Dyson series

In scattering theory, the Dyson series 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 we choose our units so that ħ (the reduced Planck constant) satisfies 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 and assume units such that the reduced Planck constant $$hbar 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 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)^2int_{t_0}^t{dt_1int_{t_0}^{t_1}{dt_2V(t_1)V(t_2)+...+(-i)^nint_{t_0}^t{dt_1int_{t_0}^{t_1}{dt_2...int_{t_0}^{t_{n-1{dt_nV(t_1)V(t_2)...V(t_n)}

Here we have $$t>t_1>t_2>...>t_n, sowe can say that the fields are time ordered, and it is useful to introduce an operator $$mathcal T called "time-ordering operator", defining:

:$$U_n(t,t_0)=(-i)^nint_{t_0}^t{dt_1int_{t_0}^{t_1}{dt_2...int_{t_0}^{t_{n-1{dt_nmathcal TV(t_1)V(t_2)...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_1int_{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_1int_{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_1int_{t_0}^t{dt_2...int_{t_0}^t{dt_nmathcal 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^{-iint_{t_0}^t{d au V( au)

The Dyson series for wavefunctions

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

:$$|psi(t) angle=sum_{n=0}^infty {(-i)^nover n!}left(prod_{k=1}^n int_{t_0}^t dt_k ight) mathcal{T}left{prod_{k=1}^n e^{iH_0 t_k}Ve^{-iH_0 t_k} ight }|psi(t_0) angle.

Returning to the Schrödinger picture, for t_f > t_i,

:$$langlepsi_f;t_f|psi_i;t_i angle=sum_{n=0}^infty (-i)^negin{matrix}underbrace{int dt_1 cdots dt_n}\t_fge t_1ge dotsge t_nge t_iend{matrix}langlepsi_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 angle.

References

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