Dyson series
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| Dyson series | |
|---|---|
| Name | Dyson series |
| Field | Quantum mechanics; Mathematical physics |
| Introduced | 1949 |
| Introduced by | Freeman Dyson |
| Related | Dyson equation; S-matrix; Time-ordered exponential; Perturbation theory |
Dyson series The Dyson series is a perturbative expansion used in Quantum electrodynamics and broader Quantum field theory that represents the time evolution operator as an iterated series of time-ordered interaction integrals. Developed in the context of scattering and many-body problems, the expansion underpins calculations of the S-matrix, Feynman diagrams, and corrections computed in Renormalization programs. The series connects to operator techniques in the Schrödinger picture and Interaction picture and provides a bridge between formal operator expressions and diagrammatic perturbation methods used in Particle physics, Condensed matter physics, and Atomic physics.
Introduction
The Dyson series appears when expressing the solution of the time-dependent Schrödinger equation in the Interaction picture where the Hamiltonian splits into a solvable free part and an interaction part; this setting is central in formulations used by Julian Schwinger, Richard Feynman, and Sin-Itiro Tomonaga for covariant perturbation theory. Freeman Dyson showed how chronological ordering of interaction Hamiltonians yields an explicit expansion for the time evolution operator that naturally corresponds to terms in the perturbative expansion of the S-matrix used by practitioners at CERN, SLAC National Accelerator Laboratory, and in theoretical treatments at institutions such as Princeton University and the Institute for Advanced Study.
Formal definition
Let H0 denote the free Hamiltonian and Hint(t) the interaction Hamiltonian in the Interaction picture; the time evolution operator U(t,t0) satisfies a first-order differential equation akin to that used in treatments by Paul Dirac and Werner Heisenberg. The Dyson series expresses U(t,t0) as an infinite sum of time-ordered integrals: U(t,t0) = 1 + sum_{n=1}^\infty (-i/\hbar)^n ∫_{t0}^t dt1 ∫_{t0}^{t1} dt2 ... ∫_{t0}^{t_{n-1}} dtn T[Hint(t1) Hint(t2) ... Hint(tn)], where T denotes chronological (time) ordering introduced in contexts analyzed by John von Neumann and applied in scattering calculations at Brookhaven National Laboratory. The time-ordering operator arranges operators according to decreasing time arguments, a construction immanent in the formulations of Oskar Klein and Lev Landau for time-dependent perturbations.
Properties and convergence
The Dyson series inherits algebraic properties from operator calculus explored by Norbert Wiener and functional analysts at Cambridge University and Harvard University. Each term corresponds to a multiple integral over a simplex in time, and combinatorial weights match those appearing in diagrammatic expansions developed by Gerard 't Hooft and Murray Gell-Mann. Convergence is subtle: in many interacting quantum field theories such as Quantum chromodynamics and perturbative Quantum electrodynamics the series is asymptotic rather than absolutely convergent, a phenomenon studied by Freeman Dyson himself and later by researchers at Princeton University and ETH Zurich. Rigorous results on convergence exist in certain models in Mathematical physics and constructive field theory, with proofs contributed by mathematicians associated with Institute for Advanced Study and Courant Institute projects, while Borel summation and resummation techniques developed by communities around Stanford University and Massachusetts Institute of Technology are often applied to extract physical predictions.
Applications in quantum mechanics
Physicists employ the Dyson series to compute transition amplitudes in contexts ranging from low-energy atomic transitions treated at Harvard-Smithsonian Center for Astrophysics to high-energy scattering at Fermilab. In Quantum optics and Atomic physics the series yields perturbative expressions for phenomena like spontaneous emission and stimulated absorption analyzed by researchers affiliated with Bell Laboratories and Max Planck Institute for Quantum Optics. In many-body physics, implementations of the Dyson expansion underpin Green's function techniques used in studies at Argonne National Laboratory and computational frameworks developed at Los Alamos National Laboratory. The series also interfaces with techniques in Condensed matter physics such as diagrammatic Monte Carlo methods used in investigations at Carnegie Mellon University and University of Illinois Urbana-Champaign.
Relation to time-ordered exponential and perturbation theory
Formally the Dyson series is the series expansion of the time-ordered exponential, often denoted T exp(- (i/\hbar) ∫_{t0}^t Hint(t') dt'). Connections to perturbation theory as systematized by Pascual Jordan and Eugene Wigner make the Dyson series the canonical bridge between operator solutions and perturbative Feynman rules developed by Richard Feynman and Gerard 't Hooft. In renormalization programs advanced by Kenneth Wilson and John C. Polkinghorne, the Dyson series provides the starting point for organizing divergences and counterterms that appear in loop integrals evaluated at facilities like DESY and theoretical groups at Imperial College London. Diagrammatic terms of the series correspond one-to-one with terms in the Feynman diagram expansion, enabling systematic order-by-order calculations.
Examples and calculations
Standard textbook calculations include first- and second-order perturbative transition amplitudes for a two-level atom interacting with a classical field, treatments found in works by Claude Cohen-Tannoudji and Steven Weinberg. In Quantum electrodynamics, the lowest-order term reproduces the Born approximation used in scattering theory developed by Max Born and Niels Bohr, while second-order terms yield loop corrections whose evaluation requires techniques introduced by Glauber and refined by researchers at CERN. Exactly solvable models—such as driven two-level systems analyzed by Lev Landau-type methods and solvable bosonic baths studied by groups at École Normale Supérieure—illustrate how truncations of the Dyson series approximate dynamics. Practical computations often combine analytic Dyson-expansion terms with numerical integration and regularization methods used at Oak Ridge National Laboratory and in computational projects at University of Cambridge.