Dyson series explained

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, can be split into a free part and an interacting part as .

The potential in the interacting picture is

V _{I} (t) = e ^{i H _{0} (t - t _{0} )/\hbar} V _{S} (t) e ^{- i H _{0} (t - t _{0} )/\hbar} , 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 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 , ^[1]

• Composition:

U(t,t _{0)} = U(t,t _{1)} U(t _{1,t} _{0),} ^[2]

• Time Reversal:

U ^{-1} (t,t _{0)} = U(t _{0,t),}

• Unitarity:

U ^{\dagger} (t,t _{0)} U(t,t _{0)=1} ^[3] and from these is possible to derive the time evolution equation of the propagator:^[4] i\hbar (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} = V(t) and thus the equation can also be written in the interaction picture asi\hbar (d{dt})/(\Psi(t)) = H _{\rm} \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{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} _{t _{0}} dt _{1V(t} _{1)} + ( - i\hbar ^{-1} ) ^{t} _{t _{0}} dt _{1} \int ^{t _{1}} _{t _{0}} dt _{2} V(t _{1)V(t} _{2)+ …} \\ & {} + ( - i\hbar ^{-1} ) ^{t} _{t _{0}} dt _{1\int} ^{t _{1}} ^{t _{0}} dt _{2} … \int ^{t _{n-1}} _{t _{0}} dt _{nV(t} _{1)V(t} _{2)} … V(t _{n)} + … . \end{align}

Here,

t _{1} > t _{2} > … > t _{n} , and so the fields are time-ordered. It is useful to introduce an operator l T , called the time-ordering operator, and to defineU _{n(t,t} ^{-1} ^{0 ) = ( - i\hbar} ) ^{n} \int ^{t} _{t _{0}} dt _{1} \int ^{t _{1}} _{t _{0}} dt _{2} … \int ^{t _{n-1}} _{t _{0}} dt _{nlTV(t} _{1)} V(t _{2) …} V(t _{n).}

The limits of the integration can be simplified. In general, given some symmetric function

K(t _{1,} t _{2,...,t} _{n),} one may define the integralsS _{n=\int} ^{t} ^{t _{0}} dt _{1\int} ^{t _{1}} ^{t _{0}} dt _{2 …} \int ^{t _{n-1}} _{t _{0}} dt _{n} K(t _{1,} t _{2,...,t} _{n).}

and

I _{n=\int} ^{t} ^{t _{0}} dt _{1\int} ^{t} ^{t _{0}} dt _{2 … \int} ^{t} ^{t _{0}} dt _{nK(t} _{1,} t _{2,...,t} _{n).}

The region of integration of the second integral can be broken in

n! sub-regions, defined by t _{1} > t _{2} > … > 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 thatS _{n} = (1)/(n!) I _{n.}

Applied to the previous identity, this gives

U _{n = (( - i \hbar ^{-1} ) ^{n})/(n!)} \int ^{t} _{t _{0}} dt _{1\int} ^{t} ^{t _{0}} dt _{2 … \int} ^{t} ^{t _{0}} dt _{n} l TV(t _{1)V(t} _{2) …} 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:^[5]

\begin{align} U(t,t _{0)&=\sum} ^{inf ty} ^{n =0} U _{n(t,t} _{0)\\ &=\sum} ^{inf ty} ^{n =0} (( - i\hbar ^{-1} ) ^{n})/(n!) \int ^{t} _{t _{0}} dt _{1\int} ^{t} ^{t _{0}} dt _{2 … \int} ^{t} ^{t _{0}} dt _{n} l TV(t _{1)V(t} _{2) …} V(t _{n)} \\ & = l T\exp{ - i\hbar ^{-1} \int ^{t{d\tau} _{t _{0}} V(\tau)}} \end{align}

This result is also called Dyson's formula.^[6] 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|\Psi(t)\rangle = \sum ^{inf ty} _{n =0} {( - i\hbar ^{-1} ) ^{n\over} n!}\underbrace{\int dt _{1} … dt _{n}} ^{t _{\rm} \ge t _{1\ge} … \ge t _{n\ge} t _{\rm}} l{T}\left\{\prod ^{n} _{k =1} e ^{iH _{0} t _{k/\hbar}} V(t _{k} )e ^{- iH _{0} t _{k/\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 _{f>t} _{i} is:\begin{align} \langle\Psi(t _{\rm} ) & \mid\Psi(t _{\rm} )\rangle = \sum ^{inf ty} _{n =0} {( - i\hbar ^{-1} ) ^{n\over} n!} x \ &\underbrace{\int dt _{1} … dt _{n}} ^{t _{\rm} \ge t _{1\ge} … \ge t _{n\ge} t _{\rm}} \langle\Psi(t _{i)\mid} e ^{- iH _{0(t} _{\rm} - t _{1)/\hbar}} V _{\rm} (t ^{- iH _{0(t} _{1-t} _{2)/\hbar}} _{1 )e} … V _{\rm} (t _{n)} e ^{- iH _{0(t} _{n-t} _{\rm} )/\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} \mid S\mid\Psi _{\rm} \rangle = \langle\Psi _{\rm} \mid\sum ^{inf ty} _{n =0} {( - i\hbar ^{-1} ) ^{n\over} n!} \underbrace{\int d ^{4 x} _{1} … d ^{4 x} _{n}} ^{t _{\rm} \ge t _{n\ge} … \ge t _{1\ge} t _{\rm}} l{T}\left\{ H _{\rm} (x _{1)H} _{\rm} (x _{2) …} H _{\rm} (x _{n)} \right\}\mid\Psi _{\rm} \rangle. 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, (Elsevier)

Notes and 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