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

Suppose that we have a Hamiltonian H, which we split into a free part H₀ and an interacting part V_S(t), i.e. H = H₀ + V_S(t).

We will work in the interaction picture here, that 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 defined by the equation:

$\Psi (t)=U(t,t_{0})\Psi (t_{0})$

is sometimes called the Dyson operator.

The evolution operator forms a unitary group with respect to the time parameter therefore we have 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:

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

We notice again that in the interaction picture the Hamiltonian is the same as the interaction potential $H_{\rm {int}}=V(t)$ and the equation can also be written in the interaction picture as:

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

This time evolution equation is not to be confused with the Tomonaga–Schwinger equation.

Consequently we can solve formally as:

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

which is ultimately a type of Volterra equation.

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 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

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

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

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

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

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

The region of integration can be broken in $n!$ sub-regions defined by $t_{1}>t_{2}>\cdots >t_{n}$, $t>t_{1}>\cdots >t_{n}$, 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

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

Returning to our previous integral, the following identity holds

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

Summing up all the terms, we obtain the Dyson series which is a simplified version of the Neumann series above and which includes the time ordered products:

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

This result is also called Dyson's formula, from here it is also possible to derive back the group laws.

Application on state vectors

One can then express the state vector at time t in terms of the state vector at time t₀, for t > t₀,

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

Then, the inner product of an initial state (t_i = t₀) with a final state (t_f = t) in the Schrödinger picture, for t_f > t_i, is as follows:

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

If we rewrite this in the Heisenberg Picture, and consider the in and out states at infinity, we can also get the S-matrix:

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

Note that the time ordering changed given we reversed 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)