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.
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 .
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).