# Stable, unstable, and virtual particles

Observable particles. In QFT, observable (hence real) particles of mass \$m\$ are conventionally defined as being associated with poles of the S-matrix at energy \$E=mc^2\$ in the rest frame of the system (Peskin/Schroeder, An introduction to QFT, p.236). If the pole is at a real energy, the mass is real and the particle is stable; if the pole is at a complex energy (in the analytic continuation of the S-matrix to the second sheet), the mass is complex and the particle is unstable. At energies larger than the real part of the mass, the imaginary part determines its decay rate and hence its lifetime (Peskin/Schroeder, p.237); at smaller energies, the unstable particle cannot form for lack of energy, but the existence of the pole is revealed by a Breit-Wigner resonance in certain cross sections. From its position and width, one can estimate the mass and the lifetime of such a particle before it has ever been observed. Indeed, many particles listed in the tables http://pdg.lbl.gov/2011/reviews/contents_sports.html by the Particle Data Group (PDG) are only resonances. The use of resonance width is the established way of interpreting real experiments in particle physics; see, e.g., this section from the PYTHIA 6.3 Manual

Stable and unstable particles. A stable particle can be created and annihilated, as there are associated creation and annihilation operators that add or remove particles to the state. According to the QFT formalism, these particles must be on-shell. This means that their momentum p is related to the real rest mass m by the relation p^2=m^2.

More precisely, it means that the 4-dimensional Fourier transform of the time-dependent single-particle wave function associated with it has a support that satisfies the on-shell relation \$p^2=m^2\$. There is no need for this wave function to be a plane wave, though these are taken as the basis functions between the scattering matrix elements are taken.

An unstable particle is represented quantitatively by a so-called Gamov state (see, e.g., arxiv/0201091), also called a Siegert state (see, e.g., here) in a complex deformation of the Hilbert space of a QFT, obtained by analytic continuation of the formulas for stable particles. In this case, as \$m\$ is complex, the mass shell consists of all complex momentum vectors \$p\$ with \$p^2=m^2\$ and \$v=p/m\$ real, and states are composed exclusively of such momentaum vectors. This is the representation in which one can take the limit of zero decay, in which the particle becomes stable (such as the neutron in the limit of negligible electromagnetic interaction), and hence the representation appropriate in the regime where the unstable particle can be observed (i.e., resolved in time).

A second representation in terms of normalizable states of real mass is given by a superposition of scattering states of their decay products, involving all energies in the range of the Breit-Wigner resonance. In this standard Hilbert space representation, the unstable particle is never formed; so this is the representation appropriate in the regime where the unstable particle reveals itself only as a resonance.

The 2010 PDG description of the Z boson (see here) discusses both descriptions in quantitative detail (p.2: Breit-Wigner approach; p.4: S-matrix approach).

All observable particles are on-shell, though the mass shell is real only for stable particles.

Virtual (= off-shell) particles. On the other hand, virtual particles are defined as internal lines in a Feynman diagram (Peskin/Schroeder, p.5, or Zeidler, QFT I Basics in mathematics and physiics, p.844). and this is their only mode of being.

Virtual particles have real mass but off-shell momenta, while multiparticle states are always composed of on-shell particles only.

The in- and out- states of the S-matrix formalism, describing decaying particles, are composed of on-shell states only, not involving any virtual particle. (Indeed, this is the reason for the name ''virtual''.)

Everyone who bothers to give a precise definition of virtual particles in terms of formulas does so in terms of internal lines of a Feynamn diagram, with off-shell momenta. The external legs must be exactly on-shell by definition of the scattering formalism. There are no off-shell states, so something off-shell (no matter how little) cannot be measured, by the very principles of quantum mechanics.

States involving virtual particles cannot be created for lack of corresponding creation operators in the theory. Thus it is impossible to represent a virtual particle by means of states.

For lack of a state, virtual particles cannot have any of the usual physical characteristics such as dynamics, detection probabilities, or decay channels. Therefore talk about ''poppoing in and out of existence during very short times'', involving their probability of decay, their life-time, their creation, and their decay when taken seriously, is scientific myth, not grounded in quantum field theory. It must be understood only figuratively.

In diagram-free approaches to QFT such as lattice gauge theory, it is impossible to make sense of the notion of a virtual particle. Even in orthodox QFT one can dispense completely with the notion of a virtual particle, as Vol. 1 of the QFT book of Weinberg demonstrates. He represents the full empirical content of QFT, carefully avoiding mentioning the notion of virtual particles.

Virtual states. In nonrelativistic scattering theory, one also meets the concept of virtual states, denoting states of real particles on the second sheet of the analytic continuation, having a well-defined but purely inmaginary energy, defined as a pole of the S-matrix. See, e.g., Thirring, A course in Mathematical Physics, Vol 3, (3.6.11).

The term virtual state is used with a different meaning in virtual state spectroscopy (see, e.g., http://people.bu.edu/teich/pdfs/PRL-80-3483-1998.pdf), and denotes there an unstable energy level above the dissociation threshold. This is equivalent with the concept of a resonance.

Virtual states have nothing to do with virtual particles, which have real energies but no associated states, though sometimes the name ''virtual state'' is associated to them. See, e.g., the thesis; the author explains on p.20 why this is a misleading terminology, but still occasionally uses this terminology in his work.

Why are virtual particles often confused with unstable particles? As we have seen, unstable particles and resonances are observable and can be characterized quantitatively in terms of states. On the other hand, virtual particles lack a state and hence have no meaningful physical properties. This raises the question why virtual particles are often confused with unstable particles, or even identified. (In particular, the Wikipedia article on virtual states is full of misunderstandings justified there by references from popular articles, without going back to sources where a physical definition and discussion is given.)

The reason, I believe, is that in many cases, the dominant contribution to a scattering cross section exhibiting a resonance comes from the exchange of a corresponding virtual particle in a Feynman diagram suggestive of a collection of world lines describing particle creation and annihilation. (Examples can be seen on the Wikipedia page for W and Z bosons.)

This space-time interpretation of Feynman diagrams is very tempting graphically, and contributes to the popularity of Feynman diagrams both among researchers and especially laypeople, though some authors - notably Weinberg in his QFT book - deliberately resist this temptation.

However, this interpretation has no physical basis. Indeed, a single Feynman diagram usually gives an infinite (and hence physically meaningless) contribution to the scattering cross section. The finite, renormalized values of the cross section are obtained only by summing infinitely many such diagrams. This shows that a Feynman diagram represents just some term in a perturbation calculation, and not a process happening in space-time. Therefore one cannot assign physical meaning to a single diagram but at best to a collection of infinitely many diagrams.

The true meaning of virtual particles. For anyone still tempted to associate a physical meaning to viertual particles as a specific quantum phenomenon, let me note that Feynman-type diagrams arise in any perturbative treatment of statistical multiparticle properties, even classically, as any textbook of statistical mechanics witnesses. But in the literature, one can find no trace of a suggestion that classical multiparticle physics is sensibly interpreted in terms of virtual particles.

Indeed, 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 indices summed over are the internal lines aka ''virtual particles''. As such sums of products occur in any expansion of expectations of multiparticle systems, they arise irrespective of the classical or quantum nature of the system.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
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