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.^{[clarification needed]}

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 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,

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

where ${\displaystyle H_{0}}$ is time-independent and ${\displaystyle V_{S}(t)}$ is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, ${\displaystyle V(t)}$ stands for ${\displaystyle V_{\text{I}}(t)}$ in what follows. We choose units such that the reduced Planck constant ħ is 1.

In the interaction picture, the evolution operator U defined by the equation:

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

is 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: ${\displaystyle U(t,t)=I,}$^[1]
• Composition: ${\displaystyle U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}),}$^[2]
• Time Reversal: ${\displaystyle U^{-1}(t,t_{0})=U(t_{0},t),}$^{[clarification needed]}
• Unitarity: ${\displaystyle U^{\dagger }(t,t_{0})U(t,t_{0})=\mathbb {1} }$^[3]

and from these is possible to derive the time evolution equation of the propagator:^[4]

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

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

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

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

Consequently we can solve formally as:

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

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:

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

Here we have ${\displaystyle t_{1}>t_{2}>\cdots >t_{n}}$, so we can say that the fields are time-ordered, and it is useful to introduce an operator ${\displaystyle {\mathcal {T}}}$ called time-ordering operator, defining

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

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

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

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

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

The region of integration can be broken in ${\displaystyle n!}$ sub-regions defined by ${\displaystyle t_{1}>t_{2}>\cdots >t_{n}}$, ${\displaystyle t>t_{1}>\cdots >t_{n}}$, etc. Due to the symmetry of K, the integral in each of these sub-regions is the same and equal to ${\displaystyle S_{n}}$ by definition. So it is true that

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

Returning to our previous integral, the following identity holds

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

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:^[5]

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

This result is also called Dyson's formula,^[6] 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,

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

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:

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

If we rewrite this in the Heisenberg Picture, and consider the in and out states at infinity, we can also get the S-matrix:^[7]

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

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)

This page was last edited on 27 March 2024, at 16:33