Propagator

Propagators are special Green's functions, ie functions of certain special solution ( partial ) differential equations as they occur in physics ( in quantum electrodynamics). As it can be interpreted as the amplitude probability that a particle propagated from x to y, which are also known as two-point propagators functions.

Depending on the differential equation with its boundary and initial conditions result in different propagators, for example, about the one-electron propagator. The term of the propagator comes from the fact that he has a propagation, ie propagation, propagation or progression of a particle or a wave describes. The famous Feynman diagrams are basically nothing more than a pictorial- geometric ( but accurate ) representation of propagators (lines) and vertices ( nodes ).

Quantum electrodynamics the quantized form of a field theory, each of which contains a Maxwell and a Dirac field, which are coupled together. Both electron and photon propagator are each represented by a 4x4 matrix, since the corresponding differential operators are also made of 4x4 matrices and propagator or Green's function, as well as differential operator are reciprocal to each other.

Schrödinger propagator

Within the time evolution of the quantum mechanics is described by the time evolution operator which is given in the case of a time-dependent Hamiltonian by:

The matrix elements of the time evolution operator

Also referred to as Green's function or ( Schrödinger ) propagator. In Feynman's formulation of quantum mechanics with path integrals to find the Feynman propagator, whose normalization is chosen just so that it matches with the Schrödinger propagator. Propagator provides the amplitude probability of finding a particle at the time when localized at the time.

Second quantization

In second quantized form, the Green function can also

Be written, where is the ground-state expectation value. This form is applicable to the Vielteilchenquantenmechanik, with only the determination of the expected value may change ( quantum field theory, solid state physics, Feynman diagram ).

What is essential is especially as the ground state is defined in the respective portion of interest in physics. In quantum field theory, the ground state is identical to the vacuum state: a state with no (real) particles (but with so-called vacuum fluctuations ). In the nuclear or atomic physics, however, the ground state already contains real particles (protons and neutrons and electrons, respectively ); also exists an additional external potential.

Another difference is in the excited states. In quantum field theory (at least for negligible coupling) to an excited state from the ground state is distinguished by the number of (real) particles; Particles are interpreted in quantum field theory even as excited states of the corresponding field. In atomic and nuclear physics, however, only the already existing particles are raised to higher energy states of the existing potential.

Both differences mean that in quantum field theory and atomic and nuclear physics distinctly different forms are used for the propagators. In contrast to the above propagator in position space usually a propagator is in quantum field theory used in momentum space (essentially the Fourier transform of the above expression with respect to space and time, it describes the probability amplitude that a particle with given energy and momentum moves ). The simplest example is the propagator for a so-called scalar field whose suggestions are particles with mass:

Here is the four-momentum of the particle.

In atomic and nuclear physics, however, often occur propagators, which indicate the probability amplitude that a system contains an additional particle in the excited state and in the excited state at the end of the beginning.

Here, the reason the above-described condition, an operator at the time of a particle in the destroyed state is an operator that produces a particle in the state at the time of order and the time operator.

Many-particle propagators

It should be pointed out briefly that, especially in atomic and nuclear physics often propagators are used which describe the propagation of not only one, but several particles simultaneously. An example would be the so-called polarization propagator.

A related concept is the so-called many-body Green's functions; but these describe A., not necessarily a spread of particles, but more general concepts ( for example, serve so-called three - point vertex functions describing the interaction of an electron with a photon).