On Virtuality

Local quantum field theories were the basis of high-energy physics in the XX century. Any such theory postulated some symmetric and covariant expression, which, according to the principle of minimal action, would describe all the variety of physical effects allowed by the model. In earlier days, this model action was used to derive the equations of motion, and a number of exact solutions have been obtained for simple cases. However, many-body equations could never be solved exactly, and approximations were necessary to obtain practically usable results. Numerous approximate theories were built in all the areas of quantum theory, from atomic physics, to quantum gravity. Some of them invoked phenomenological elements; more "rigorous" theorizing largely exploited the standard technical trick, successive approximations. The perturbation-theory approach splits the full action into two parts, one of them describing a simpler system with already known physical properties, and the other part treated as a small correction to the "zero-order" behavior, so that the motion of the "perturbed" system could be expanded in a power series: in the first order, perturbation couples the elements of the simplified system just once, the second order terms account for the possible double couplings, and so on. As diagrams were invented as a convenient technique for constructing all kinds of perturbation series, equations of motion have gradually come to a deep neglect, while approximate solutions can be immediately obtained using the form of action, without deriving any equations of motion. Physicists went in for computing higher-order diagrams, and nobody much cared for the original strongly coupled theory, without separating zero-order action from perturbation.

There was a price to pay.

Thus, perturbation series utterly refused to converge, producing infinite values in all the orders above the very first. For most fundamental interactions, the perturbation was not small, so that the series could only be considered as asymptotic. Infrared and ultraviolet divergences haunted quantum field theory from the very beginning; many ingenious tricks were invented to overcome this difficulty and obtain finite solutions. Physicists learned to "renormalize" the theory up to any given order, removing all the divergences. Perturbation theory has been complemented with renormalization theory, and the best we can expect is to prove that our theory is renormalizable to any finite order. A fine new criterion of physical sense!

Second, physicists got accustomed to treating successive approximations as the only way to describe physical reality; they just cannot think but in the terms of perturbation theory, interpreting all the natural phenomena from that angle. Mathematical tricks take the place of physical reasoning, with various artificial constructions presented as the true picture of the physical world. This is how the idea of a virtual particle has penetrated science.

In any order of perturbation, the expression for the transition amplitude (or any other appropriate quantity, like S-matrix, K-matrix, propagators, evolution operators, density matrix etc.) takes the form that can be formally interpreted as a superposition of all possible transitions from the initial to final state via a sequence of intermediate states, coupled to each other by the perturbation. The intermediate states do not need to preserve the energy (or any other quantum numbers) of the initial and final states. This means that it is only the initial and final states that can be considered as physical, in the sense that they have the correct symmetry and obey conservation laws; the intermediate states are free to violate the physical symmetries and hence they will be unobservable—that is, virtual. For instance, in atomic scattering, an unbound state of the system (target + projectile) may exhibit a complex resonance structure due to formation of virtual bound states "embedded in continuum". From the perturbation-theory viewpoint, this can be interpreted as temporary formation of a compound particle from two incident particles (target and projectile), followed by subsequent dissociation into the original (or some other) set of particles. That is, the number of particles in the system may change from physical to virtual states and between virtual states. In quantum field theory, the same technique applied to gauge fields (and primarily the electromagnetic field in quantum electrodynamics) lead to the picturesque idea of the physical vacuum as a sea of virtual massive particles born and annihilating back at any moment. This picture of particles popping out "from nothing" and disappearing "into nothing" is a perfect source of inspiration for idealistic philosophers and theologians, as it provides enough room for spiritualism in-between the acts of spontaneous birth/decay. The apparent violation of causality in the virtual states is easy to attribute to the will of a deity, or any other existential abstraction of consciousness.

However, such virtual particles are nothing like true particles, but rather an artifact of a specific computation technique. Yes, such abstractions are helpful in solving complex physical problems, as they may lead to practically acceptable results. Still, at least in principle, the same results could be obtained in many other ways. Thus, perturbation-theory expressions usually involve summation (integration) over all the intermediate states assumed to form a complete set. It is well known, that one can always choose a different basis set for such a summation, without influencing the results. For instance, there are mathematical tricks that replace a continuous spectrum of intermediate states with a discrete set (Sturmian expansions); such intermediate states can hardly be interpreted in terms of any physical fields or particles. This illustrates the absurdity of putting too much confidence in the vulgar descriptions of physical processes from the perturbation-theory viewpoint. Such figurative explanations are necessary to visualize the abstract mathematical procedures, thus rendering them more tractable; however, they do not necessarily correspond to any physical reality, no more than the complex conglomeration of rods and wheels invented by Maxwell to visualize his theory of electromagnetism.

From non-perturbative solutions of the equations of motion, one can obtain the same complex structures as in the perturbation-theory scheme, but without any need for virtual states or particles. This approach is physically attractive, and it has been widely used in atomic and molecular calculations (the so-called "close coupling" approach). Unfortunately, in many cases, we just cannot derive the equations of motion, and even if we can, their direct solution for many-body systems with variable number of particles is practically unfeasible. That is why we will always need perturbation theory, but as a mere computation tool rather than the language of explanation.

While the picture of virtual particles as independent entities is basically unphysical, there is a more adequate idea applicable to the description of complex structure formation in physical systems due to collective effects. It is well known that the dynamics of nonlinear media can result in various shapes existing for a long time, and eventually dissipating (like phonons, solitons etc.). In this view, resonance spectra observed in quantum physics can be explained as quasi-stable collective modes of motion in many-body systems. The system moves as if there were virtual particles or states, though this resemblance can never be complete. Quantum states are determined for the whole system, and they evolve according to the system's equations of motion; however, for some time, the structure of the state can resemble a collection of distinct particles coupled by relatively weak interactions. When the lifetime of such dynamic formations is much less than the characteristic time of system evolution, we speak about virtual particles. As long as external fields remain weak as compared to the system's inner interactions, they do not resolve individual "virtual" structures, and all we see is their combined effect (resonances). However, a strong external field can penetrate a virtual structure, transforming it into an observable entity. For instance, a drop-like bulge can be formed at the mouth of a water tube; it may vibrate, or change its size and shape for some time, still joined to the rest of the water in the tube; then gravity or mechanical shock will complete the formation of the falling drop. In this sense, one can say that virtual structures are incomplete structures, or quasi-structures. Obviously, their incompleteness is determined by their relation to the whole: virtual structures are always substructures of something better coupled. In physics, we usually deal with abstract quasi-closed systems, which can be deemed to be structurally complete; their substructures will, in general, be virtual, in the physical sense, regardless of the formal representation. Reality and virtuality are well separable on this level. However, in a wider scope, any possible structure can only exist within the same and only world, and everything could be considered as virtual in certain respects, when we are not interested, for a while, in its connectedness to the rest of the Universe.

13 Feb 2007