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In scattering theory, a part of mathematical physics, the **Dyson series**, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams.

This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10^{−10}. This close agreement holds because the coupling constant (also known as the fine-structure constant) of QED is much less than 1.^{[clarification needed]}

In the interaction picture, a Hamiltonian H, can be split into a *free* part *H*_{0} and an *interacting part* *V*_{S}(*t*) as *H* = *H*_{0} + *V*_{S}(*t*).

The potential in the interacting picture is

where is time-independent and is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, stands for in what follows.

In the interaction picture, the **evolution operator** U is defined by the equation:

This is sometimes called the **Dyson operator**.

The evolution operator forms a unitary group with respect to the time parameter. It has the group properties:

- Identity and normalization:
^{[1]} - Composition:
^{[2]} - Time Reversal:
^{[clarification needed]} - Unitarity:
^{[3]}

and from these is possible to derive the time evolution equation of the propagator:^{[4]}

In the interaction picture, the Hamiltonian is the same as the interaction potential and thus the equation can also be written in the interaction picture as

*Caution*: this time evolution equation is not to be confused with the Tomonaga–Schwinger equation.

The formal solution is

which is ultimately a type of Volterra integral.

An iterative solution of the Volterra equation above leads to the following Neumann series:

Here, , and so the fields are time-ordered. It is useful to introduce an operator , called the *time-ordering operator*, and to define

The limits of the integration can be simplified. In general, given some symmetric function one may define the integrals

and

The region of integration of the second integral can be broken in sub-regions, defined by . Due to the symmetry of , the integral in each of these sub-regions is the same and equal to by definition. It follows that

Applied to the previous identity, this gives

Summing up all the terms, the Dyson series is obtained. It is a simplified version of the Neumann series above and which includes the time ordered products; it is the path-ordered exponential:^{[5]}

This result is also called Dyson's formula.^{[6]} The group laws can be derived from this formula.

The state vector at time can be expressed in terms of the state vector at time , for as

The inner product of an initial state at with a final state at in the Schrödinger picture, for is:

The *S*-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:^{[7]}

Note that the time ordering was reversed in the scalar product.

**^**Sakurai, Modern Quantum mechanics, 2.1.10**^**Sakurai, Modern Quantum mechanics, 2.1.12**^**Sakurai, Modern Quantum mechanics, 2.1.11**^**Sakurai, Modern Quantum mechanics, 2.1 pp. 69-71**^**Sakurai, Modern Quantum Mechanics, 2.1.33, pp. 72**^**Tong 3.20, http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf**^**Dyson (1949), "The S-matrix in quantum electrodynamics",*Physical Review*,**75**(11): 1736–1755, Bibcode:1949PhRv...75.1736D, doi:10.1103/PhysRev.75.1736

- Charles J. Joachain,
*Quantum collision theory*, North-Holland Publishing, 1975, ISBN 0-444-86773-2 (Elsevier)