Zusammenfassung
We present a generalization of Luescher's relation between the finite-volume
spectrum and scattering amplitudes to the case of three particles. We consider
a relativistic scalar field theory in which the couplings are arbitrary aside
from a Z2 symmetry that removes vertices with an odd number of particles. The
theory is assumed to have two-particle phase shifts that are bounded by \pi/2
in the regime of elastic scattering. We determine the spectrum of the
finite-volume theory from the poles in the odd-particle-number finite-volume
correlator, which we analyze to all orders in perturbation theory. We show that
it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard
infinite-volume three-to-three K-matrix. A key feature of our result is the
need to subtract physical singularities in the three-to-three amplitude and
thus deal with a divergence-free quantity. This allows our initial, formal
result to be truncated to a finite dimensional determinant equation. At
present, the relation of the three-to-three K-matrix to the corresponding
scattering amplitude is not known, although previous results in the
non-relativistic limit suggest that such a relation exists.
Nutzer