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S -matrix

The S-matrix, also known as the scattering matrix, is a unitary operator in quantum mechanics and quantum field theory that relates the asymptotic initial states of particles at t \to -\infty to their asymptotic final states at t \to +\infty, thereby encoding the probabilities and amplitudes for all possible outcomes in scattering processes.^[1] It is defined as S = \lim_{t_\text{out}, t_\text{in} \to \mp \infty} U(t_\text{out}, t_\text{in}), where U is the time-evolution operator, ensuring that the total probability is conserved through its unitarity property S^\dagger S = 1.^[1] In practice, the S-matrix elements \langle f | S | i \rangle provide the transition amplitudes from initial state |i\rangle to final state |f\rangle, often expressed as S = 1 + iT, where T is the transition matrix related to interaction strengths.^[2] The concept of the S-matrix was first introduced by John Archibald Wheeler in 1937 as a tool to describe nuclear reactions using resonating group structures, marking an early attempt to systematize scattering without full wave function solutions.^[3] It gained prominence in the 1940s through Werner Heisenberg's development of S-matrix theory, proposed as an alternative framework to quantum field theory for handling high-energy particle interactions, emphasizing observable scattering data over unobservable fields.^[4] Heisenberg's approach, detailed in papers from 1943 onward, aimed to resolve divergences in quantum electrodynamics by focusing solely on the S-matrix's analytic properties derived from causality and unitarity.^[5] This theory influenced subsequent work, including the bootstrap models of the 1960s and Regge theory, before being largely integrated into standard quantum field theory by the 1970s.^[6] Key properties of the S-matrix include its analyticity in the complex energy plane, which stems from causality (effects cannot precede causes) and locality of interactions, leading to dispersion relations that connect real and imaginary parts of amplitudes.^[6] Unitarity not only conserves probability but also implies the optical theorem, $2 \Im \langle f | T | i \rangle = \sum_n \langle f | T^\dagger | n \rangle \langle n | T | i \rangle, linking the forward scattering amplitude's imaginary part to total cross-sections.^[1] Poincaré invariance ensures the S-matrix commutes with Lorentz transformations, making it a covariant object suitable for relativistic theories.^[1] In modern physics, the S-matrix remains central to perturbative calculations in quantum chromodynamics and electroweak theory, as well as non-perturbative approaches like the S-matrix bootstrap, where it serves as the primary observable for constraining effective field theories without relying on underlying Lagrangians.^[4] Recent applications extend to amplitude methods in collider experiments and even connections to string theory and chaos in quantum field theory. As of 2024, there has been a resurgence in S-matrix research, highlighted by workshops like the S-Matrix Marathon and advances in the S-matrix bootstrap program.^[7]^[8]

Historical Development

Origins in Early Quantum Mechanics

The concept of the S-matrix emerged from early efforts in quantum mechanics to describe scattering processes without relying on the full solution of the Schrödinger equation, building on foundational ideas about transition probabilities. In 1927, Paul Dirac introduced key precursors through his work on the quantum mechanics of collision processes, where he developed methods to compute scattering amplitudes, laying the groundwork for descriptions of particle transitions.^[9] This approach gained practical momentum in nuclear physics with John Archibald Wheeler's 1937 proposal of the S-matrix as a unitary operator mapping initial to final states in scattering reactions. Wheeler formulated the S-matrix within the resonating group structure method to analyze light nuclei and nuclear interactions, allowing predictions of reaction outcomes—such as cross-sections for particle collisions—directly from asymptotic wave functions, bypassing the computationally intensive task of solving the complete many-body Schrödinger equation for complex nuclear systems.^[10] Wheeler's non-relativistic framework was significantly extended by Werner Heisenberg during World War II, amid growing frustrations with the infinities plaguing perturbative calculations in quantum field theory (QFT). In papers published in Zeitschrift für Physik from 1943 to 1944, Heisenberg proposed the S-matrix as a fundamental tool for relativistic particle interactions, prioritizing observable scattering amplitudes over unphysical intermediate states and emphasizing principles like causality and unitarity to circumvent renormalization issues.^[11] Heisenberg introduced the unitary property as a core postulate, ensuring probability conservation in transitions between free-particle states, which became central to the theory's predictive power for high-energy processes.^[11]

Evolution in Quantum Field Theory

In the post-World War II era, the S-matrix formalism underwent significant refinement within relativistic quantum field theory (QFT), particularly through the development of dispersion relations in the 1950s. These relations, derived from causality and unitarity principles, established the analyticity of the S-matrix in the complex energy plane, enabling non-perturbative predictions for scattering amplitudes without relying on underlying field Lagrangians. Pioneering contributions came from Geoffrey Chew and Julian Schwinger, who applied dispersion relations to pion-nucleon scattering, connecting asymptotic behavior at high energies to low-energy experimental data and resolving issues in strong interaction phenomenology.^[12]^[13] A hallmark of this evolution was the bootstrap hypothesis, proposed by Geoffrey Chew in the early 1960s, which posited that the S-matrix could be self-consistently determined solely from its general principles—such as unitarity, crossing symmetry, and analyticity—without invoking elementary particles or fundamental fields. This approach aimed to derive particle masses, widths, and coupling constants as bound states emerging from the S-matrix dynamics itself, offering a democratic view of hadrons as composite excitations. The hypothesis gained traction in strong interaction physics, where it integrated with Regge theory, developed in the late 1950s and 1960s by Tullio Regge and others, to model high-energy scattering via Regge poles and trajectories that interpolated between resonances and particle exchanges. Regge theory successfully described forward scattering peaks and total cross-section behaviors in hadron collisions, providing a framework for extrapolating experimental data beyond perturbative regimes.^[14]^[15]^[16]^[17] The S-matrix approach demonstrated early empirical success in interpreting 1950s experiments on pion photoproduction, where dispersion relations predicted multipole amplitudes that matched observations from photon-nucleon interactions at energies up to several hundred MeV, validating the method's non-perturbative power before the dominance of Feynman diagram techniques. By the 1970s, however, the rise of quantum chromodynamics (QCD) as the standard theory of strong interactions—confirmed by asymptotic freedom and deep inelastic scattering data—led to a decline in pure S-matrix methods, as QCD provided a field-theoretic basis for confinement and quark-gluon dynamics. Despite this, S-matrix principles persisted in effective field theories for low-energy phenomena and saw a revival in the 21st century through the AdS/CFT correspondence, where bulk scattering amplitudes in anti-de Sitter space map to boundary conformal field theory correlators, bridging holography and S-matrix analyticity.^[18]^[19]^[4]^[20]

Conceptual Foundations

Motivation from Scattering Processes

In scattering processes, particles interact over long distances, transitioning from initial free-particle states to final states that are also asymptotically free, allowing the S-matrix to characterize these interactions without resolving the full dynamics of the collision.^[21] This approach arises from the practical difficulties in solving the time-dependent Schrödinger equation for multi-particle systems, where the complexity of entangled wavefunctions and time evolution makes exact solutions infeasible, shifting emphasis to observable transition probabilities between distant asymptotic regimes.^[22] As time approaches ±∞, interactions effectively vanish, defining incoming waves from the past and outgoing waves to the future, which the S-matrix maps directly.^[21] The elements of the S-matrix, denoted as S_{fi} = \langle f | S | i \rangle, where |i\rangle and |f\rangle are initial and final asymptotic states, yield transition probabilities |S_{fi}|^2 that quantify the likelihood of specific outcomes in scattering experiments.^[23] These probabilities underpin the calculation of differential and total cross-sections, essential for predicting reaction rates and comparing theory with experimental data, such as in particle accelerators where cross-sections measure interaction strengths.^[24] Historically, the S-matrix concept emerged in nuclear physics to describe collision outcomes in light nuclei without full wavefunction computations, as introduced by John Archibald Wheeler in 1937 for resonating group structures. In quantum field theory, Werner Heisenberg further motivated its development in 1943 to circumvent ultraviolet divergences plaguing perturbative calculations, by parameterizing observables like scattering amplitudes directly through an integral formulation that avoids differential equations and focuses solely on asymptotic behaviors.^[5] This strategy ensured finite, unitarity-constrained results for physical processes, preserving probability conservation amid theoretical uncertainties.^[25]

General Role and Applications

The S-matrix serves as the fundamental operator in quantum field theory that encodes the probabilities for transitions between free-particle asymptotic states, allowing direct computation of key observables in scattering experiments. Specifically, the differential cross-section for a scattering process is proportional to the square of the magnitude of the relevant S-matrix element, \frac{d\sigma}{d\Omega} \propto | \langle f | S | i \rangle |^2, where |i\rangle and |f\rangle denote the initial and final states, while the total cross-section is obtained by integrating this over all solid angles.^[26] These relations enable precise predictions of interaction strengths without relying on intermediate off-shell states. In perturbative quantum field theory, the S-matrix can be expanded using the Dyson series to compute these elements order by order in the coupling constant. Beyond cross-sections, the S-matrix connects to other particle physics observables, including decay rates via the optical theorem, which relates the imaginary part of the forward scattering amplitude to the total cross-section and thus to sum rules over decay channels. Resonances appear as poles in the complex energy plane of the S-matrix, determining their positions and widths, while partial-wave analysis extracts phase shifts that characterize the energy dependence of interactions.^[27] These features are crucial for interpreting collider data and modeling unstable particles. The S-matrix finds broad applications across physics disciplines. In nuclear physics, it is used to analyze neutron-proton scattering, where phase shifts derived from S-matrix elements fit experimental data to constrain nucleon-nucleon potentials.^[28] In condensed matter physics, the scattering matrix approach describes quantum electron transport in mesoscopic systems, relating transmission probabilities to conductance via Landauer-Büttiker formalism.^[29] Analogously, in optics, the S-matrix framework for wave propagation through multilayer structures parallels the Jones matrix, which handles polarization transformations in birefringent media.^[30] In axiomatic quantum field theory, the S-matrix is constrained by fundamental postulates: unitarity ensures probability conservation, analyticity arises from causality and locality, and crossing symmetry relates scattering in different channels, providing a non-perturbative foundation independent of Lagrangian details.^[6] Modern applications leverage these properties in effective field theories for low-energy quantum chromodynamics (QCD), where S-matrix bootstrap methods bound low-energy constants in chiral perturbation theory to describe pion scattering and nucleon interactions.088) Additionally, S-matrix unitarity imposes bounds on beyond-Standard-Model physics, guiding searches for new particles at the Large Hadron Collider (LHC) by constraining deviations in high-energy scattering amplitudes.

S-matrix in Non-Relativistic Quantum Mechanics

Definition in One Dimension

In one-dimensional non-relativistic quantum mechanics, the S-matrix provides a compact description of particle scattering by a localized potential V(x), relating the asymptotic forms of the incoming and outgoing wavefunctions. The scattering problem is governed by the time-independent Schrödinger equation,

-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi(x) = E \psi(x),

where E > 0 is the energy of the incident particle, m is its mass, and \hbar is the reduced Planck's constant. For large |x|, where V(x) \to 0, the solutions reduce to free-particle plane waves with wave number k = \sqrt{2mE}/\hbar. The general asymptotic behavior of the wavefunction \psi(x) is thus

\psi(x) \sim A e^{ikx} + B e^{-ikx} \quad (x \to -\infty),

\psi(x) \sim C e^{ikx} + D e^{-ikx} \quad (x \to +\infty),

where A and D represent incoming amplitudes from the left and right, respectively, while B and C are outgoing amplitudes to the left and right.^[21]^[31] The S-matrix is defined as the $2 \times 2 unitary matrix that maps the incoming amplitudes to the outgoing ones:

\begin{pmatrix} B \\ C \end{pmatrix} = S \begin{pmatrix} A \\ D \end{pmatrix},

with

S = \begin{pmatrix} r & t' \\ t & r' \end{pmatrix}.

Here, t (or t') is the transmission coefficient, giving the amplitude for the particle to pass through the potential from left to right (or right to left), while r (or r') is the reflection coefficient, giving the amplitude for backscattering from left (or right). For time-reversal invariant and real potentials, t = t' and |r|^2 + |t|^2 = 1, ensuring conservation of probability current.^[21]^[31] The explicit form of S is obtained by solving the Schrödinger equation numerically or analytically, matching the wavefunction and its derivative at boundaries within the potential region to determine the coefficients.^[21]

Unitary Property

In one-dimensional quantum mechanics, the unitarity of the S-matrix arises directly from the conservation of probability, as encoded in the continuity equation derived from the Schrödinger equation. The probability density is \rho(x, t) = |\psi(x, t)|^2, and the continuity equation is \frac{\partial \rho}{\partial t} + \frac{\partial j}{\partial x} = 0, where the probability current is j(x, t) = \frac{\hbar}{2mi} \left( \psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} \right).^[21] For stationary scattering states of the form \psi(x, t) = \psi(x) e^{-iEt/\hbar}, the density \rho is time-independent, implying \frac{\partial j}{\partial x} = 0 and thus a constant current j throughout space.^[21] Consider a scattering state with incidence from the left, where the asymptotic wave function is \psi(x) \sim e^{ikx} + r e^{-ikx} as x \to -\infty and \psi(x) \sim t e^{ikx} as x \to +\infty. The current at x \to -\infty evaluates to j(-\infty) = \frac{\hbar k}{m} (1 - |r|^2), while at x \to +\infty it is j(+\infty) = \frac{\hbar k}{m} |t|^2. Equating these constant currents yields |t|^2 + |r|^2 = 1.^[21] A similar relation holds for incidence from the right. The full S-matrix, which linearly relates the outgoing wave amplitudes to the incoming ones as \begin{pmatrix} b_L \\ b_R \end{pmatrix} = S \begin{pmatrix} a_L \\ a_R \end{pmatrix} (with a_L, a_R incoming and b_L, b_R outgoing amplitudes from left and right), takes the form S = \begin{pmatrix} t & r' \\ r & t' \end{pmatrix}. Unitarity S^\dagger S = I follows from these flux conservation conditions, ensuring the norms of incoming and outgoing states are preserved.^[21] This unitarity has direct implications for elastic scattering: the transmission probability T = |t|^2 and reflection probability R = |r|^2 satisfy T + R = 1, conserving the total incident probability flux.^[21] In one dimension, unitarity also implies the optical theorem, which relates the imaginary part of the forward transmission amplitude \operatorname{Im} t(k) to the total scattering cross-section (analogous to the reflection probability in 1D). Specifically, \sigma_\text{tot} = \frac{2}{k} \operatorname{Im} f^{(+)}(k, +), where f^{(+)}(k, +) is the forward scattering amplitude related to t(k).^[32] For multi-channel scattering, such as when the particle has internal degrees of freedom (e.g., spin), the S-matrix generalizes to a larger matrix acting on the space of open channels at a given energy; unitarity S^\dagger S = I holds in this Hilbert space, or block-unitarity for degenerate channel subspaces, again ensuring probability conservation across all accessible channels.^[33] Time-reversal symmetry further preserves this unitarity by relating the S-matrix to its transpose in a basis of time-reversed states.^[21]

Time-Reversal Symmetry

In one-dimensional non-relativistic quantum mechanics, time-reversal symmetry constrains the S-matrix for scattering processes in systems without magnetic fields or absorption. The time-reversal operator T, an antiunitary operator, acts on a wave function as T \psi(x, t) = \psi^*(x, -t), where * denotes complex conjugation, effectively reversing the direction of time while complex-conjugating the state.^[34] For a time-independent Hamiltonian H that commutes with T, satisfying T H T^{-1} = H, this invariance implies that if \psi(x, t) is a solution to the Schrödinger equation, then T \psi(x, t) is also a solution.^[34] Applying T to the asymptotic scattering states maps an incident wave from the left (or right) to the time-reversed outgoing wave, leading to specific relations among the S-matrix elements for real-valued, time-independent potentials. In particular, the transmission coefficient is the same from both sides, t = t'. For potentials even under spatial reversal, V(-x) = V(x), the reflection coefficients satisfy r = -r', where r and r' are the amplitudes for reflection from the left and right, respectively.^[35] These relations ensure the symmetry of the S-matrix, S(k) = S^T(-k), for real-valued potentials.^[36] The derivation proceeds by considering the time-reversed scattering solution: under T, the roles of incoming and outgoing waves are interchanged, and for a real potential, the reversed process corresponds to incidence from the opposite direction, yielding the equality of transmission amplitudes and the phase relation between reflections.^[35] This symmetry holds as long as the potential is real and time-independent, preserving the invariance T H T^{-1} = H.^[36] Time-reversal symmetry is broken by mechanisms such as absorption, introduced via an imaginary component in the potential (e.g., V(x) + i W(x)), or by magnetic fields, which couple through a vector potential odd under T, resulting in non-reciprocal scattering where t \neq t' and the reflection coefficients become unrelated.^[35] In such cases, transmission may differ directionally, and the S-matrix loses its symmetry.^[36] In relativistic quantum field theory, these constraints extend through the CPT theorem, which guarantees that any local, Lorentz-invariant theory is invariant under the combined charge conjugation (C), parity (P), and time-reversal (T) transformations, imposing analogous symmetries on the S-matrix elements.^[37] Unitarity serves as a complementary constraint, ensuring flux conservation independently of time reversal.^[36]

Scattering Coefficients

In one-dimensional quantum mechanics, the scattering coefficients are defined in terms of the elements of the S-matrix, which connects the amplitudes of incoming and outgoing plane waves on either side of the scattering potential. For an incident wave from the left, the asymptotic wave function is expressed as \psi(x) \to e^{ikx} + r e^{-ikx} for x \to -\infty and \psi(x) \to t e^{ikx} for x \to +\infty, where r is the reflection amplitude and t is the transmission amplitude. The transmission coefficient T = |t|^2 represents the probability that the incident particle is transmitted through the potential, while the reflection coefficient R = |r|^2 is the probability of reflection.^[21] The unitary property of the S-matrix, arising from the conservation of probability current, imposes the constraint T + R = 1 for elastic scattering processes. This relation ensures that the incident flux is fully accounted for by transmission and reflection alone, without loss. Experimentally, these coefficients are determined by measuring the ratios of probability currents in the asymptotic regions: the incident current is proportional to |1|^2 v, the reflected to |r|^2 v, and the transmitted to |t|^2 v, where v = \hbar k / m is the particle velocity, yielding T and R directly from current measurements.^[21] The energy dependence of the scattering coefficients is a key feature, with T(E) varying smoothly or oscillatory for energies above any potential barrier, reflecting interference effects in the scattering process. Near specific energies, T(E) displays sharp resonant peaks, corresponding to quasi-bound states where the particle is temporarily trapped before transmission or reflection; these resonances manifest as poles of the S-matrix in the complex energy plane just above the real axis. In one dimension, the basis states are plane waves, providing a direct analogy to the partial-wave decomposition in higher dimensions, where the S-matrix is expanded in spherical harmonics and the l=0 (s-wave) component mirrors the 1D plane-wave scattering.^[38]^[21] The optical theorem in one dimension links the imaginary part of the forward scattering amplitude to the total scattering probability (reflection coefficient in 1D), providing an additional consistency check on unitarity.^[39]

Optical Theorem in One Dimension

The optical theorem in one dimension is the analog of the relation in higher dimensions that connects the total scattering cross section to the imaginary part of the forward scattering amplitude. In the one-dimensional case, derived from S-matrix theory, the theorem states that the total cross section σ_tot = (2/k) Im f(0), where k is the wave number and f(0) is the forward scattering amplitude in a flux-normalized form. This relation holds for short-range potentials and reflects the conservation of probability flux. The derivation follows directly from the unitarity of the S-matrix, S^\dagger S = I, which ensures the conservation of probability in scattering processes. For an initial state |i\rangle, the unitarity condition implies \sum_f |S_{fi}|^2 = 1, where the sum is over all final states f. In the elastic scattering limit, the forward amplitude corresponds to the diagonal S-matrix element S_{ii}, and the imaginary part arises from the discontinuity across the cut in the complex energy plane. Specifically, 2 Im S_{ii} = \sum_{f \neq i} |S_{fi}|^2, linking the imaginary part of the forward amplitude to the total probability of transition to other states, interpreted as the "cross section" comprising transmission and reflection probabilities. In one dimension, with scattering coefficients t for transmission and r for reflection satisfying |t|^2 + |r|^2 = 1, this yields the flux-normalized form where the imaginary part of the forward amplitude (related to t - 1 in asymptotic wave conventions) equals the total scattering probability normalized by incident flux. A similar flux-normalized expression involves the reflection coefficient as (2/k) Im r(-k) = |t(k)|^2, where r(-k) is the reflection amplitude for incident waves with momentum - \hbar k, connecting the imaginary part to the transmission probability under time-reversal symmetry.^[40] This theorem admits an interpretation in terms of shadow scattering or the extinction paradox within one-dimensional waveguides, where the scatterer creates a forward "shadow" due to diffraction, leading to an extinction equal to twice the geometric "size" at high energies—half from actual scattering and half from the diffractive shadow. In 1D quantum or wave propagation, this manifests as the interference between incident and scattered waves reducing the transmitted flux, with the theorem quantifying the extinction via the forward imaginary part.^[41] The theorem applies specifically to elastic scattering processes without absorption, where the S-matrix remains unitary due to probability conservation. In cases with absorption or inelastic channels, unitarity is relaxed (||S|| ≤ 1), and the relation generalizes to include absorption in the total extinction, but the pure elastic form requires no loss mechanisms.

Examples in One-Dimensional Potentials

Transfer Matrix Relation

In one-dimensional quantum scattering, the transfer matrix provides an alternative parameterization to the S-matrix by relating the coefficients of the wavefunction on the left and right sides of a scattering potential. Consider the asymptotic form of the wavefunction: to the left of the potential (x → -∞), ψ(x) = A e^{ikx} + B e^{-ikx}, where A is the amplitude of the incoming wave from the left and B is the reflected wave; to the right (x → +∞), ψ(x) = a e^{ikx} + b e^{-ikx}, with a the transmitted wave and b the incoming wave from the right. The transfer matrix M connects these coefficients via

\begin{pmatrix} A \\ B \end{pmatrix} = M \begin{pmatrix} a \\ b \end{pmatrix},

where M is a 2×2 matrix.^[42]^[43] The unitarity of the S-matrix implies that det M = 1, ensuring conservation of probability current.^[43]^[44] The transfer matrix relates directly to the S-matrix, which typically expresses outgoing amplitudes in terms of incoming ones as \begin{pmatrix} B \ a \end{pmatrix} = S \begin{pmatrix} A \ b \end{pmatrix}. By inverting the transfer matrix relation and adjusting for the direction conventions (swapping incoming and outgoing labels), the S-matrix elements can be expressed in terms of M^{-1}, such as S_{11} = -M_{21}/M_{11} and S_{21} = 1/M_{11} for incidence from the left.^[42]^[44] This connection highlights the equivalence of the two formalisms while emphasizing different aspects of the scattering process. A key advantage of the transfer matrix approach lies in its multiplicative structure for composite systems, such as layered potentials, where the total M is the product of individual transfer matrices for each layer, making it ideal for numerical computations of scattering in complex, piecewise-defined potentials.^[43]^[42] In systems invariant under time reversal, such as those with real-valued potentials, the transfer matrix exhibits symmetry properties M_{11} = M_{22} and M_{12} = -M_{21}, reflecting the reciprocity of transmission and the relation between left and right reflections.^[43]^[44]

Finite Square Well

The finite square well potential serves as a key example for computing the S-matrix in one-dimensional non-relativistic quantum mechanics, particularly illustrating bound states and resonant scattering. Defined as V(x) = -V_0 for |x| < a and V(x) = 0 elsewhere, with V_0 > 0 and a > 0, this attractive potential supports a finite number of bound states for -V_0 < E < 0 and scattering states for E > 0.^[45] For scattering states (E > 0), the time-independent Schrödinger equation -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi is solved piecewise. In the exterior regions (|x| > a), the wave number is k = \sqrt{2mE}/\hbar, yielding plane wave solutions. Inside the well (|x| < a), the effective wave number is \kappa = \sqrt{2m(E + V_0)}/\hbar, also yielding oscillatory solutions. The general wave function for incidence from the left is

S = \lim_{T \to +\infty, , t \to -\infty} U_I(T, t),

where $ U_I(T, t) $ denotes the time-evolution operator in the [interaction picture](/page/Interaction_picture), given by the time-ordered exponential

U_I(T, t) = \mathcal{T} \exp\left( -i \int_t^T V_I(t') , dt' \right).

Here, $ V_I(t) = e^{i H_0 t} V e^{-i H_0 t} $ is the interaction part of the [Hamiltonian](/page/Hamiltonian) in the [interaction picture](/page/Interaction_picture), with $ H_0 $ the free [Hamiltonian](/page/Hamiltonian) and $ V $ the interaction potential.[](https://pages.uoregon.edu/soper/QFT/interactions.pdf) This limit assumes that interactions vanish sufficiently rapidly at early and late times to allow free-particle asymptotics.[](http://sites.science.oregonstate.edu/~stetza/COURSES/ph654/ShortBook.pdf) The S-matrix acts on asymptotic in-states as $ |f, \text{out} \rangle = S |i, \text{in} \rangle $, connecting the [Hilbert space](/page/Hilbert_space) of free incoming particles to that of free outgoing particles.[](http://sites.science.oregonstate.edu/~stetza/COURSES/ph654/ShortBook.pdf) Its unitarity follows directly from the unitarity of the full time-evolution operator, yielding $ S^\dagger S = I $, which preserves probabilities and implies [conservation](/page/Conservation) of particle number in the absence of [absorption](/page/Absorption).[](https://pages.uoregon.edu/soper/QFT/interactions.pdf) Locality of interactions further imposes the cluster [decomposition](/page/Decomposition) property on S, ensuring that distant systems evolve independently.[](https://arxiv.org/pdf/math-ph/0509047.pdf) The physical content of the S-matrix resides in its matrix elements $ \langle f, \text{out} | S | i, \text{in} \rangle $, which encode the scattering amplitudes for transitions from initial state $ |i, \text{in} \rangle $ to final state $ |f, \text{out} \rangle $.[](https://pages.uoregon.edu/soper/QFT/interactions.pdf) These amplitudes include a forward-scattering delta function plus interaction contributions. The S-matrix relates to the transition operator (T-matrix) via $ S = I + i T $, where the T-matrix isolates the non-trivial scattering effects; its elements connect to momentum-space amplitudes through the [LSZ reduction formula](/page/LSZ_reduction_formula), which extracts on-shell matrix elements from correlation functions without deriving the full procedure here.[](https://pages.uoregon.edu/soper/QFT/interactions.pdf) Causality in [quantum field theory](/page/Quantum_field_theory) implies analytic properties for the S-matrix: it is analytic in the upper half of the [complex](/page/Complex) momentum-squared plane (away from bound-state poles), arising from the time-ordered nature of propagators and the iε prescription that enforces retarded response. ## Perturbative Methods ### Time Evolution Operator In the [interaction picture](/page/Interaction_picture) of [quantum field theory](/page/Quantum_field_theory), the [time evolution](/page/Time_evolution) operator $ U(t_2, t_1) $ describes the dynamics of states under the influence of an [interaction Hamiltonian](/page/Hamiltonian) $ V_I(t) $, which is the interaction term transformed to evolve with the free [Hamiltonian](/page/Hamiltonian) $ H_0 $. This operator is formally defined as the time-ordered exponential

U(t_2, t_1) = \mathcal{T} \exp\left( -i \int_{t_1}^{t_2} V_I(t) , dt \right),

where $ \mathcal{T} $ denotes the time-ordering operation, ensuring that operators at later times are placed to the left of those at earlier times in the expansion. This non-perturbative expression, introduced by [Freeman Dyson](/page/Freeman_Dyson), captures the full [time evolution](/page/Time_evolution) in the [interaction picture](/page/Interaction_picture) without assuming weak coupling. The S-matrix, which encodes scattering amplitudes between asymptotic states, is obtained as the limit $ S = U(\infty, -\infty) $, assuming the interaction vanishes at early and late times. Key properties of $ U(t_2, t_1) $ include unitarity, satisfying $ U^\dagger(t_2, t_1) U(t_2, t_1) = 1 $, which follows from the Hermitian nature of the total Hamiltonian and ensures probability conservation in [scattering](/page/Scattering) processes, and the initial condition $ U(t, t) = 1 $. Dyson's formulation of time-ordering provides a rigorous way to handle the non-commutativity of operators at different times, enabling consistent perturbative calculations while defining the operator exactly. The [interaction picture](/page/Interaction_picture) simplifies the treatment of perturbations by separating the free evolution $ e^{-i H_0 t} $ from the [interaction](/page/Interaction) effects, in contrast to the full [time evolution](/page/Time_evolution) operator $ U_\text{full}(t) = e^{-i H t} $, where $ H = H_0 + V $. To justify the asymptotic limits in the S-matrix, the [adiabatic theorem](/page/Adiabatic_theorem) is invoked, ensuring that states evolve freely under $ H_0 $ as $ t \to \pm \infty $ when the [interaction](/page/Interaction) is [switched on](/page/Switched_On) and off sufficiently slowly, thus connecting initial and final free-particle states to the interacting dynamics.[](https://doi.org/10.1007/BF01035766) This framework underpins the identification of in- and out-states as those evolved by $ U $ from the distant past and future, respectively. ### Dyson Series Expansion The Dyson series provides a perturbative expansion for the S-matrix, which is defined as the time-evolution [operator](/page/Operator) $ S = U(\infty, -\infty) $ in the [interaction picture](/page/Interaction_picture) of relativistic [quantum field theory](/page/Quantum_field_theory), where the full [Hamiltonian](/page/Hamiltonian) is split into free and interaction parts. This expansion arises from iteratively solving the time-dependent [Schrödinger equation](/page/Schrödinger_equation) in the [interaction picture](/page/Interaction_picture), yielding a [power series](/page/Power_series) in the [coupling constant](/page/Coupling_constant) of the interaction potential $ V_I(t) $. The series begins with the identity [operator](/page/Operator) and incorporates higher-order corrections through multiple time integrals of the interaction, ensuring proper chronological ordering to handle non-commuting operators at different times.[](https://doi.org/10.1103/PhysRev.75.1736) The iterative form of the expansion for the time-evolution operator $ U(t, t_0) $ is given by

U(t, t_0) = 1 + \sum_{n=1}^\infty U_n(t, t_0),

where the first few terms are $ U_1(t, t_0) = -i \int_{t_0}^t dt_1 \, V_I(t_1) $ and $ U_2(t, t_0) = (-i)^2 \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 \, T \left[ V_I(t_1) V_I(t_2) \right] $, with $ T $ denoting the time-ordering operator that arranges products of operators in decreasing order of their time arguments. In general, the $ n $-th order term is \begin{equation} U_n(t, t_0) = \frac{(-i)^n}{n!} \int_{t_0}^t dt_1 \cdots \int_{t_0}^t dt_n \, T \left[ V_I(t_1) \cdots V_I(t_n) \right], \end{equation} where the factor of $ 1/n! $ accounts for the indistinguishability of the integration variables under time-ordering, and the integrals extend over all times with the $ T $-product enforcing the correct operator sequence. This form was derived by [Dyson](/page/Dyson) to systematically compute S-matrix elements order by order in the [fine-structure constant](/page/Fine-structure_constant).[](https://doi.org/10.1103/PhysRev.75.1736) To evaluate matrix elements of these time-ordered products between free-[field](/page/Field) states, [Wick's theorem](/page/Wick's_theorem) is applied, which decomposes the $ T $-product into a sum of normal-ordered terms plus all possible full contractions (pairings of [field](/page/Field) operators) using free-[field](/page/Field) propagators. In the free-[field](/page/Field) limit, only fully contracted terms contribute to connected diagrams, facilitating the computation of [scattering](/page/Scattering) amplitudes. This theorem, essential for higher-order calculations, reduces the complexity by replacing operator products with c-number contractions. The convergence of the Dyson series is asymptotic rather than absolute, meaning it provides accurate approximations for weak couplings but diverges for strong interactions; nonetheless, it serves as the foundation for renormalization procedures in [quantum electrodynamics](/page/Quantum_electrodynamics). Diagrammatically, each term in the series corresponds to Feynman graphs, where vertices represent interactions and lines denote propagators, offering a visual and computational tool for summing infinite sets of diagrams at fixed order.[](https://doi.org/10.1103/PhysRev.75.1736) The equivalence between the nested integral form and the time-ordered [integral](/page/Integral) form is proven using Heaviside step functions (theta functions) to express the time-ordering operator explicitly. For the second-order term, for instance,

T \left[ V_I(t_1) V_I(t_2) \right] = \theta(t_1 - t_2) V_I(t_1) V_I(t_2) + \theta(t_2 - t_1) V_I(t_2) V_I(t_1),

and integrating over all $ t_1, t_2 \in [t_0, t] $ yields twice the nested [integral](/page/Integral) $ \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 V_I(t_1) V_I(t_2) $ (up to the symmetric part if operators commute), with the $ 1/2! $ factor restoring equality; this generalizes to arbitrary $ n $ by summing over all $ n! $ permutations weighted by theta functions.[](http://scipp.ucsc.edu/~haber/ph215/TimeOrderedExp.pdf) ### Relation to Transition Operators In perturbative [quantum field theory](/page/Quantum_field_theory), the S-matrix relates asymptotic in- and out-states and is decomposed as $ S = 1 + i T $, where $ T $ is the [transition](/page/Transition) operator (or T-matrix operator) that encodes the non-trivial [scattering](/page/Scattering) interactions.[](https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) The [leading-order term](/page/Leading-order_term) in the perturbative expansion of $ T $ is $ T^{(1)} = -i \int_{-\infty}^{\infty} dt \, V_I(t) $, with $ V_I(t) $ the interaction Hamiltonian in the [interaction picture](/page/Interaction_picture), and higher-order contributions arise from connected diagrams that capture the irreducible interactions between particles, excluding disconnected [vacuum](/page/Vacuum) fluctuations.[](https://www.damtp.cam.ac.uk/user/tong/qft/qfthtml/S3.html) This decomposition ensures that the identity part of $ S $ accounts for no-scattering forward propagation, while $ iT $ isolates the transition amplitudes. The matrix elements of the T-operator are given by $ \langle f | T | i \rangle = i (2\pi)^4 \delta^4(P_f - P_i) M_{fi} $, where $ |i\rangle $ and $ |f\rangle $ are initial and final multi-particle states with total four-momenta $ P_i $ and $ P_f $, respectively, and $ M_{fi} $ is the Lorentz-invariant [scattering amplitude](/page/Scattering_amplitude).[](https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) This form arises from the extraction of connected Feynman diagrams in the perturbative expansion, which automatically excludes [vacuum](/page/Vacuum) bubbles—disconnected loops that do not contribute to physical [scattering](/page/Scattering) processes but appear in the full time-evolution [operator](/page/Operator). The connected [structure](/page/Structure) of $ T $ is enforced by normalizing the S-matrix elements against the [vacuum](/page/Vacuum) persistence amplitude, ensuring unitarity and [causality](/page/Causality) in the theory.[](https://www.damtp.cam.ac.uk/user/tong/qft/qfthtml/S3.html) Historically, the T-matrix formalism bridges old-fashioned [perturbation theory](/page/Perturbation_theory), which relies on non-covariant time-ordered expansions and [energy conservation](/page/Energy_conservation) at each [vertex](/page/Vertex), with modern covariant methods using Feynman diagrams and manifest Lorentz-invariant [propagators](/page/Propagator). In old-fashioned approaches, intermediate states are summed over positive and negative energies without immediate covariance, leading to equivalent results for S-matrix elements but complicating higher-order calculations; covariant [perturbation theory](/page/Perturbation_theory), via the [Dyson series](/page/Dyson_series), generates the T-operator orders more systematically while preserving [relativity](/page/Relativity). In contemporary applications, the T-matrix serves as the irreducible two-particle kernel in the Bethe-Salpeter equation, describing bound-state wave functions as $ \Psi(P, q) = G(q, P) T(P, q) \Psi(P, q) $, where $ G $ is the two-particle [propagator](/page/Propagator) and $ T $ sums irreducible interactions beyond ladder approximations.[](https://theses.hal.science/tel-04544120v1/file/2023TOU30275.pdf)

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