In
scattering theory, a part of
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, the Dyson series, formulated by
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
, is a
perturbative
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for whi ...
expansion of the
time evolution operator
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
in the
interaction picture. Each term can be represented by a sum of
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s.
This series diverges
asymptotically, but in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) at the second order the difference from experimental
data
In the pursuit of knowledge, data (; ) is a collection of discrete Value_(semiotics), values that convey information, describing quantity, qualitative property, quality, fact, statistics, other basic units of meaning, or simply sequences of sy ...
is in the order of 10
−10. This close agreement holds because the coupling constant (also known as the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
) of
QED is much less than 1.
Notice that in this article
Planck units are used, so that ''ħ'' = 1 (where ''ħ'' is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
).
The Dyson operator
Suppose that we have a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, which we split into a ''free'' part and an ''interacting part'' , i.e. .
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
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
.
To avoid subscripts,
stands for
in what follows.
We choose units such that the reduced Planck constant is 1.
In the interaction picture, the evolution operator defined by the equation
:
is called the Dyson operator.
We have
:
:
:
and hence the time evolution equation of the propagator:
:
This is not to be confused with the
Tomonaga–Schwinger equation
Consequently:
:
Which is ultimately a type of
Volterra equation In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is
: f(t) = \int_a^t K(t,s)\,x(s) ...
Derivation of the Dyson series
This leads to the following
Neumann series A Neumann series is a mathematical series of the form
: \sum_^\infty T^k
where T is an operator and T^k := T^\circ its k times repeated application.
This generalizes the geometric series.
The series is named after the mathematician Carl Neumann ...
:
:
Here we have
, so we can say that the fields are
time-ordered
In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter:
:\mathcal P \left\
\equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_).
H ...
, and it is useful to introduce an operator
called ''
time-ordering
In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter:
:\mathcal P \left\
\equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_).
H ...
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 Dyson's theorem for the Dyson series:
:
Application on State Vectors
One can then express the state vector at time ''t'' in terms of the state vector at time ''t''
0, for ''t'' > ''t''
0,
:
Then, the inner product of an initial state (''t''
i = ''t''
0) with a final state (''t''
f = ''t'') in the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
, for ''t''
f > ''t''
i, is as follows:
:
See also
*
Schwinger–Dyson equation
The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Eulerâ ...
*
Magnus series In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it fur ...
*
Picard iteration
In numerical analysis, fixed-point iteration is a method of computing fixed point (mathematics), fixed points of a function.
More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of ...
References
*
Charles J. Joachain, ''Quantum collision theory'', North-Holland Publishing, 1975, {{ISBN, 0-444-86773-2 (Elsevier)
Scattering theory
Quantum field theory
Freeman Dyson