Dyson series

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.

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 , which we split into a ''free'' part and an ''interacting part'' , i.e. .

We will work in the interaction picture here, that is,
:$V_(t) = \mathrm^ V_(t) \mathrm^,$
where $H_0$ is time-independent and $V_(t)$ is the possibly time-dependent interacting part of the Schrödinger picture.
To avoid subscripts, $V(t)$ stands for $V_\text(t)$ in what follows.
We choose units such that the reduced Planck constant is 1.

In the interaction picture, the evolution operator defined by the equation
:$\Psi(t) = U(t,t_0) \Psi(t_0)$
is called the Dyson operator.

We have
:$U(t,t) = I,$
:$U(t,t_0) = U(t,t_1) U(t_1,t_0),$
:$U^(t,t_0) = U(t_0,t),$
and hence the time evolution equation of the propagator:
:$i\frac d U(t,t_0)\Psi(t_0) = V(t) U(t,t_0)\Psi(t_0).$
This is not to be confused with the Tomonaga–Schwinger equation

Consequently:
:$U(t,t_0)=1 - i \int_^t.$
Which is ultimately a type of Volterra equation

Derivation of the Dyson series

This leads to the following Neumann series:

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

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

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

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

:$S_n=\int_^t dt_1\int_^ dt_2\cdots \int_^ dt_n \, K(t_1, t_2,\dots,t_n).$

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

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

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

:$S_n = \fracI_n.$

Returning to our previous integral, the following identity holds

:$U_n=\frac\int_^t dt_1\int_^t dt_2\cdots\int_^t dt_n \, \mathcal TV(t_1)V(t_2)\cdots V(t_n).$

Summing up all the terms, we obtain Dyson's theorem for the Dyson series:

:$U(t,t_0)=\sum_^\infty U_n(t,t_0)=\mathcal Te^.$

Application on State Vectors

One can then express the state vector at time ''t'' in terms of the state vector at time ''t''₀, for ''t'' > ''t''₀,

:$, \Psi(t)\rangle=\sum_^\infty \underbrace_\, \mathcal\left\, \Psi(t_0)\rangle.$

Then, the inner product of an initial state (''t''_i = ''t''₀) with a final state (''t''_f = ''t'') in the Schrödinger picture, for ''t''_f > ''t''_i, is as follows:

:$\langle\Psi(t_i)\mid\Psi(t_f)\rangle=\sum_^\infty \underbrace_\, \langle\Psi(t_i)\mid e^V_S(t_1)e^\cdots V_S(t_n) e^\mid\Psi(t_i)\rangle.$

See also

*Schwinger–Dyson equation
* Magnus series
* Picard iteration

References

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

Scattering theory
Quantum field theory
Freeman Dyson