Particles with Identical Quantum Numbers in Dispersion Theory and Field Theory

The general structure of the scattering amplitude is expressed in terms of the one-particle reducible and irreducible parts, when there are several particles present having the same quantum numbers in the channel in addition to the physical and unphysical cuts. A comparison with field theory is made to obtain the propagator, the vertex functions, and the matrix of the wave-function renormalization constant Z in terms of the $N$ and $D$ functions of the $\frac{N}{D}$ method by making use of the Lehmann representation of the propagator. We then prove the equivalence between composite particles defined in the $\frac{N}{D}$ method and elementary particles with a singular matrix of the wave-function renormalization constant, i.e., detZ=0 in a full field theory under the approximation of keeping only up to the two-particle intermediate states. We show also that Z becomes singular when the one-particle reducible part $A\left(s\right)\mathrm{}$ of the scattering amplitude does not decrease as fast as $s^{-1\mathrm{}}$ at high energies, i.e., $B\equiv -{\lim }_{s\to \infty }{s}^{-1\mathrm{}}{A}^{-1\mathrm{}}\mathrm{}\left(s\right)=0\mathrm{}$. In particular, the condition $B=0\mathrm{}$ so that detZ=0 makes all particles to become composites of other particles, while $B\ne 0$ but with detZ-0 allows mixture of the elementary and composite states. The one- and two-particle cases are discussed in detail to illustrate the compositeness condition detZ=0.