Physics:Dyson series

Short description: Expansion of the time evolution operator

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.

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,

[math]\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})}, }[/math]

where [math]\displaystyle{ H_0 }[/math] is time-independent and [math]\displaystyle{ V_{S}(t) }[/math] is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, [math]\displaystyle{ V(t) }[/math] stands for [math]\displaystyle{ V_\text{I}(t) }[/math] 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:

[math]\displaystyle{ \Psi(t) = U(t,t_0) \Psi(t_0) }[/math]

is called the Dyson operator.

We have a few properties:

• Identity and normalization: [math]\displaystyle{ U(t,t) = I, }[/math]
• Composition: [math]\displaystyle{ U(t,t_0) = U(t,t_1) U(t_1,t_0), }[/math]
• Time Reversal: [math]\displaystyle{ U^{-1}(t,t_0) = U(t_0,t), }[/math]
• Unitarity: [math]\displaystyle{ U^{\dagger}(t,t_0) U(t,t_0)=\mathbb{1} }[/math]

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

[math]\displaystyle{ i\frac d{dt} U(t,t_0)\Psi(t_0) = V(t) U(t,t_0)\Psi(t_0). }[/math]

We notice again that in the interaction picture the Hamiltonian is the same as the interaction potential [math]\displaystyle{ V(t) }[/math]. This equation is not to be confused with the Tomonaga–Schwinger equation

Consequently:

[math]\displaystyle{ U(t,t_0)=1 - i \int_{t_0}^t{dt_1\ V(t_1)U(t_1,t_0)}, }[/math]

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:

[math]\displaystyle{ \begin{align} U(t,t_0) = {} & 1 - i \int_{t_0}^t dt_1V(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_nV(t_1)V(t_2) \cdots V(t_n) +\cdots. \end{align} }[/math]

Here we have [math]\displaystyle{ t_1 \gt t_2 \gt \cdots \gt t_n }[/math], so we can say that the fields are time-ordered, and it is useful to introduce an operator [math]\displaystyle{ \mathcal T }[/math] called time-ordering operator, defining

[math]\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 TV(t_1) V(t_2)\cdots V(t_n). }[/math]

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

[math]\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). }[/math]

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

[math]\displaystyle{ I_n=\int_{t_0}^t dt_1\int_{t_0}^t dt_2\cdots\int_{t_0}^t dt_nK(t_1, t_2,\dots,t_n). }[/math]

The region of integration can be broken in [math]\displaystyle{ n! }[/math] sub-regions defined by [math]\displaystyle{ t_1 \gt t_2 \gt \cdots \gt t_n }[/math], [math]\displaystyle{ t \gt t_1 \gt \cdots \gt t_n }[/math], etc. Due to the symmetry of K, the integral in each of these sub-regions is the same and equal to [math]\displaystyle{ S_n }[/math] by definition. So it is true that

[math]\displaystyle{ S_n = \frac{1}{n!}I_n. }[/math]

Returning to our previous integral, the following identity holds

[math]\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 TV(t_1)V(t_2)\cdots V(t_n). }[/math]

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:

[math]\displaystyle{ U(t,t_0)=\sum_{n=0}^\infty U_n(t,t_0)=\mathcal Te^{-i\int_{t_0}^t{d\tau V(\tau)}}. }[/math]

This result is also called Dyson's formula.

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,

[math]\displaystyle{ |\Psi(t)\rangle=\sum_{n=0}^\infty {(-i)^n\over n!}\underbrace{\int dt_1 \cdots dt_n}_{t_{\rm f}\,\ge\, t_1\,\ge\, \cdots\, \ge\, t_n\,\ge\, 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. }[/math]

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:

[math]\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}\,\ge\, t_1\,\ge\, \cdots\, \ge\, t_n\,\ge\, 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. }[/math]

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)