LSZ reduction formula

LSZ reduction formula
Name	LSZ reduction formula
Type	Quantum field theory
Field	Particle physics
Discovered by	Harry Lehmann, Kurt Symanzik, Wolfhart Zimmermann
Year	1955

LSZ reduction formula. In quantum field theory, the LSZ reduction formula is a fundamental result that provides a rigorous link between the S-matrix elements, which describe scattering processes in particle physics, and the vacuum expectation values of time-ordered products of quantum field operators, known as correlation functions. Developed independently by Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann in the mid-1950s, it serves as a cornerstone for calculating scattering amplitudes from the underlying Lagrangian of a theory. The formula effectively "reduces" the problem of computing probabilities for asymptotic particle states to the evaluation of certain Green's functions, thereby connecting the abstract formalism of quantum fields to measurable experimental outcomes in facilities like CERN.

Overview and motivation

Prior to the work of Lehmann, Symanzik, and Zimmermann, calculating scattering cross-sections in quantum field theory relied heavily on perturbation theory and the intuitive but mathematically loose Feynman diagram approach. The primary motivation was to establish a more rigorous foundation for the S-matrix program, connecting the in states and out states of distant, non-interacting particles to the dynamics encoded in correlation functions. This formalism was crucial for advancing axiomatic quantum field theory and provided a systematic procedure within frameworks like Yang-Mills theory and quantum electrodynamics. The development occurred during a period of intense activity following the success of Feynman, Schwinger, and Tomonaga in renormalizing quantum electrodynamics, with significant contributions also emerging from the work of Murray Gell-Mann and Marvin Goldberger.

Derivation from correlation functions

The derivation begins by considering the Heisenberg picture field operator \(\phi(x)\) for a given quantum field, such as a scalar field in \(\phi^4\) theory. One defines asymptotic creation and annihilation operators for in states and out states using the LSZ asymptotic condition, which assumes particles behave as free fields at times \(t \to \pm\infty\). The key step involves inserting a complete set of states between these asymptotic operators and the time-ordered product, then using the Klein-Gordon equation (or its counterpart for Dirac fields or vector fields) to convert the matrix elements. This process introduces wavefunction renormalization factors, denoted \(Z\), which account for the difference between the interacting field and the asymptotic free field. The derivation rigorously employs the Lehmann-Symanzik-Zimmermann axioms, including microcausality and the spectral condition.

Mathematical statement

For a scalar field theory, the LSZ reduction formula for the S-matrix element connecting \(n\) initial particles to \(m\) final particles is expressed as: \[ \langle p_1', \ldots, p_m' | S | p_1, \ldots, p_n \rangle = \left( \frac{i}{\sqrt{Z}} \right)^{m+n} \prod_{i=1}^m \int d^4x_i' e^{-ip_i' \cdot x_i'} (\partial_{x_i'}^2 + m^2) \prod_{j=1}^n \int d^4x_j e^{ip_j \cdot x_j} (\partial_{x_j}^2 + m^2) \langle \Omega | T \phi(x_1') \ldots \phi(x_m') \phi(x_1) \ldots \phi(x_n) | \Omega \rangle. \] Here, \(| \Omega \rangle\) is the vacuum state, \(T\) denotes the time-ordered product, \(m\) is the physical mass of the particle, and the differential operators \((\partial^2 + m^2)\) project onto the mass shell. For theories with spinor fields like quantum electrodynamics, the formula incorporates Dirac equation operators, while in non-abelian gauge theory such as quantum chromodynamics, it must account for ghost fields and BRST symmetry.

Connection to scattering amplitudes

The formula directly yields the scattering amplitude by applying the LSZ reduction procedure to the correlation function, which is typically computed via functional integral methods or perturbation theory. Each external particle leg in a Feynman diagram corresponds to an application of the operator \((\partial^2 + m^2)\), which amputates the external propagators and enforces the on-shell condition. This amputation is crucial for obtaining finite, physical results and is intimately related to the optical theorem and unitarity of the S-matrix. In practical calculations within the Standard Model, this connection allows physicists at laboratories like Fermilab and SLAC to predict cross-sections for processes involving Higgs boson production or top quark decays from the underlying Lagrangian.

Examples and applications

A classic example is the computation of the Møller scattering amplitude in quantum electrodynamics, where the LSZ formula reduces the problem to evaluating the vacuum expectation value of a time-ordered product of four Dirac field operators. In quantum chromodynamics, it is applied to processes like deep inelastic scattering observed at the HERA accelerator, linking parton distribution functions to fundamental correlation functions. The formula is also essential for deriving Feynman rules in gauge theory, including the Glashow-Weinberg-Salam model, and for proving foundational results like the Goldstone theorem in the context of spontaneous symmetry breaking. Modern applications extend to calculations in supersymmetry and string theory, where it aids in determining scattering amplitudes for graviton interactions.

Relation to other formalisms

The LSZ reduction formula is deeply connected to the path integral formulation of quantum field theory, where correlation functions are generated by functional derivatives of the partition function. It complements the Haag-Ruelle scattering theory, which provides a more rigorous framework for asymptotic states in axiomatic quantum field theory. The formula also underlies the Cutkosky rules for computing discontinuities of amplitudes and is a precursor to modern on-shell methods like the BCFW recursion relation developed by Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten. Furthermore, it relates to the Polchinski equation in the renormalization group flow and shares conceptual ground with the Bogoliubov-Shirkov approach in constructive quantum field theory.

Category:Quantum field theory Category:Scattering theory Category:Theoretical physics