Dyson series

Dyson operator

In the interaction picture, a Hamiltonian H, can be split into a free part H0 and an interacting part VS(t) as .

The potential in the interacting picture is :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}, where H_0 is time-independent and V_{\mathrm S}(t) is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, V(t) stands for V_\mathrm{I}(t) in what follows.

In the interaction picture, the evolution operator U is defined by the equation: :\Psi(t) = U(t,t_0) \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,
• 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^{\dagger}(t,t_0) U(t,t_0)=\mathbb{1} and from these is possible to derive the time evolution equation of the propagator: :i\hbar\frac d{dt} U(t,t_0)\Psi(t_0) = V(t) U(t,t_0)\Psi(t_0). In the interaction picture, the Hamiltonian is the same as the interaction potential H_{\rm int}=V(t) and thus the equation can also be written in the interaction picture as :i\hbar \frac d{dt} \Psi(t) = H_{\rm int}\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\hbar^{-1} \int_{t_0}^t{dt_1\ V(t_1)U(t_1,t_0)}, 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:

: \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 \cdots t_n, and so the fields are time-ordered. It is useful to introduce an operator \mathcal T, called the time-ordering operator, and to define

: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).

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 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).

and

: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).

The region of integration of the second integral can be broken in n! sub-regions, defined by t_1 t_2 \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 = \frac{1}{n!}I_n.

Applied to the previous identity, this gives

: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).

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{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. 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 tt_0, as

:|\Psi(t)\rangle=\sum_{n=0}^\infty {(-i\hbar^{-1})^n\over n!}\underbrace{\int dt_1 \cdots dt_n}{t,\ge, t_1,\ge, \cdots, \ge, t_n,\ge, t_0}, \mathcal{T}\left{\prod{k=1}^n e^{iH_0 (t_k-t_0)/\hbar}V_{\rm S}(t_{k})e^{-iH_0 (t_k-t_0)/\hbar}\right }|\Psi(t_0)\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_ft_i is:

:\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_1-t_i)/\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:

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

References

• Charles J. Joachain, Quantum collision theory, North-Holland Publishing, 1975, (Elsevier)

References

1. Sakurai, Modern Quantum mechanics, 2.1.10
2. Sakurai, Modern Quantum mechanics, 2.1.12
3. Sakurai, Modern Quantum mechanics, 2.1.11
4. Sakurai, Modern Quantum mechanics, 2.1 pp. 69-71
5. Sakurai, Modern Quantum Mechanics, 2.1.33, pp. 72
6. Tong 3.20, http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf
7. Dyson. (1949). "The S-matrix in quantum electrodynamics". Physical Review.