The S-matrix maps incoming asymptotic states to outgoing asymptotic states. Its connected transition matrix elements contain an overall energy-momentum Dirac delta function and an invariant scattering amplitude. In quantum field theory, the LSZ reduction formula obtains these elements from amputated field correlators.
A contribution to the unitarity relation for the S-matrix obtained by placing a set of intermediate lines on shell and summing their intermediate states. In a physical-state formulation only physical states enter the sum. When covariant gauge boson index sums include auxiliary polarizations, matching Faddeev-Popov ghost field contributions are needed to reproduce the physical result in a Non-Abelian gauge theory.
A pole of a scattering matrix arising from the numerator rather than a zero of the incoming Jost function can fail to correspond to a bound state. For , the pole at is associated with the incoming zero; the pole at comes from the numerator. The regular Eckart realization has . The distinction and this explicit example are discussed in dipot.ulb.ac.be/dspace/bitstream/2013/373712/4/2306.12216.pdf.
For radial quantum scattering, the incoming coefficient of the regular solution defines the Jost function, with convention . A zero at , , removes the exponentially growing component and produces a bound state, subject to regularity and normalizability. Poles of the outgoing numerator alone need not produce bound states.
For real , this scalar S-matrix block satisfies and Hermitian analyticity of a two-particle S-matrix. It therefore has modulus one for real rapidity differences. For its pole at has residue and admits a direct-channel bound-state pole interpretation. Its crossed block is . The endpoints and are removable degeneracies giving constant amplitudes, so a particle cannot be inferred there from the uncancelled denominator alone. This block specifies an amplitude; a complete theory must also obey the remaining crossing symmetry and bootstrap fusion constraints.
For relativistic two-body scattering in one spatial dimension, this strip is the standard domain containing bound-state and crossed-channel poles. A stable-particle interpretation requires appropriate pole kinematics and residues; not every singularity automatically represents a new particle. Crossing relates rapidity to .
A physical-strip pole of a diagonal S-matrix can come from exchange of an existing particle in the crossed channel. For external masses , use the momentum difference, rather than the sum, to identify the exchanged mass through the displayed invariant. Crossing symmetry turns the pole at into a direct pole at of the particle-antiparticle channel. For , the pole of at has and exchanges the existing charge-one particle; it is not a new charge-three bound state.
At relative rapidity , the analytically continued sum of the constituent four-momentum vectors is on the bound-state mass shell with the displayed mass. For equal constituents, rapidities sum to . A pole at is at threshold rather than a strictly bound state.
In a real species basis, the two-particle S-matrix satisfies under the usual scattering analyticity assumptions. On the real rapidity axis this relates inverse-rapidity amplitudes to complex conjugates. Combined with the algebraic inverse relation , it gives physical unitarity.
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The S-matrix, or scattering matrix, is a fundamental concept in quantum mechanics and quantum field theory that describes how the initial states of a physical system evolve into final states through scattering processes. It encapsulates the probabilities of transitioning from one set of quantum states to another due to interactions. In more detail: 1. **Definitions**: The S-matrix relates the "in" states (initial states of particles before interaction) to the "out" states (final states after interaction).