# Interpreting Feynman diagrams

Informally, especially in the popular literature, virtual paricles are viewed as transmitting the fundamental forces in quantum field theory. The weak force is transmitted by virtual Zs and Ws. The strong force is transmitted by virtual gluons. The electromagnetic force is transmitted by virtual photons. In the eyes of their afficionados, this ''proves'' the existence of virtual particles.

The physics underlying this figurative speech are Feynman diagrams, primarily the simplest tree diagrams that encode the low order perturbative contributions of interactions to the classical limit of scattering experiments. (Thus they are really a manifestation of classical perturbative field theory, not of quantum fields. Quantum corrections involve at least one loop.)

Feynman diagrams describe how the terms in a series expansion of the S-matrix elements arise in a perturbative treatment of the interactions as linear combinations of multiple integrals. Each such multiple integral is a product of vertex contributions and propagators, and each propagator depends on a 4-momentum vector that is integrated over. In additon, there is a dependence on the momenta of the ingoing (prepared) and outgoing (in principle detectable) particles. The structure of each such integral can be represented by a Feynman diagram. This is done by associating with each vertex a node of the diagram and with each momentum a line; for ingoing momenta an external line ending in a node, for outgoing momenta an external line starting in a node, and for propagator momenta an internal line between two nodes.

The resulting diagrams can be given a very vivid but superficial interpretation as the worldlines of particles that undergo a metamorphosis (creation, deflection, or decay) at the vertices. In this interpretation, the in- and outgoing lines are the worldlines of the prepared and detected particles, respectively, and the others are dubbed virtual particles, not being real but required by this interpretation. This interpretation is related to - and indeed historically originated with - Feynman's 1945 intuition that all particles take all possible paths with a probability amplitute given by the path integral density. Unfortunately, such a view is naturally related only to the formal, unrenormalized path integral. But there all contributions of diagrams containing loops are infinite, defying a probability interpretation.

According to the definition in terms of Feynman diagrams, a virtual particle has specific values of 4-momentum, spin, and charges, characterizing the form and variables in its defining propagator. As the 4-momentum is integrated over all of \$R^4\$, there is no mass shell constraint, hence virtual particles are off-shell.

Beyond this, formal quantum field theory is unable to assign any property or probability to a virtual particle. This would require to assign to them states, for which there is no place in the QFT formalism.

However, the interpretation requires them to exist in space and time, hence they are equipped by imagination with all sorts of miraculous properties that complete the picture to something plausible. (See, for example, the Wikipedia article on virtual particles.) Being dressed with a fuzzy notion of quantum fluctuations, where the Heisenberg uncertainty relation allegedly allows one to borrow for a very short time energy from the quantum bank, these properties have a superficial appearance of being scientific. But they are completely unphysical as there is neither a way to test them experimentally nor one to derive them from formal properties of virtual particles.

The long list of manifestations of virtual particles mentioned in the Wikipedia article cited are in fact manifestations of computed scattering matrix elements. They manifest the correctness of the formulas for the multiple integrals associated with Feynman diagrams, but they are silent about the validity of the claims about virtual particles.

Though QFT computations generally use the momentum representation, there is also a (physically nearly useless) Fourier-transformed complementary picture of Feynman diagrams using space-time positions in place of 4-momenta. In this version, the integration is over all of space-time, so virtual particles now have space-time positions but no dynamics, hence no world lines. For in physics, dynamics is always tied to states and an equation of motion. No such thing exists for virtual particles.

## Classical Feynman diagrams

For anyone tempted to associate a physical meaning to virtual particles as a specific quantum phenomenon, let me note that

shows that the perturbation theory for any classical field theory leads to an expansion into Feynman diagrams very similar to those for quantum field theories, except that only tree diagrams occur.

The reason for this similarity in the classical and the quantum case is that Feynman diagrams are nothing else than a graphical notation for writing down products of tensors with many indices summed via the Einstein summation convention. The indices of the results are the external lines aka ''real particles'', while the index pairs summed over are the internal lines aka ''virtual particles''. As such sums of products occur in any multiparticle expansion of expectations, they arise irrespective of the classical or quantum nature of the system.

Feynman-type diagrams therefore arise in any perturbative treatment of statistical multiparticle properties, even classically, as any textbook of statistical mechanics witnesses.

If the picture of virtual particles derived from Feynman diagrams had any intrinsic validity, one should conclude that associated to every classical field there are classical virtual particles behaving just like their quantum analogues, except that (due to the lack of loop diagrams) there are no virtual creation/annihilation patterns. But in the literature, one can find not the slightest trace of a suggestion that classical field theory is sensibly interpreted in terms of virtual particles.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ