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]}

Notice that in this article Planck units are used, so that *ħ* = 1 (where *ħ* is the reduced Planck constant).

##
The Dyson operator

Suppose that we have a Hamiltonian H, which we split into a *free* part *H* = *H*_{0} and an *interacting part* *V*_{S}(t), i.e. *H* = *H*_{0} + *V*_{S}(t).

We will work in the interaction picture here, that is,

- $V_{I}(t)=\mathrm {e} ^{\mathrm {i} H_{0}(t-t_{0})}V_{S}(t)\mathrm {e} ^{-\mathrm {i} H_{0}(t-t_{0})},$

where $V_{S}(t)$ is the possibly time-dependent interacting part of the Schrödinger picture.
To avoid subscripts, $V(t)$ stands for $V_{\text{I))(t)$ in what follows.
We choose units such that the reduced Planck constant ħ is 1.

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

- $\Psi (t)=U(t,t_{0})\Psi (t_{0})$

is called the **Dyson operator**.

We have

- $U(t,t)=I,$
- $U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}),$
- $U^{-1}(t,t_{0})=U(t_{0},t),$

and hence the Tomonaga–Schwinger equation,

- $i{\frac {d}{dt))U(t,t_{0})\Psi (t_{0})=V(t)U(t,t_{0})\Psi (t_{0}).$

Consequently,

- $U(t,t_{0})=1-i\int _{t_{0))^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})}.$

##
Derivation of the Dyson series

This leads to the following Neumann series:

- ${\begin{aligned}U(t,t_{0})={}&1-i\int _{t_{0))^{t}dt_{1}V(t_{1})+(-i)^{2}\int _{t_{0))^{t}dt_{1}\int _{t_{0))^{t_{1))\,dt_{2}V(t_{1})V(t_{2})+\cdots \\&{}+(-i)^{n}\int _{t_{0))^{t}dt_{1}\int _{t_{0))^{t_{1))dt_{2}\cdots \int _{t_{0))^{t_{n-1))dt_{n}V(t_{1})V(t_{2})\cdots V(t_{n})+\cdots .\end{aligned))$

Here we have $t_{1}>t_{2}>\cdots >t_{n))$, so we can say that the fields are time-ordered, and it is useful to introduce an operator ${\mathcal {T))$ called *time-ordering operator*, defining

- $U_{n}(t,t_{0})=(-i)^{n}\int _{t_{0))^{t}dt_{1}\int _{t_{0))^{t_{1))dt_{2}\cdots \int _{t_{0))^{t_{n-1))dt_{n}\,{\mathcal {T))V(t_{1})V(t_{2})\cdots V(t_{n}).$

We can now try to make this integration simpler. In fact, by the following example:

- $S_{n}=\int _{t_{0))^{t}dt_{1}\int _{t_{0))^{t_{1))dt_{2}\cdots \int _{t_{0))^{t_{n-1))dt_{n}\,K(t_{1},t_{2},\dots ,t_{n}).$

Assume that *K* is symmetric in its arguments and define (look at integration limits):

- $I_{n}=\int _{t_{0))^{t}dt_{1}\int _{t_{0))^{t}dt_{2}\cdots \int _{t_{0))^{t}dt_{n}K(t_{1},t_{2},\dots ,t_{n}).$

The region of integration can be broken in $n!$ sub-regions defined by $t_{1}>t_{2}>\cdots >t_{n))$, $t_{2}>t_{1}>\cdots >t_{n))$, etc. Due to the symmetry of *K*, the integral in each of these sub-regions is the same and equal to $S_{n))$ by definition. So it is true that

- $S_{n}={\frac {1}{n!))I_{n}.$

Returning to our previous integral, the following identity holds

- $U_{n}={\frac {(-i)^{n)){n!))\int _{t_{0))^{t}dt_{1}\int _{t_{0))^{t}dt_{2}\cdots \int _{t_{0))^{t}dt_{n}\,{\mathcal {T))V(t_{1})V(t_{2})\cdots V(t_{n}).$

Summing up all the terms, we obtain Dyson's theorem for the **Dyson series**:^{[clarification needed]}

- $U(t,t_{0})=\sum _{n=0}^{\infty }U_{n}(t,t_{0})={\mathcal {T))e^{-i\int _{t_{0))^{t}{d\tau V(\tau ))).$

##
Wavefunctions

Then, going back to the wavefunction for *t* > *t*_{0},

- $|\Psi (t)\rangle =\sum _{n=0}^{\infty }{(-i)^{n} \over n!}\left(\prod _{k=1}^{n}\int _{t_{0))^{t}dt_{k}\right){\mathcal {T))\left\{\prod _{k=1}^{n}e^{iH_{0}t_{k))V(t_{k})e^{-iH_{0}t_{k))\right\}|\Psi (t_{0})\rangle .$

Returning to the Schrödinger picture, for *t*_{f} > *t*_{i},

- $\langle \psi _{\rm {f));t_{\rm {f))\mid \psi _{\rm {i));t_{\rm {i))\rangle =\sum _{n=0}^{\infty }(-i)^{n}\underbrace {\int dt_{1}\cdots dt_{n)) _{t_{\rm {f))\,\geq \,t_{1}\,\geq \,\cdots \,\geq \,t_{n}\,\geq \,t_{\rm {i))}\,\langle \psi _{\rm {f));t_{\rm {f))\mid e^{-iH_{0}(t_{\rm {f))-t_{1})}V_{S}(t_{1})e^{-iH_{0}(t_{1}-t_{2})}\cdots V_{S}(t_{n})e^{-iH_{0}(t_{n}-t_{\rm {i)))}\mid \psi _{\rm {i));t_{\rm {i))\rangle .$