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.

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

[math]\displaystyle{ 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}, }[/math]

where [math]\displaystyle{ H_0 }[/math] is time-independent and [math]\displaystyle{ V_{\mathrm 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_\mathrm{I}(t) }[/math] in what follows.

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

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

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: [math]\displaystyle{ U(t,t) = 1, }[/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\hbar\frac d{dt} U(t,t_0)\Psi(t_0) = V(t) U(t,t_0)\Psi(t_0). }[/math]

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

[math]\displaystyle{ i\hbar \frac d{dt} \Psi(t) = H_{\rm int}\Psi(t) }[/math]

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

The formal solution is

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

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:

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

Here, [math]\displaystyle{ t_1 \gt t_2 \gt \cdots \gt t_n }[/math], and so the fields are time-ordered. It is useful to introduce an operator [math]\displaystyle{ \mathcal T }[/math], called the time-ordering operator, and to define

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

The limits of the integration can be simplified. In general, given some symmetric function [math]\displaystyle{ K(t_1, t_2,\dots,t_n), }[/math] one may define the integrals

[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]

and

[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 of the second integral 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]. Due to the symmetry of [math]\displaystyle{ K }[/math], the integral in each of these sub-regions is the same and equal to [math]\displaystyle{ S_n }[/math] by definition. It follows that

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

Applied to the previous identity, this gives

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

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:

[math]\displaystyle{ \begin{align} 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 TV(t_1)V(t_2)\cdots V(t_n) \\ &=\mathcal T\exp{-i\hbar^{-1}\int_{t_0}^t{d\tau V(\tau)}} \end{align} }[/math]

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 [math]\displaystyle{ t }[/math] can be expressed in terms of the state vector at time [math]\displaystyle{ t_0 }[/math], for [math]\displaystyle{ t\gt t_0, }[/math] as

[math]\displaystyle{ |\Psi(t)\rangle=\sum_{n=0}^\infty {(-i\hbar^{-1})^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/\hbar}V(t_{k})e^{-iH_0 t_k/\hbar}\right \}|\Psi(t_0)\rangle. }[/math]

The inner product of an initial state at [math]\displaystyle{ t_i=t_0 }[/math] with a final state at [math]\displaystyle{ t_f=t }[/math] in the Schrödinger picture, for [math]\displaystyle{ t_f\gt t_i }[/math] is:

[math]\displaystyle{ \begin{align} \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}\,\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)/\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{align} }[/math]

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

[math]\displaystyle{ \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^4x_1 \cdots d^4x_n}_{t_{\rm out}\,\ge\, t_n\,\ge\, \cdots\, \ge\, t_1\,\ge\, 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. }[/math]

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)