Physics:Dyson series

Short description: Expansion of the time evolution operator

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.

Dyson operator

In the interaction picture, a Hamiltonian H, can be split into a free part H₀ and an interacting part V_S(t) as H = H₀ + V_S(t).

The potential in the interacting picture is

$V_{\mathrm{I}}(t)={\mathrm{e}}^{\mathrm{i}{H}_0(t-t_0)/\hslash }{V}_{\mathrm{S}}(t){\mathrm{e}}^{-\mathrm{i}{H}_0(t-t_0)/\hslash },$

where $H_0$ is time-independent and $V_{\mathrm{S}}(t)$ is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, $V(t)$ stands for $V_{\mathrm{I}}(t)$ in what follows.

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

$\mathrm{\Psi }(t)=U(t,t_0)\mathrm{\Psi }(t_0)$

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: $U(t,t)=1,$
• Composition: $U(t,t_0)=U(t,t_1)U(t_1,t_0),$
• Time Reversal: $U^{-1}(t,t_0)=U(t_0,t),$
• Unitarity: $U^{†}(t,t_0)U(t,t_0)=\mathbb{𝟙}$

and from these is possible to derive the time evolution equation of the propagator:

$i\hslash \frac{d}{dt}U(t,t_0)\mathrm{\Psi }(t_0)=V(t)U(t,t_0)\mathrm{\Psi }(t_0).$

In the interaction picture, the Hamiltonian is the same as the interaction potential $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=V(t)$ and thus the equation can also be written in the interaction picture as

$i\hslash \frac{d}{dt}\mathrm{\Psi }(t)=H_{\mathrm{i}\mathrm{n}\mathrm{t}}\mathrm{\Psi }(t)$

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

The formal solution is

$U(t,t_0)=1-i{\hslash }^{-1}\underset{t_0}{\overset{t}{\int }}d{t}_{1}V(t_1)U(t_1,t_0),$

which is ultimately a type of Volterra integral.

Derivation of the Dyson series

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

$\begin{array}{rl}U(t,t_0)=& 1-i{\hslash }^{-1}\underset{t_0}{\overset{t}{\int }}d{t}_{1}V(t_1)+(-i{\hslash }^{-1}{)}^2\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t_1}{\int }}d{t}_{2}V(t_1)V(t_2)+\cdots \\ & +(-i{\hslash }^{-1}{)}^n\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t_1}{\int }}d{t}_2\cdots \underset{t_0}{\overset{t_{n-1}}{\int }}d{t}_{n}V(t_1)V(t_2)\cdots V(t_n)+\cdots .\end{array}$

Here, $t_1>t_2>\cdots >t_n$, and so the fields are time-ordered. It is useful to introduce an operator ${𝒯}$, called the time-ordering operator, and to define

$U_n(t,t_0)=(-i{\hslash }^{-1}{)}^n\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t_1}{\int }}d{t}_2\cdots \underset{t_0}{\overset{t_{n-1}}{\int }}d{t}_n{𝒯}V(t_1)V(t_2)\cdots V(t_n).$

The limits of the integration can be simplified. In general, given some symmetric function $K(t_1,t_2,\dots ,t_n),$ one may define the integrals

$S_n=\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t_1}{\int }}d{t}_2\cdots \underset{t_0}{\overset{t_{n-1}}{\int }}d{t}_{n}K(t_1,t_2,\dots ,t_n).$

and

$I_n=\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t}{\int }}d{t}_2\cdots \underset{t_0}{\overset{t}{\int }}d{t}_{n}K(t_1,t_2,\dots ,t_n).$

The region of integration of the second integral can be broken in $n!$ sub-regions, defined by $t_1>t_2>\cdots >t_n$. Due to the symmetry of $K$, the integral in each of these sub-regions is the same and equal to $S_n$ by definition. It follows that

$S_n=\frac{1}{n!}{I}_n.$

Applied to the previous identity, this gives

$U_n=\frac{(-i{\hslash }^{-1}{)}^n}{n!}\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t}{\int }}d{t}_2\cdots \underset{t_0}{\overset{t}{\int }}d{t}_n{𝒯}V(t_1)V(t_2)\cdots V(t_n).$

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:

$\begin{array}{rl}U(t,t_0)& =\sum _{n=0}^{\mathrm{\infty }}{U}_n(t,t_0)\\ & =\sum _{n=0}^{\mathrm{\infty }}\frac{(-i{\hslash }^{-1}{)}^n}{n!}\underset{t_0}{\overset{t}{\int }}d{t}_1\underset{t_0}{\overset{t}{\int }}d{t}_2\cdots \underset{t_0}{\overset{t}{\int }}d{t}_n{𝒯}V(t_1)V(t_2)\cdots V(t_n)\\ & ={𝒯}\exp -i{\hslash }^{-1}\underset{t_0}{\overset{t}{\int }}d\tau V(\tau )\end{array}$

This result is also called Dyson's formula. The group laws can be derived from this formula.

Application on state vectors

The state vector at time $t$ can be expressed in terms of the state vector at time $t_0$, for $t>t_0,$ as

$|\mathrm{\Psi }(t)⟩=\sum _{n=0}^{\mathrm{\infty }}\frac{(-i{\hslash }^{-1}{)}^n}{n!}\underset{t_{\mathrm{f}}\ge t_1\ge \cdots \ge t_n\ge t_{\mathrm{i}}}{\underbrace{\int d{t}_1\cdots d{t}_n}}{𝒯}\{\prod _{k=1}^n{e}^{i{H}_0{t}_k/\hslash }V(t_k)e^{-i{H}_0{t}_k/\hslash }\}|\mathrm{\Psi }(t_0)⟩.$

The inner product of an initial state at $t_i=t_0$ with a final state at $t_f=t$ in the Schrödinger picture, for $t_f>t_i$ is:

$\begin{array}{rl}⟨\mathrm{\Psi }(t_{\mathrm{i}})& \mid \mathrm{\Psi }(t_{\mathrm{f}})⟩=\sum _{n=0}^{\mathrm{\infty }}\frac{(-i{\hslash }^{-1}{)}^n}{n!}\times \\ & \underset{t_{\mathrm{f}}\ge t_1\ge \cdots \ge t_n\ge t_{\mathrm{i}}}{\underbrace{\int d{t}_1\cdots d{t}_n}}⟨\mathrm{\Psi }(t_i)\mid e^{-i{H}_0(t_{\mathrm{f}}-t_1)/\hslash }{V}_{\mathrm{S}}(t_1)e^{-i{H}_0(t_1-t_2)/\hslash }\cdots V_{\mathrm{S}}(t_n)e^{-i{H}_0(t_n-t_{\mathrm{i}})/\hslash }\mid \mathrm{\Psi }(t_i)⟩\end{array}$

The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:

$⟨{\mathrm{\Psi }}_{\mathrm{o}\mathrm{u}\mathrm{t}}\mid S\mid {\mathrm{\Psi }}_{\mathrm{i}\mathrm{n}}⟩=⟨{\mathrm{\Psi }}_{\mathrm{o}\mathrm{u}\mathrm{t}}\mid \sum _{n=0}^{\mathrm{\infty }}\frac{(-i{\hslash }^{-1}{)}^n}{n!}\underset{t_{\mathrm{o}\mathrm{u}\mathrm{t}}\ge t_n\ge \cdots \ge t_1\ge t_{\mathrm{i}\mathrm{n}}}{\underbrace{\int d^4{x}_1\cdots d^4{x}_n}}{𝒯}\{H_{\mathrm{i}\mathrm{n}\mathrm{t}}(x_1)H_{\mathrm{i}\mathrm{n}\mathrm{t}}(x_2)\cdots H_{\mathrm{i}\mathrm{n}\mathrm{t}}(x_n)\}\mid {\mathrm{\Psi }}_{\mathrm{i}\mathrm{n}}⟩.$

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

See also

• Schwinger–Dyson equation
• Magnus series
• Peano–Baker series
• Picard iteration

References

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