LSZ reduction formula
The Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a foundational theorem in quantum field theory that relates S-matrix elements—describing the probabilities of particle scattering processes—to the time-ordered correlation functions (or Green's functions) of interacting quantum fields.[1] Developed in 1955 by physicists Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann, the formula provides a rigorous method to extract physically observable scattering amplitudes from theoretical computations of vacuum expectation values, such as \langle \Omega | T \hat{\Phi}(x_1) \cdots \hat{\Phi}(x_n) | \Omega \rangle, where \hat{\Phi} denotes Heisenberg-picture field operators and |\Omega\rangle is the vacuum state.[1][2] At its core, the LSZ formula operates in momentum space by identifying poles in the Fourier-transformed correlation functions when external momenta approach the physical mass shell (p_i^2 \to m_{\text{phys}}^2), effectively amputating external propagators to isolate the connected S-matrix element.[3] The S-matrix amplitude \langle p_1 \cdots p_k | \hat{S} | p_{k+1} \cdots p_n \rangle is obtained from the Fourier-transformed correlation function by multiplying by factors such as \prod_{i=1}^n \sqrt{Z_i} (p_i^2 - m^2 + i \epsilon), where Z_i are field renormalization constants ensuring the connection to asymptotic creation and annihilation operators for free-particle states.[3] This procedure relies on the existence of asymptotic fields that behave like free fields at early and late times, enabling the definition of in- and out-states for scattering theory.[2] The LSZ formalism is indispensable in perturbative quantum field theory, as it bridges abstract field correlations—computed via Feynman diagrams—with experimental predictions for processes like electron-positron scattering or deep inelastic collisions.[2] It assumes stable, massive particles and requires modifications for massless fields, unstable resonances, or non-perturbative regimes, but remains a cornerstone for validating quantum electrodynamics and the Standard Model.[2] Extensions of the formula have been applied to noncommutative field theories and condensed matter systems, underscoring its versatility in modern theoretical physics.[3]Background and Context
Historical Development
The LSZ reduction formula originated in a collaborative effort by three prominent German physicists—Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann—in their 1955 paper, which sought to rigorously link perturbative quantum field theory calculations to the physically observable scattering processes described by S-matrix elements.[1] This work addressed the challenges of infinities and divergences plaguing early quantum field theories by providing an axiomatic framework for extracting scattering amplitudes from correlation functions, marking a pivotal advancement in making quantum field theory predictive for particle interactions.[1] Harry Lehmann (1924–1998), born in Güstrow, Mecklenburg, was a key figure in post-war German theoretical physics, renowned for his foundational contributions to dispersion relations and correlation functions in quantum field theory during his time at the Max Planck Institute and later at the University of Hamburg.[4] Kurt Symanzik (1923–1983), originating from Lyck in East Prussia and educated at universities in Göttingen and Munich after World War II, became celebrated for his innovative Euclidean methods in quantum field theory, which facilitated the study of field configurations in imaginary time. Wolfhart Zimmermann (1928–2016), born in Freiburg im Breisgau and trained at the University of Göttingen, brought his expertise in renormalization techniques to the collaboration, having developed rigorous mathematical approaches to handling divergences that would later influence the BPHZ renormalization scheme.[5] The LSZ formula evolved from earlier S-matrix concepts introduced by Werner Heisenberg in the early 1940s as a way to focus on observable scattering outcomes without relying on unphysical intermediate states, building on the 1945 absorber theory of John Archibald Wheeler and Richard Feynman that emphasized radiation reaction and causality in quantum electrodynamics.[6] This progression culminated in the 1955 axiomatic formulation, which provided a precise mathematical bridge between time-ordered vacuum expectation values and the unitary S-matrix. The formula was first detailed in the paper "Zur Formulierung quantisierter Feldtheorien," published in Il Nuovo Cimento, where it established the direct connection between correlation functions and scattering amplitudes essential for perturbative computations.[1]Role in S-Matrix Theory
The S-matrix serves as a unitary operator in quantum field theory that encodes the transition probabilities between initial and final asymptotic states observed in scattering experiments. It represents the evolution of incoming particle configurations into outgoing ones as time progresses from t \to -\infty to t \to +\infty, capturing the full dynamics of interactions in a relativistic setting.[3] The LSZ reduction formula presupposes the existence of time-ordered correlation functions, or vacuum expectation values (VEVs), defined in the Heisenberg picture to describe field operator correlations. It further relies on the asymptotic completeness hypothesis, which asserts that free-particle states fully span the Hilbert space at spatial and temporal infinities, allowing interactions to become negligible far from the scattering region.[3] In interacting quantum field theories, direct calculation of S-matrix elements proves intractable owing to the non-perturbative nature of the full Hamiltonian; the LSZ formula mitigates this by expressing those elements in terms of VEVs of field operators, which lend themselves to systematic evaluation through perturbation theory and diagrammatic techniques. This conceptual bridge, originally developed in 1955 by Lehmann, Symanzik, and Zimmermann, transforms abstract field correlations into concrete predictions for physical processes. Central to its utility, the scattering amplitude \langle \mathrm{out} | \mathrm{in} \rangle extracted via LSZ connects to experimentally verifiable cross-sections through the optical theorem, a consequence of S-matrix unitarity that equates the imaginary part of the forward amplitude to the total cross-section summed over intermediate states.[7] Moreover, by incorporating asymptotic fields rather than bare interacting ones, the formula circumvents challenges posed by Haag's theorem—such as the non-unitary equivalence between free and interacting representations—thus maintaining consistency in quantum field theories featuring non-trivial vacuum structures.[8]Asymptotic States and Fields
In and Out States
In quantum field theory, the in states represent idealized configurations of non-interacting particles prepared in the distant past, as time approaches t \to -\infty. These states evolve under the full interacting Hamiltonian H to describe the initial conditions for scattering processes. Formally, a single-particle in state with momentum \mathbf{p} is defined as |p, \mathrm{in}\rangle = \sqrt{2 E_{\mathbf{p}}} \lim_{t \to -\infty} e^{i H t} a^\dagger(\mathbf{p}) |0\rangle, where a^\dagger(\mathbf{p}) is the creation operator for free particles acting on the interacting vacuum |0\rangle, and E_{\mathbf{p}} = \sqrt{\mathbf{p}^2 + m^2}.[1] The out states, in contrast, describe free-particle configurations emerging in the distant future, as t \to +\infty, after interactions have occurred. These states are obtained analogously, but using the annihilation operator and forward time evolution: \langle p, \mathrm{out}| = \sqrt{2 E_{\mathbf{p}}} \lim_{t \to +\infty} \langle 0| a(\mathbf{p}) e^{-i H t}. The S-matrix elements, which encode transition amplitudes between initial and final states, are given by \langle \mathrm{out}| S | \mathrm{in} \rangle. Multi-particle in and out states are constructed by applying multiple creation or annihilation operators to the vacuum, forming a Fock space basis.[1] A key assumption underlying this framework is asymptotic completeness, which posits that the full Hilbert space of the interacting theory is spanned by the multiparticle in and out states in the asymptotic limits. This ensures that scattering processes can be fully described within the subspace of these free-particle states. The in and out states are normalized relativistically to preserve Lorentz invariance: \langle \mathbf{p} | \mathbf{q} \rangle = (2\pi)^3 2 E_{\mathbf{p}} \delta^3(\mathbf{p} - \mathbf{q}), where E_{\mathbf{p}} = \sqrt{\mathbf{p}^2 + m^2} is the energy of a particle with mass m. This normalization convention facilitates covariant calculations in relativistic quantum field theory.[1]Asymptotic Field Operators
In quantum field theory, the interacting Heisenberg field operator \phi(x) for a scalar field approaches the incoming asymptotic free field \phi_{\rm in}(x) as t \to -\infty and the outgoing asymptotic free field \phi_{\rm out}(x) as t \to +\infty, with radiation terms becoming negligible at large |t|, where interactions become negligible. Specifically, \phi_{\rm in/out}(x) satisfy the free Klein-Gordon equation (\square + m^2) \phi_{\rm in/out} = 0. This approximation underpins the LSZ formalism by enabling the connection between correlation functions and scattering amplitudes through free-particle-like behavior at large times.[9] The explicit form of the incoming asymptotic field operator is given by the mode expansion \phi_{\rm in}(x) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 \omega_p}} \left[ a_{\rm in}(\mathbf{p}) e^{-i p \cdot x} + a_{\rm in}^\dagger(\mathbf{p}) e^{i p \cdot x} \right], where \omega_p = \sqrt{\mathbf{p}^2 + m^2} and p \cdot x = \omega_p t - \mathbf{p} \cdot \mathbf{x}. The outgoing field \phi_{\rm out}(x) has an analogous expansion but with the corresponding asymptotic creation and annihilation operators a_{\rm out}^\dagger(\mathbf{p}) and a_{\rm out}(\mathbf{p}), which satisfy the same canonical commutation relations. These operators act on the respective Fock spaces to create asymptotic particle states.[9][10] A sketch of the derivation relies on the interaction picture, where the field operators evolve freely under the unperturbed Hamiltonian, while the state vectors account for interactions via the S-matrix; in the asymptotic limits, the full Heisenberg fields \phi(x) approach the free interaction-picture fields \phi_{\rm in/out}(x) that satisfy the Klein-Gordon equation without sources. The creation and annihilation operators obey canonical commutation relations [a_{\rm in}(\mathbf{p}), a_{\rm in}^\dagger(\mathbf{q})] = (2\pi)^3 \delta^3(\mathbf{p} - \mathbf{q}), with all other commutators vanishing, which are preserved under the asymptotic limit due to the cluster decomposition property. Similarly for the out operators.[9][10] The radiation terms, which account for the non-asymptotic contributions from interactions, can be expressed using the retarded Green's function as \int d^4 y \, \Delta_{\rm ret}(x - y) j(y), where j(y) is the current or source term arising from the interaction Lagrangian, ensuring causality by propagating disturbances only forward in time. These terms become negligible at infinity, justifying the dominance of the free asymptotic fields.[10]Formulation of the LSZ Formula
Scalar Fields
The LSZ reduction formula for scalar fields provides a precise connection between S-matrix elements, which describe scattering processes in quantum field theory, and the time-ordered vacuum expectation values of products of scalar field operators, known as correlation functions. This relation allows the computation of physical scattering amplitudes from the theory's Green's functions, typically evaluated perturbatively via Feynman diagrams. For a process involving m incoming particles with momenta q_j ( j = 1, \dots, m ) and n outgoing particles with momenta p_i ( i = 1, \dots, n ), the formula expresses the S-matrix element in terms of integrals over the positions of the interacting fields. The full LSZ formula for this n-particle scattering in scalar field theory is \begin{equation}\langle p_1 \dots p_n ,\text{out} \mid q_1 \dots q_m ,\text{in} \rangle = \left( \prod_{i=1}^n \int d^4 x_i , e^{i p_i \cdot x_i} (\square_{x_i} + m^2) \frac{1}{\sqrt{Z}} \right) \left( \prod_{j=1}^m \int d^4 x_j , e^{-i q_j \cdot x_j} (\square_{x_j} + m^2) \frac{1}{\sqrt{Z}} \right) \langle 0 | T { \phi(x_1) \dots \phi(x_{n+m}) } | 0 \rangle,
\end{equation} where \square = \partial^\mu \partial_\mu is the d'Alembertian operator, m is the physical mass of the scalar particles, Z is the wave function renormalization constant, \phi(x) denotes the scalar field operator, and T indicates time-ordering. This expression assumes the use of asymptotic field operators \phi_{\text{in}} and \phi_{\text{out}} to create the multi-particle states from the vacuum, with the renormalization factors $1/\sqrt{Z} accounting for the overlap between interacting and free fields. The formula was originally derived by Lehmann, Symanzik, and Zimmermann as part of their axiomatic approach to quantizing field theories, ensuring consistency with causality and spectral conditions. The derivation proceeds by first Fourier transforming the position-space correlation function \langle 0 | T \{ \phi(x_1) \dots \phi(x_{n+m}) \} | 0 \rangle into momentum space, which introduces delta functions enforcing overall energy-momentum conservation among the external particles and reveals the analytic structure of the amplitude. In momentum space, the connected correlation function (the amputated Green's function) exhibits poles at the physical on-shell momenta p_i^2 = q_j^2 = m^2, corresponding to the propagators of the external legs in Feynman diagrams. Applying the Klein-Gordon operator (\square + m^2) to each external field in position space isolates the residue at these poles, effectively amputating the external propagators and projecting onto the physical subspace. This step removes the singular free-particle propagation factors, leaving the finite scattering matrix element. Equivalently, in the momentum-space formulation, the amputation of external legs is performed by dividing the full momentum-space correlation function by the product of the external propagators, specifically multiplying by $1 / [i \Delta(p^2 - m^2)] for each outgoing leg and $1 / [-i \Delta(q^2 - m^2)] for each incoming leg, where \Delta(k^2) = 1/(k^2 - m^2 + i\epsilon) is the Feynman propagator. The resulting expression yields the on-shell S-matrix element, with the residues at the poles providing the physical amplitudes free of infrared and collinear divergences associated with off-shell propagators. This procedure assumes the momenta are on-shell (p_i^2 = q_j^2 = m^2) and that the theory satisfies the necessary asymptotic completeness and cluster decomposition properties.