Dyson series

In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative series, and each term is represented by 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. 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 H₀ and an interacting part V, i.e. H = H₀ + V.

We will work in the interaction picture here and assume 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:

$\begin{array}{lcl} U(t,t_0) & = & 1 - i \int_{t_0}^{t}{dt_1V(t_1)}+(-i)^2\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2V(t_1)V(t_2)}}+\cdots \\ & &{} + (-i)^n\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2 \cdots \int_{t_0}^{t_{n-1}}{dt_nV(t_1)V(t_2) \cdots V(t_n)}}} +\cdots. \end{array}$

Here we have t₁ > t₂ > ..., > 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

$U_n(t,t_0)=(-i)^n\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2\cdots\int_{t_0}^{t_{n-1}}{dt_n\mathcal TV(t_1)V(t_2)\cdots V(t_n)}}}.$

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

$S_n=\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2\cdots\int_{t_0}^{t_{n-1}}{dt_nK(t_1, t_2,\dots,t_n)}}}.$

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

$I_n=\int_{t_0}^t{dt_1\int_{t_0}^{t}{dt_2\cdots\int_{t_0}^t{dt_nK(t_1, t_2,\dots,t_n)}}}.$

The region of integration can be broken in n! sub-regions defined by t₁ > t₂ > ... > t_n, t₂ > t₁ > ... > 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, it holds the identity

$U_n=\frac{(-i)^n}{n!}\int_{t_0}^t{dt_1\int_{t_0}^t{dt_2\cdots\int_{t_0}^t{dt_n\mathcal TV(t_1)V(t_2)\cdots V(t_n)}}}.$

Summing up all the terms, we obtain the Dyson series:

$U(t,t_0)=\sum_{n=0}^\infty U_n(t,t_0)=\mathcal Te^{-i\int_{t_0}^t{d\tau V(\tau)}}.$

The Dyson series for wavefunctions

Then, going back to the wavefunction for t > t₀,

$|\Psi(t)\rangle=\sum_{n=0}^\infty {(-i)^n\over n!}\left(\prod_{k=1}^n \int_{t_0}^t dt_k\right) \mathcal{T}\left\{\prod_{k=1}^n e^{iH_0 t_k}Ve^{-iH_0 t_k}\right \}|\Psi(t_0)\rangle.$

Returning to the Schrödinger picture, for t_f > t_i,

$\langle\psi_f;t_f\mid\psi_i;t_i\rangle=\sum_{n=0}^\infty (-i)^n \underbrace{\int dt_1 \cdots dt_n}_{t_f\,\ge\, t_1\,\ge\, \cdots\, \ge\, t_n\,\ge\, t_i}\, \langle\psi_f;t_f\mid e^{-iH_0(t_f-t_1)}Ve^{-iH_0(t_1-t_2)}\cdots Ve^{-iH_0(t_n-t_i)}\mid\psi_i;t_i\rangle.$

See also

• Magnus series
• Picard iteration

References

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