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 H, can be split into a free part H₀ and an interacting part V_S(t) as H = H₀ + V_S(t).

The potential in the interacting picture is

$V_{\mathrm {I} }(t)=\mathrm {e} ^{\mathrm {i} H_{0}(t-t_{0})/\hbar }V_{\mathrm {S} }(t)\mathrm {e} ^{-\mathrm {i} H_{0}(t-t_{0})/\hbar },$

where $H_{0}$ is time-independent and $V_{\mathrm {S} }(t)$ is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, $V(t)$ stands for $V_{\mathrm {I} }(t)$ in what follows.

In the interaction picture, the evolution operator U 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: $U^{-1}(t,t_{0})=U(t_{0},t),$
• Unitarity: $U^{\dagger }(t,t_{0})U(t,t_{0})=\mathbb {1}$

and from these is possible to derive the time evolution equation of the propagator:

$i\hbar {\frac {d}{dt}}U(t,t_{0})\Psi (t_{0})=V(t)U(t,t_{0})\Psi (t_{0}).$

In the interaction picture, the Hamiltonian is the same as the interaction potential $H_{\rm {int}}=V(t)$ and thus the equation can also be written in the interaction picture as

$i\hbar {\frac {d}{dt}}\Psi (t)=H_{\rm {int}}\Psi (t)$

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

The formal solution is

$U(t,t_{0})=1-i\hbar ^{-1}\int _{t_{0}}^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})},$

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:

${\begin{aligned}U(t,t_{0})={}&1-i\hbar ^{-1}\int _{t_{0}}^{t}dt_{1}V(t_{1})+(-i\hbar ^{-1})^{2}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}\,dt_{2}V(t_{1})V(t_{2})+\cdots \\&{}+(-i\hbar ^{-1})^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}V(t_{1})V(t_{2})\cdots V(t_{n})+\cdots .\end{aligned}}$

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

$U_{n}(t,t_{0})=(-i\hbar ^{-1})^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n}).$

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

$S_{n}=\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\,K(t_{1},t_{2},\dots ,t_{n}).$

and

$I_{n}=\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}K(t_{1},t_{2},\dots ,t_{n}).$

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

$S_{n}={\frac {1}{n!}}I_{n}.$

Applied to the previous identity, this gives

$U_{n}={\frac {(-i\hbar ^{-1})^{n}}{n!}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n}).$

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:

${\begin{aligned}U(t,t_{0})&=\sum _{n=0}^{\infty }U_{n}(t,t_{0})\\&=\sum _{n=0}^{\infty }{\frac {(-i\hbar ^{-1})^{n}}{n!}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n})\\&={\mathcal {T}}\exp {-i\hbar ^{-1}\int _{t_{0}}^{t}{d\tau V(\tau )}}\end{aligned}}$

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

$|\Psi (t)\rangle =\sum _{n=0}^{\infty }{(-i\hbar ^{-1})^{n} \over n!}\underbrace {\int dt_{1}\cdots dt_{n}} _{t_{\rm {f}}\,\geq \,t_{1}\,\geq \,\cdots \,\geq \,t_{n}\,\geq \,t_{\rm {i}}}\,{\mathcal {T}}\left\{\prod _{k=1}^{n}e^{iH_{0}t_{k}/\hbar }V(t_{k})e^{-iH_{0}t_{k}/\hbar }\right\}|\Psi (t_{0})\rangle .$

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:

${\begin{aligned}\langle \Psi (t_{\rm {i}})&\mid \Psi (t_{\rm {f}})\rangle =\sum _{n=0}^{\infty }{(-i\hbar ^{-1})^{n} \over n!}\times \\&\underbrace {\int dt_{1}\cdots dt_{n}} _{t_{\rm {f}}\,\geq \,t_{1}\,\geq \,\cdots \,\geq \,t_{n}\,\geq \,t_{\rm {i}}}\,\langle \Psi (t_{i})\mid e^{-iH_{0}(t_{\rm {f}}-t_{1})/\hbar }V_{\rm {S}}(t_{1})e^{-iH_{0}(t_{1}-t_{2})/\hbar }\cdots V_{\rm {S}}(t_{n})e^{-iH_{0}(t_{n}-t_{\rm {i}})/\hbar }\mid \Psi (t_{i})\rangle \end{aligned}}$

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

$\langle \Psi _{\rm {out}}\mid S\mid \Psi _{\rm {in}}\rangle =\langle \Psi _{\rm {out}}\mid \sum _{n=0}^{\infty }{(-i\hbar ^{-1})^{n} \over n!}\underbrace {\int d^{4}x_{1}\cdots d^{4}x_{n}} _{t_{\rm {out}}\,\geq \,t_{n}\,\geq \,\cdots \,\geq \,t_{1}\,\geq \,t_{\rm {in}}}\,{\mathcal {T}}\left\{H_{\rm {int}}(x_{1})H_{\rm {int}}(x_{2})\cdots H_{\rm {int}}(x_{n})\right\}\mid \Psi _{\rm {in}}\rangle .$

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)