Dyson series · Lablypedia

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_{I} (t) = e^{i H_{0} (t - t_{0}) / ℏ} V_{S} (t) e^{- i H_{0} (t - t_{0}) / ℏ},

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

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

Ψ (t) = U (t, t_{0}) Ψ (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: $U^{†} (t, t_{0}) U (t, t_{0}) = 1$

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

i ℏ \frac{d}{d t} U (t, t_{0}) Ψ (t_{0}) = V (t) U (t, t_{0}) Ψ (t_{0}) .

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

i ℏ \frac{d}{d t} Ψ (t) = H_{i n t} Ψ (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 ℏ^{- 1} \int_{t_{0}}^{t} d t_{1} V (t_{1}) U (t_{1}, t_{0}),

which is ultimately a type of Volterra integral.

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Derivation of the Dyson series

An iterative solution of the Volterra equation above leads to the following Neumann series:

\begin{matrix} U (t, t_{0}) = & 1 - i ℏ^{- 1} \int_{t_{0}}^{t} d t_{1} V (t_{1}) + (- i ℏ^{- 1})^{2} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t_{1}} d t_{2} V (t_{1}) V (t_{2}) + \dots \\ + (- i ℏ^{- 1})^{n} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t_{1}} d t_{2} \dots \int_{t_{0}}^{t_{n - 1}} d t_{n} V (t_{1}) V (t_{2}) \dots V (t_{n}) + \dots . \end{matrix}

Here, $t_{1} > t_{2} > \dots > t_{n}$ , and so the fields are time-ordered. It is useful to introduce an operator $T$ , called the time-ordering operator, and to define

U_{n} (t, t_{0}) = (- i ℏ^{- 1})^{n} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t_{1}} d t_{2} \dots \int_{t_{0}}^{t_{n - 1}} d t_{n} T V (t_{1}) V (t_{2}) \dots 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} d t_{1} \int_{t_{0}}^{t_{1}} d t_{2} \dots \int_{t_{0}}^{t_{n - 1}} d t_{n} K (t_{1}, t_{2}, \dots, t_{n}) .

and

I_{n} = \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t} d t_{2} \dots \int_{t_{0}}^{t} d t_{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 $t_{1} > t_{2} > \dots > t_{n}$ . 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 ℏ^{- 1})^{n}}{n!} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t} d t_{2} \dots \int_{t_{0}}^{t} d t_{n} T V (t_{1}) V (t_{2}) \dots 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{matrix} U (t, t_{0}) & = \sum_{n = 0}^{\infty} U_{n} (t, t_{0}) \\ = \sum_{n = 0}^{\infty} \frac{(- i ℏ^{- 1})^{n}}{n!} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t} d t_{2} \dots \int_{t_{0}}^{t} d t_{n} T V (t_{1}) V (t_{2}) \dots V (t_{n}) \\ = T \exp - i ℏ^{- 1} \int_{t_{0}}^{t} d τ V (τ) \end{matrix}

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

| Ψ (t) ⟩ = \sum_{n = 0}^{\infty} \frac{(- i ℏ^{- 1})^{n}}{n!} \underset{t \geq t_{1} \geq \dots \geq t_{n} \geq t_{0}}{\underset{⏟}{\int d t_{1} \dots d t_{n}}} T {\prod_{k = 1}^{n} e^{i H_{0} (t_{k} - t_{0}) / ℏ} V_{S} (t_{k}) e^{- i H_{0} (t_{k} - t_{0}) / ℏ}} | Ψ (t_{0}) ⟩ .

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{matrix} ⟨ Ψ (t_{i}) & ∣ Ψ (t_{f}) ⟩ = \sum_{n = 0}^{\infty} \frac{(- i ℏ^{- 1})^{n}}{n!} \times \\ \underset{t_{f} \geq t_{1} \geq \dots \geq t_{n} \geq t_{i}}{\underset{⏟}{\int d t_{1} \dots d t_{n}}} ⟨ Ψ (t_{i}) ∣ e^{i H_{0} (t_{1} - t_{i}) / ℏ} V_{S} (t_{1}) e^{- i H_{0} (t_{1} - t_{2}) / ℏ} \dots V_{S} (t_{n}) e^{- i H_{0} (t_{n} - t_{i}) / ℏ} ∣ Ψ (t_{i}) ⟩ \end{matrix}

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

⟨ Ψ_{o u t} ∣ S ∣ Ψ_{i n} ⟩ = ⟨ Ψ_{o u t} ∣ \sum_{n = 0}^{\infty} \frac{(- i ℏ^{- 1})^{n}}{n!} \underset{t_{o u t} \geq t_{n} \geq \dots \geq t_{1} \geq t_{i n}}{\underset{⏟}{\int d^{4} x_{1} \dots d^{4} x_{n}}} T {H_{i n t} (x_{1}) H_{i n t} (x_{2}) \dots H_{i n t} (x_{n})} ∣ Ψ_{i n} ⟩ .

Note that the time ordering was reversed in the scalar product.

Dyson operator

Derivation of the Dyson series

Application on state vectors

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