Dyson series

In QED (also known as the fine structure constant) is much less than 1. Notice that in this article ħ = 1.

The Dyson operator

We suppose 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.

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

Ψ(t) = U (t, t 0)Ψ(t 0)

is called 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 then (Tomonaga-Schwinger equation)

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

Thus:

$U(t,t_0)=1 - i \int_{t_0}^t{dt_1\ V(t_1)U(t_1,t_0)}.$

Derivation of the Dyson series

This leads to the following Neumann series:

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

If we assume that $t > t 1 > t 2 > ... > t n$ we can say that the fields are time ordered, and so it is useful to introduce an operator called time-ordering operator. Defining:

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

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

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

If K is symmetric in its arguments, we can define (look at integration limits):

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

And so it is true that:

$S_n=\frac{1}{n!}K_n$

Returning to our previous integral, it holds the identity:

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

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,

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

References

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

Scattering theory

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Dyson_series". A list of authors is available in Wikipedia.