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.[clarification needed]

Notice that in this article Planck units are used, so that ħ = 1 (where ħ is the reduced Planck constant).

The Dyson operator edit

Suppose that we have a Hamiltonian H, which we split into a free part H0 and an interacting part VS(t), i.e. H = H0 + VS(t).

We will work in the interaction picture here, that is,

 

where   is time-independent and   is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts,   stands for   in what follows. We choose units such that the reduced Planck constant ħ is 1.

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

 

is called the Dyson operator.

We have a few properties:

  • Identity and normalization:  [1]
  • Composition:  [2]
  • Time Reversal:  [clarification needed]
  • Unitarity:  [3]

and from these is possible to derive the time evolution equation of the propagator:[4]

 

We notice again that in the interaction picture the Hamiltonian is the same as the interaction potential  . This equation is not to be confused with the Tomonaga–Schwinger equation

Consequently:

 

which is ultimately a type of Volterra equation.

Derivation of the Dyson series edit

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

 

Here we have  , so we can say that the fields are time-ordered, and it is useful to introduce an operator   called time-ordering operator, defining

 

We can now try to make this integration simpler. In fact, by the following example:

 

Assume that K is symmetric in its arguments and define (look at integration limits):

 

The region of integration can be broken in   sub-regions defined by  ,  , etc. Due to the symmetry of K, the integral in each of these sub-regions is the same and equal to   by definition. So it is true that

 

Returning to our previous integral, the following identity holds

 

Summing up all the terms, we obtain the Dyson series which is a simplified version of the Neumann series above and which includes the time ordered products:[5]

 

This result is also called Dyson's formula.[6]

Application on state vectors edit

One can then express the state vector at time t in terms of the state vector at time t0, for t > t0,

 

Then, the inner product of an initial state (ti = t0) with a final state (tf = t) in the Schrödinger picture, for tf > ti, is as follows:

 

See also edit

References edit

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