Propagator

Propagators are special Green functions , i.e. special solution functions of certain partial differential equations , as they occur in physics (e.g. in quantum electrodynamics ). Since propagators are singular at two points , they are also called two-point functions. They can be interpreted as the probability amplitude that a particle or a wave propagates from  x to  y , i.e. H. spreads, propagates or progresses. Depending on the differential equation with its boundary and initial conditions , different propagators result, for example the one-electron propagator. ${\displaystyle G}$

In Feynman diagrams , propagators are represented graphically and geometrically (but precisely) as lines (and vertices as nodes).

Quantum electrodynamics is the quantized form of a field theory , which contains a Maxwell and a Dirac field that are coupled to one another. Both the electron and the photon propagator are each represented by a 4 × 4 matrix , since the associated differential operators also consist of 4 × 4 matrices and the propagator or Green function and differential operator are reciprocal .

Schrödinger propagator

Within quantum mechanics , the time evolution is described by the time evolution operator , which in the case of a time-independent Hamilton operator is given by: ${\displaystyle U}$ ${\displaystyle H}$

${\displaystyle U(t;t_{0}):=e^{-{\frac {\mathrm {i} }{\hbar }}(t-t_{0})H}}$

The matrix elements of the time evolution operator

${\displaystyle G(x,t|x_{0},t_{0}):=\langle x|U(t;t_{0})|x_{0}\rangle ,}$

is also known as Green's function or (Schrödinger) propagator .

In Feynman's formulation of quantum mechanics with path integrals one finds the Feynman propagator , the normalization of which is chosen so that it agrees with the Schrödinger propagator. The propagator gives the probability amplitude one at the time, with localized particles at the time in to find. ${\displaystyle t_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle t}$ ${\displaystyle x}$

Second quantization

In the second quantized form , the Green function can also be written as

${\displaystyle G(x,t|x_{0},t_{0})=\langle {\hat {\psi }}(x,t)\;{\hat {\psi }}^{\dagger }(x_{0},t_{0})\rangle }$

where stands for the expected value of the ground state . This form can be transferred to many-particle quantum mechanics, whereby only the determination of the expected value may change ( solid-state physics , Feynman diagram ). ${\displaystyle \langle \cdots \rangle }$

Atomic and Nuclear Physics

In atomic and nuclear physics , the ground state in the system under consideration already contains real particles ( protons and neutrons or electrons ); there is also an additional external potential . In excited states , only the particles that are already present are lifted into energetically higher states of the existing potential.

Usually a propagator is used in the local area . Propagators often appear, which indicate the probability amplitude that a system contains an additional particle in the excited state at the beginning and in the excited state at the end : ${\displaystyle q}$ ${\displaystyle p}$

${\displaystyle G_{pq}(t,t'):=\langle 0\,|\,{\hat {T}}[{\hat {\psi }}_{p}(t)\;{\hat {\psi }}_{q}^{\dagger }(t')]\,|\,0\rangle }$

Here is

• ${\displaystyle |0\rangle }$ the basic state described above
• ${\displaystyle {\hat {T}}}$the timing operator
• ${\displaystyle {\hat {\psi }}_{p}(t)}$an operator that is currently destroying a particle in the state ${\displaystyle t}$ ${\displaystyle p}$
• ${\displaystyle {\hat {\psi }}_{q}^{\dagger }(t')}$an operator that currently creates a particle in the state . ${\displaystyle t'}$ ${\displaystyle q}$

Quantum field theory

In quantum field theory , the ground state is identical to the vacuum state : without real particles, but with vacuum fluctuations . At least for negligible coupling, an excited state differs from the ground state by the number of (real) particles; Particles are even interpreted as excitation states of the associated field.

Usually a propagator is 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 will move with a given energy and momentum ). The simplest example is the propagator for a scalar field whose excitations are particles with mass : ${\displaystyle m}$

${\displaystyle G(p)={\frac {\mathrm {i} }{p^{2}-m^{2}+\mathrm {i} \epsilon }}}$

Here is the quadruple momentum of the particle. ${\displaystyle p}$

Multi-particle propagators

In atomic and nuclear physics in particular, propagators are often used that describe the propagation of not just one, but several particles at the same time. One example of this is the polarization propagator .

A related concept are many-body Green functions ; but these describe i. A. not necessarily a propagation of particles, but more general concepts. For example, so-called three-point vertex functions are used to describe the interaction of an electron with a photon .

Individual evidence

1. ↑ "The entity called the kernel here is often called the“ propagator ”or the“ Green's function. ”" Quantum Mechanics and Path Integrals, Richard P. Feynman and Albert R. Hibbs, ISBN 0486134636 in the notes
2. ↑ Techniques and Applications of Path Integration, LS Schulman, Courier Dover Publications, 2012, ISBN 0486137023 , p. 3,4 Google Books