Dyson series#The Dyson operator

In the interaction picture, a Hamiltonian {{mvar|H}}, can be split into a free part {{math|H₀}} and an interacting part {{math|V_S(t)}} as {{math|H {{=}} H₀ + V_S(t)}}.

The potential in the interacting picture is

:$V\_\{\backslash mathrm\; I\}(t)\; =\; \backslash mathrm\{e\}^\{\backslash mathrm\{i\}\; H\_\{0\}(t\; -\; t\_\{0\})/\backslash hbar\}\; V\_\{\backslash mathrm\; S\}(t)\; \backslash mathrm\{e\}^\{-\backslash mathrm\{i\}\; H\_\{0\}\; (t\; -\; t\_\{0\})/\backslash hbar\},$

where $H\_0$ is time-independent and $V\_\{\backslash mathrm\; S\}(t)$ is the possibly time-dependent interacting part of the Schrödinger picture.

To avoid subscripts, $V(t)$ stands for $V\_\backslash mathrm\{I\}(t)$ in what follows.

In the interaction picture, the evolution operator {{mvar| U}} is defined by the equation:

:$\backslash Psi(t)\; =\; U(t,t\_0)\; \backslash Psi(t\_0)$

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: $U(t,t)\; =\; 1,$Sakurai, Modern Quantum mechanics, 2.1.10
• Composition: $U(t,t\_0)\; =\; U(t,t\_1)\; U(t\_1,t\_0),$Sakurai, Modern Quantum mechanics, 2.1.12
• Time Reversal: $U^\{-1\}(t,t\_0)\; =\; U(t\_0,t),${{Clarify|reason=Not sure if this hypothesis is needed, why this is not in Sakurai ?|date=August 2023}}
• Unitarity: $U^\{\backslash dagger\}(t,t\_0)\; U(t,t\_0)=\backslash mathbb\{1\}$Sakurai, Modern Quantum mechanics, 2.1.11

and from these is possible to derive the time evolution equation of the propagator:Sakurai, Modern Quantum mechanics, 2.1 pp. 69-71

:$i\backslash hbar\backslash frac\; d\{dt\}\; U(t,t\_0)\backslash Psi(t\_0)\; =\; V(t)\; U(t,t\_0)\backslash Psi(t\_0).$

In the interaction picture, the Hamiltonian is the same as the interaction potential $H\_\{\backslash rm\; int\}=V(t)$ and thus the equation can also be written in the interaction picture as

:$i\backslash hbar\; \backslash frac\; d\{dt\}\; \backslash Psi(t)\; =\; H\_\{\backslash rm\; int\}\backslash Psi(t)$

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

The formal solution is

:$U(t,t\_0)=1\; -\; i\backslash hbar^\{-1\}\; \backslash int\_\{t\_0\}^t\{dt\_1\backslash \; V(t\_1)U(t\_1,t\_0)\},$

which is ultimately a type of Volterra integral.

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

:$$

\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}

Here, $t\_1\; >\; t\_2\; >\; \backslash cdots\; >\; t\_n$, and so the fields are time-ordered. It is useful to introduce an operator $\backslash mathcal\; T$, called the time-ordering operator, and to define

:$U\_n(t,t\_0)=(-i\backslash hbar^\{-1\}\; )^n\; \backslash int\_\{t\_0\}^t\; dt\_1\; \backslash int\_\{t\_0\}^\{t\_1\}\; dt\_2\; \backslash cdots\; \backslash int\_\{t\_0\}^\{t\_\{n-1\}\}\; dt\_n\backslash ,\backslash mathcal\; TV(t\_1)\; V(t\_2)\backslash cdots\; V(t\_n).$

The limits of the integration can be simplified. In general, given some symmetric function $K(t\_1,\; t\_2,\backslash dots,t\_n),$ one may define the integrals

:$S\_n=\backslash int\_\{t\_0\}^t\; dt\_1\backslash int\_\{t\_0\}^\{t\_1\}\; dt\_2\backslash cdots\; \backslash int\_\{t\_0\}^\{t\_\{n-1\}\}\; dt\_n\; \backslash ,\; K(t\_1,\; t\_2,\backslash dots,t\_n).$

and

:$I\_n=\backslash int\_\{t\_0\}^t\; dt\_1\backslash int\_\{t\_0\}^t\; dt\_2\backslash cdots\backslash int\_\{t\_0\}^t\; dt\_nK(t\_1,\; t\_2,\backslash dots,t\_n).$

The region of integration of the second integral can be broken in $n!$ sub-regions, defined by $t\_1\; >\; t\_2\; >\; \backslash cdots\; >\; t\_n$. 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

:$S\_n\; =\; \backslash frac\{1\}\{n!\}I\_n.$

Applied to the previous identity, this gives

:$U\_n=\backslash frac\{(-i\; \backslash hbar^\{-1\})^n\}\{n!\}\backslash int\_\{t\_0\}^t\; dt\_1\backslash int\_\{t\_0\}^t\; dt\_2\backslash cdots\backslash int\_\{t\_0\}^t\; dt\_n\; \backslash ,\; \backslash mathcal\; TV(t\_1)V(t\_2)\backslash cdots\; V(t\_n).$

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:Sakurai, Modern Quantum Mechanics, 2.1.33, pp. 72

:$\backslash begin\{align\}$

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}

This result is also called Dyson's formula. Tong 3.20, http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf The group laws can be derived from this formula.

The state vector at time $t$ can be expressed in terms of the state vector at time $t\_0$, for $t>t\_0,$ as

:$|\backslash Psi(t)\backslash rangle=\backslash sum\_\{n=0\}^\backslash infty\; \{(-i\backslash hbar^\{-1\})^n\backslash over\; n!\}\backslash underbrace\{\backslash int\; dt\_1\; \backslash cdots\; dt\_n\}\_\{t\_\{\backslash rm\; f\}\backslash ,\backslash ge\backslash ,\; t\_1\backslash ,\backslash ge\backslash ,\; \backslash cdots\backslash ,\; \backslash ge\backslash ,\; t\_n\backslash ,\backslash ge\backslash ,\; t\_\{\backslash rm\; i\}\}\backslash ,\; \backslash mathcal\{T\}\backslash left\backslash \{\backslash prod\_\{k=1\}^n\; e^\{iH\_0\; t\_k/\backslash hbar\}V(t\_\{k\})e^\{-iH\_0\; t\_k/\backslash hbar\}\backslash right\; \backslash \}|\backslash Psi(t\_0)\backslash rangle.$

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:

:$\backslash begin\{align\}$

\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}

The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:{{citation|title=The S-matrix in quantum electrodynamics|journal=Physical Review|year=1949|author=Dyson|volume=75 |issue=11 |pages=1736–1755 |doi=10.1103/PhysRev.75.1736 |bibcode=1949PhRv...75.1736D |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.75.1736|url-access=subscription}}

:$\backslash langle\backslash Psi\_\{\backslash rm\; out\}\; \backslash mid\; S\backslash mid\backslash Psi\_\{\backslash rm\; in\}\backslash rangle=\; \backslash langle\backslash Psi\_\{\backslash rm\; out\}\backslash mid\backslash sum\_\{n=0\}^\backslash infty\; \{(-i\backslash hbar^\{-1\})^n\backslash over\; n!\}\; \backslash underbrace\{\backslash int\; d^4x\_1\; \backslash cdots\; d^4x\_n\}\_\{t\_\{\backslash rm\; out\}\backslash ,\backslash ge\backslash ,\; t\_n\backslash ,\backslash ge\backslash ,\; \backslash cdots\backslash ,\; \backslash ge\backslash ,\; t\_1\backslash ,\backslash ge\backslash ,\; t\_\{\backslash rm\; in\}\}\backslash ,\; \backslash mathcal\{T\}\backslash left\backslash \{\; H\_\{\backslash rm\; int\}(x\_1)H\_\{\backslash rm\; int\}(x\_2)\backslash cdots\; H\_\{\backslash rm\; int\}(x\_n)\; \backslash right\backslash \}\backslash mid\backslash Psi\_\{\backslash rm\; in\}\backslash rangle.$

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

• Schwinger–Dyson equation
• Magnus series
• Peano–Baker series
• Picard iteration

{{reflist}}

• Charles J. Joachain, Quantum collision theory, North-Holland Publishing, 1975, {{ISBN|0-444-86773-2}} (Elsevier)

Category:Scattering theory

Category:Quantum field theory

Category:Freeman Dyson