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Are virtual particles and decaying particles the same?
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Decaying particles and resonances are used synonymously in the
literature; they are complementary views of the same unstable state.
A very sharp resonance has a long lifetime relative to a scattering
event, hence behaves like a particle in scattering. It is regarded
as a real object if it lives long enough that its trace in a wire
chamber is detectable, or if its decay products are detectable at
places significantly different from the place where it was created.
On the other hand, a very broad resonance has a very short lifetime
and cannot be differentiated well from the scattering event producing
it; so the idealization defining the scattering event is no longer
valid, and one would not regard the resonance as a particle.
Of course, there is an intermediate grey regime where different people
apply different judgment. This can be seen, e.g., in discussions
concerning the tables of the Particle Data Group.
The only difference between a short-living particle and a stable
particle is the fact that the stable particle has a real rest mass,
while the mass m of the resonance has a small imaginary part.
Note that states with complex masses can be handled well in a rigged
Hilbert space (= Gelfand triple) formulation of quantum mechanics.
Resonances appear as so-called Siegert (or Gamov) states.
A good reference on resonances (not well covered in textbooks) is
V.I. Kukulin et al.,
Theory of Resonances,
Kluwer, Dordrecht 1989.
For rigged Hilbert spaces (treated in Appendix A of Kukulin), see also
quant-ph/9805063 and for its functional analysis ramifications,
K. Maurin,
General Eigenfunction Expansions and Unitary Representations of
Topological Groups,
PWN Polish Sci. Publ., Warsaw 1968.
But a very short-living particle is not the same as a virtual
particle. Often it is a complicated, nearly bound state of other
particles. On the other hand, virtual particles are essentally always
elementary. (There are exceptions when deriving Bethe-Salpeter equations
and the like for the approximate calculations of bound states and
resonances, where one creates an effective theory in which the latter
are treated as elementary.)
Conceptually, an unstable elementary particle is clearly distinguished
from a virtual particle. In perturbation theory, unstable elementary
particles are modelled exactly like stable particles,
namely as external lines in a Feynman diagram.
Virtual particles in Feynman diagrams are exactly those parts
of the diagram which are not given by external lines.
In particular, what is real and what is virtual is not affected
by a diagram rotation - this only affects what is input
and what is output.
The difference can also be seen in the mathematical representation.
In an effective theory where the resonance (e.g., the neutron or a
meson) is regarded as an elementary object, the resonance appears
in in/out states as a real particle, with complex on shell momentum
satisfying p^2=m^2 with a complex mass, but in internal Feynman
diagrams as a virtual particle with real mass, almost always off-shell,
i.e., violating this equation.
There are also some unstable elementary particles like the weak
gauge bosons. Usually, one observes a 4-fermion interaction and the
gauge bosons are virtual. But at high energy = very short scales,
one can in principle observe the gauge bosons and make them real.
This means that they now appear as external lines in the corresponding
perturbative calculations, which displays their nonvirtual nature.
In any case, from a mathematical point of view, one must choose the
framework. Either one works in a Hilbert space, then masses are real
and there are no unstable particles (since these 'are' poles on the
so-called 'unphysical' sheet); in this case, there are no asymptotic
gauge bosons and all are therefore virtual.
Or one works in a rigged Hilbert space and deforms the inner product;
this makes part of the 'unphysical' sheet visible; then the gauge
bosons have complex masses and there exist unstable particles
corresponding to in/out gauge bosons which are real.
The modeling framework therefore decides which language is appropriate.