| S‑matrix | |
|---|---|
| Name | S‑matrix |
| Caption | Schematic depiction of scattering: incoming states |
| Field | Quantum field theory |
| Introduced | 1940s |
| Notable users | Werner Heisenberg, John Archibald Wheeler, Richard Feynman |
S‑matrix
The S‑matrix, or scattering matrix, is an operator that relates the asymptotic incoming and outgoing states of a physical system undergoing interaction, encoding probabilities for transition processes in quantum mechanics and quantum field theory. It provides a framework for calculating observable scattering amplitudes without requiring explicit solutions of time‑dependent dynamics and underpins experimental predictions in high‑energy physics. The S‑matrix formalism is central to the interpretation of cross sections measured at facilities such as the CERN Large Hadron Collider.
The S‑matrix is defined as the linear operator S that maps the remote past "in" Hilbert space of free particle states to the remote future "out" Hilbert space: |ψ_out⟩ = S |ψ_in⟩. In experiments the matrix elements of S between specific asymptotic states yield transition amplitudes whose squared moduli determine measurable cross sections and decay rates. The formalism bypasses detailed time evolution during interactions by assuming that interactions are localized in spacetime and that free particle states exist asymptotically, an assumption used in the LSZ reduction formula and in the construction of scattering states in axiomatic quantum field theory.
Mathematically the S‑matrix acts on the Fock space of a quantum field theory and is constructed from the time‑evolution operator U(t, t0) in the interaction picture via the limit S = lim_{T→∞} U(T, −T). In perturbative contexts S is often written as a time‑ordered exponential of the interaction Hamiltonian H_I(t): S = T exp(−i ∫_{−∞}^{∞} H_I(t) dt). The relation of S to correlation functions is established through the LSZ reduction formula, which expresses S‑matrix elements in terms of time‑ordered Green's functions computed from the path integral or operator methods. The Hilbert space structure involves asymptotic completeness, which asserts that the "in" and "out" scattering states span the physical subspace of the theory.
Key mathematical constraints on the S‑matrix include unitarity, analyticity, and crossing symmetry. Unitarity (S†S = 1) embodies conservation of probability and leads to the optical theorem, which relates the forward scattering amplitude to the total cross section. Analyticity properties of scattering amplitudes as functions of complexified kinematic invariants such as the Mandelstam variables (s, t, u) enable dispersion relations and connect low‑ and high‑energy behavior; these ideas were central to the S‑matrix program and the development of Regge theory. Crossing symmetry relates amplitudes for different channels by analytic continuation, reflecting particle–antiparticle interchange. In relativistic contexts, Lorentz invariance constrains S‑matrix tensor structures and phase space integrals.
S‑matrix elements are labeled by asymptotic particle quantum numbers (momenta, spins, internal charges) and are often written ⟨β_out|α_in⟩ = S_{βα}. In practical calculations one isolates the nontrivial part, the T‑matrix, via S = 1 + i T, where the identity corresponds to no interaction. The T‑matrix elements are connected to measurable observables by formulas involving flux and phase space factors; for example, two‑body scattering differential cross sections are expressed in terms of |T|^2. In nonrelativistic quantum mechanics the Born approximation provides a first‑order approximation to T, while in relativistic scattering the same structure appears in lowest‑order perturbation theory computed from Feynman rules.
In perturbation theory the S‑matrix is expanded as a series in a coupling constant. Each term corresponds to integrals over intermediate virtual states and is represented diagrammatically by Feynman diagrams. Renormalization procedures developed by Julian Schwinger, Richard Feynman, and Sin-Itiro Tomonaga remove ultraviolet divergences to yield finite S‑matrix elements order by order. Loop diagrams encode virtual corrections and give rise to anomalous thresholds and branch cuts in analytic structure. Modern computational methods—such as the unitarity method, on‑shell recursion (BCFW), and the use of spinor helicity variables—exploit S‑matrix properties to compute amplitudes more efficiently than traditional Feynman graph summation.
The S‑matrix is the principal object used to connect theoretical models to experiments at colliders like the Large Electron–Positron Collider and the LHC. It appears in calculations for processes in the Standard Model (electroweak scattering, Quantum chromodynamics jets, Higgs production) and in searches for physics beyond the Standard Model (resonances, effective field theories). In condensed matter physics scattering matrices describe electron transport and are used in the Landauer–Büttiker formalism for mesoscopic conductors. In string theory the S‑matrix approach inspired early dual models and remains a means to compute string scattering amplitudes via worldsheet path integrals; influential works include the Veneziano amplitude and developments by Gabriele Veneziano and Miguel Virasoro.
The concept of the S‑matrix was advanced in the 1930s–1950s by figures working on nuclear and particle scattering. Werner Heisenberg proposed an S‑matrix program emphasizing observable quantities; the modern diagrammatic perturbative approach was formalized by Richard Feynman, Julian Schwinger, and Sin‑itiro Tomonaga during development of quantum electrodynamics. Subsequent contributors who shaped analytic S‑matrix methods and dispersion theory include Lev Landau, Geoffrey Chew, Vladimir Gribov, and Tullio Regge. Later techniques for on‑shell methods and amplitude bootstrap were developed by researchers such as Edward Witten and modern amplitude communities based at institutions like Perimeter Institute and SLAC National Accelerator Laboratory.