it as soon as possible. Physicists talk about virtual particles as illustrative language for internal lines in so-called Feynman diagrams. A Feynman diagram is a mnemonic graphical representation of a multiple integral contributing to a scattering amplitude in a collision process between real (i.e., measurable) particles. The collision process is characterized by ingoing and outgoing real particles, represented in the Feynman diagram as external lines, which suggests an interpretation of the remaining, internal, lines as sort of short living intermediated products of the collision, made up of virtual (i.e., nonmeasurable) particles. While this gives the Feynman diagrams an intuitive interpretation, it is impossible to give this intuition a deeper foundation in terms of processes happening in space and time. The attempt to do so leads to a phantastic view of the microscopic world. In this phantastic world view, all sorts of unobservable, nonphysical behavior (e.g., imaginary mass, violation of the conservation of energy, violation of causality, traveling faster than light or backwards in time, popping in and out of the vacuum via ''vacuum fluctuations'', etc.) must be postulated in order to reconcile the mathematical properties of virtual particles with their alleged existence in space and time. None of these speculative aspects can be verified by experiment, which places them outside the realm of science and into the realm of fiction. Since virtual particles are unobservable, one can attribute to them whatever properties one likes, without any real consequence or testability. This explains the phantastic aura surrounding virtual particles, and it also explains their name - they are called virtual since they are not real in any strong sense of the word. If one clearly distinguishes between reality and virtual reality, one finds that the physics of the former is much more rational than that of the latter, where everything goes, and where (as the Wikipedia article on virtual particles shows) inconsistent statements stand undisputed side by side. On the other hand, those accustomed to the view that virtual particles are ''really there'' have later a difficult time unlearning it when they want to get real understanding and want to work with the concepts. Below the surface talk, nothing but internal lines in diagrams is associated with the concept. No states, no positions, no motion (forward or backward in time), no times, no creation or annihilation - nothing. People are sometimes invoking Heisenberg's uncertainty relation that allegedly allows the violation of conservation of energy for a very short time, thus apparently making room for seemingly nonphysical processes. However, the uncertainty relation is based on the existence of operators satisfying the canonical commutation rule, and while there are such operators for spatial position and spatial momentum, there are no such operators for time and energy, or for 4-position and 4-momentum. Indeed, there is no time operator in either quantum mechanics or quantum field theory, and since the energy operator (the Hamiltonian) of a physical system is always bounded below, it cannot be part of a pair of operators satisfying the canonical commutation rule. Therefore the time-energy uncertainty relation is without a formal basis. What can be verified with high accuracy are physical effects derivable from the scattering theory of the particles, i.e., from the fully summed and renormalized perturbative calculations involving an evaluation of the multiple integrals represented by the Feynman diagrams. Plenty of experiments establish without doubt the correctness of the scattering theory and the phenomena predicted by it, such as Coulomb scattering and the Casimir effect. But (in spite of frequent claims in the popular physics literature and sources from the internet) none of these experiments verify anything of the unobservable phantastic scenarios frequently associated with virtual particles. The claims simply rest on taking the successes of perturbation theory with its Feynman diagrams as proof of the validity of the virtual particle picture. But these successes are based on the multiple integral interpretation of the Feynman diagrams, not on their virtual particle interpretation. No evidence at all exists that the latter has any roots in space and time. There is plenty of evidence that sums of Feynman diagrams, interpreted as renormalized multidimensional integrals, correctly predict many phenomena. But to interpret this as evidence for the existence of virtual particles manifesting themselves in space and time is stretching the interpretation too far -- something perhaps acceptable at the layman's level to provide some sort of intuition for otherwise too abstract things (which is what one can find in popularizing accounts by some well-known physicists), but unacceptable on a more scientific level. Indeed, there are strong arguments that loudly speak against assigning reality to virtual particles. If virtual particles were real, they would leave their trace in all methods of predicting certain phenomena, and they would assign the same properties to the virtual particles no matter which approximation method is used. However, the literature readily shows that the details of Feynman diagrams strongly depend on the perturbation scheme used: In light front calculations, one gets a completely different set of diagrams than in the more traditional covariant form. And in nonperturbative approaches such as lattice gauge theory or conformal field theory, the predictions do not involve virtual particles at all. How can anything be real if its existence depends on a particular way of viewing the world? How can an experiment (verifying the Casimir effect, say) can be said to prove the existence of virtual particles if the same experiment can be explained by a method of calculation not involving virtual particles at all? The nonexistence of virtual particles in nonperturbative calculations (whether conformal field theory or lattice gauge theory) is proof that the virtual particle concept is an artifact of perturbation theory. Something whose existence depends on the method of calculation cannot exist in a strong sense of the word. Thus from the scientific point of view, the concept of virtual particles is quite shallow. The latter is not the level of science but the level on which science can be presented to laymen. For them, physicists put their intuition (which often is quite imprecise) into ordinary language since it is intended only to give a rough orientation of what happens. But when they do real science, they ignore the superficial imagery and work on the level of formulas - not virtual particles. The huge differences in the answers given by experts to questions such as ''Are virtual particles really there?''only shows how vague these questions are: Each physicist understands them in a different way, because they have a different conception of the meaing of 'real'. See the thread on the Physics Forum at http://www.physicsforums.com/showthread.php?t=75307 where a number of well-known physicists are cited with widely differing answers related to this question. Therefore virtual particles ''exist'' as lines on paper, as intuition in people's minds, as superficial but catchy allusions to images that make abstract things concrete, but not as tangible, verifiable entities. On the level of physics, virtual particles are quite similar to what ghosts are on the level of ordinary experience. One cannot ascribe to them most properties that real things have. One can only ascribe to them the properties of internal lines in diagrams and multidimensional integrals in perturbative computations. Once one attempts to ascribe to them more, one gets nonsense. A real particle is generally taken as an elementary system (described by an irreducible representation of the Poincare group) separated well enough from the environment to be tractable with creation and annihilation operators (e.g., as an in or out state in scattering). This separation makes it distinguishable enough from the environment to merit the designation ''particle''. Note that it is only an approximate concept, but a very useful one. When the separation gets poorer (as during scattering or in a many-body context), the notion of a real particle becomes less and less useful. In particular, in the solid state, one has no longer identifiable particles but only so-called quasi-particles. Again their characteristics is that they are described by (effective) creation and annihilation operators. On the other hand, there are no creation and annihilation operators for virtual particles, not even in theory. This makes them unreal - they cannot be created or annihilated, not even in theory. They can only be used to write down Feynman diagrams! That calculations of perturbative effects involve integrals corresponding to internal lines of Feynman diagrams (which may be interpreted loosely as virtual particles) doesn't make these virtual particle real. (Nowhere in physics is reality ascribed to diagrams related to mathematical techniques that help one evaluate the terms of a series.) One cannot write down a state vector containing a virtual particle - a physical Hilbert space does not contain such states. But one can easily write down state vectors for the usual, real objects, such as quarks, nuclei, electrons, or photons. What is left of the virtual particle concept when stripped of the phantastic superstition surrounding it? On the scientific level, the formal definition of a virtual particle is as an object associated with an internal line of a Feynman diagram. As such, a virtual particle corresponds to a particular representation of the Poincare group, hence has a definite value of rest mass, spin or helicity, and maybe other quantum number such as charge or color. Associated with the internal line is a 4-dimensional momentum variable p over which an integration is performed. The fact that we have an internal line means that the virtual particle is ''exchanged'' between other (virtual or real) particles; because of the effective force resulting in the scattering, one says that the virtual particles ''mediate'' the force between the real particles in volved in the scattering process. The mommentum is not specified but takes all possible values consistent with the boundary conditions imposed by the scattering process. Energy-momentum conservation is part of the formal framework of Feynman diagrams; it allows one in the simplest (H-like) exchange diagram between two real particles to relate the possible momenta of the virtual particle to the measurable ingoing and outgoing momenta. If the ingoing momenta are p and p' then the outgoing momenta are p+q and p'-q, where q is the momentum exchanged, i..e, the momentum transported by the virtual particle. In particular, one can determine q from measurements. That's all; this makes up virtual particles and their alleged influence on real (observable) particles. Everything else is superstition. And this much is needed to relate virtual particles to QFT: without the interpretation as internal lines of Feynman diagrams, QFT would be completely silent about virtual particles, and none of the successes of QFT could be interpreted as evidence for virtual particles. On the surface, this virtual particle picture is attractive and has an appearance of explanatory power. This is the reason why it is frequently used as an illustrative tool for laymen and when introducing the abstract formalism of Feynman diagrams. But the limitations of the virtual particle picture become immediate when one realizes that the H-like Feynman diagram only gives the lowest-order contribution to the scattering amplitude. Higher-order contributions involve more complex diagrams with the same external lines but arbitrary internal structure involving loops, constrained only by the general principles of conservation of charges. Thus, depending on the Feynman diagram considered, the two particles exchange one or more virtual particles, which again may exchange one or more virtual particles, etc.. Unlike in the simple case of the H-like diagram, the exchange momenta in the more complex diagrams are no longer determined by the external momenta; so there are infinitely many different situations that are possible. Which of these possible exchanges actually happened? It is impossible to answer to this, since the exchange is unobservable, ficticious. A common intuitive response is to argue that in quantum theory, things happen simultaneously in a superposition of all the possibilities. But unlike in the simple nonrelativistic quantum systems used to illustrate the superposition principle, it is impossible in relativistic theories such as QED to calculate probabilities for the possible exchange momenta defining the virtual particles. The reason is that the Feynman integrals containing loops all diverge! Thus there is not even a formal support for assigning probabilities of virtual particles exisitng in a specified number and with specified momenta. Of course, there are renormalization prescriptions for rendering the scattering amplitudes finite, but these prescriptions don't render the contributions of individual diagrams finite but only the total sum with a given number of loops. Another common response is to fall back on Feynman's path integral, according to which the scattering amplitude of a single particle in an external potential is represented as an integral over all posssible paths of the particle. This suggests that the scattering amplitude of a collision process is represented as an integral over all posssible histories of the input, allowing for pair creation and destruction. The histories appearing in this way look like Feynman diagrams, except that they live in space and time. But in the path integral, all histories appear, whether or not they satisfy energy-momentum conservation, while they can be interpreted as Feynman diagrams only in case of energy-momentum conservation. Thus the correspondence is only apparent. Moreover, the path integrals actually used in quantum field theory are always integrals over histories of a quantum field, not of a collection of particles. The virtual particle view is dynamically not coherent. There is no theory how the state of a virtual particle changes with time, not even in the simplest situations. Virtual particles make sense only at a very superficial level comparable to a billiard ball view of quantum particles. Both are very inadequate to describe reality. Trying to interpret virtual particles in space and time by giving the quantum field formalism a well-defined Hamiltonian formulation also failed. The Feynman diagram techniques only tell about the behavior of a process starting in the infinite past and proceeding to the infinite future, and says nothing at all about the time in between. Nobody so far succeeded to give a valid definition of QED at finite times. Since virtual particles are defined only in terms of the Feynman diagrams, they describe asymptotic properties of the scattering, not an actual motion (which would be described by some process at finite times). Thus virtual particles don't ''move''. They are ''exchanged'', but it makes no scientific sense sense to talk about their motion, their speed, or about the direction they travel, This is meaningless talk, and asking about such properties is like asking for the speed of a ghost. Thus it seems impossible to place the superficial virtual particle picture on a sound scientific footing. It is a picture valid only if restricted to the superficial level where no detailed inquiries are made. It is like ordinary people using the word ghost to describe a fleeting but fear-provoking experience. It makes sense only as long as you don't ask about their precise properties. But once you start asking how fast a ghost is traveling, things no longer make sense, since the concept of a ghost is not intended to be applied literally.