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.

Dyson operator

In the interaction picture, a Hamiltonian , can be split into a ''free'' part and an ''interacting part'' as .

The potential in the interacting picture 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 $V_\mathrm(t)$ in what follows.

In the interaction picture, the evolution operator is defined by the equation:
:$\Psi(t) = U(t,t_0) \Psi(t_0)$
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: $U^(t,t_0) = U(t_0,t),$
* Unitarity: $U^(t,t_0) U(t,t_0)=\mathbb$
and from these is possible to derive the time evolution equation of the propagator:
:$i\hbar\frac d U(t,t_0)\Psi(t_0) = V(t) U(t,t_0)\Psi(t_0).$
In the interaction picture, the Hamiltonian is the same as the interaction potential $H_=V(t)$ and thus the equation can also be written in the interaction picture as
:$i\hbar \frac d \Psi(t) = H_\Psi(t)$

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

The formal solution is
:$U(t,t_0)=1 - i\hbar^ \int_^t,$
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:

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

Here, $t_1 > t_2 > \cdots > t_n$, and so the fields are time-ordered. It is useful to introduce an operator $\mathcal T$, called the '' time-ordering operator'', and to define

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

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

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

and

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

The region of integration 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

:$S_n = \fracI_n.$

Applied to the previous identity, this gives

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

Summing up all the terms, 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:

:$\begin U(t,t_0)&=\sum_^\infty U_n(t,t_0)\\ &=\sum_^\infty \frac\int_^t dt_1\int_^t dt_2\cdots\int_^t dt_n \, \mathcal TV(t_1)V(t_2)\cdots V(t_n) \\ &=\mathcal T\exp \end$

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

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

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:

:$\begin \langle\Psi(t_) & \mid\Psi(t_)\rangle=\sum_^\infty \times \\ &\underbrace_\, \langle\Psi(t_i)\mid e^V_(t_1)e^\cdots V_(t_n) e^\mid\Psi(t_i)\rangle \end$

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

:$\langle\Psi_ \mid S\mid\Psi_\rangle= \langle\Psi_\mid\sum_^\infty \underbrace_\, \mathcal\left\\mid\Psi_\rangle.$
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)

Scattering theory
Quantum field theory
Freeman Dyson