Short description: Mathematical objects describing the propagation of quantum fields and particles between space-time points

Propagators in quantum field theory describe how quantum fields and their excitations propagate between different points in space-time.^[1] They are central to the calculation of physical processes, particularly in perturbation theory and Feynman diagrams.

Propagation of a quantum field excitation between two space-time points

Definition

A propagator is a correlation function that gives the amplitude for a field to propagate from one point to another. For a scalar field, the time-ordered propagator is defined as: $D_{F} (x - y) = ⟨ 0 | T {ϕ (x) ϕ (y)} | 0 ⟩$

where:

$T$ denotes time ordering
$| 0 ⟩$ is the vacuum state

This function encodes how disturbances in the field travel through space-time.^[2]

Momentum-space representation

It is often useful to express propagators in momentum space: $D_{F} (p) = \frac{i}{p^{2} - m^{2} + i ϵ}$

where:

$p^{2} = p_{μ} p^{μ}$
$m$ is the mass of the particle
$i ϵ$ ensures proper boundary conditions

This form is widely used in calculations involving scattering amplitudes.^[1]

Physical interpretation

The propagator represents the probability amplitude for a particle to travel between two space-time points. However, in quantum field theory this is not a classical trajectory but a sum over all possible paths.

It can also be interpreted as describing the propagation of virtual particles that mediate interactions.^[3]

Role in Feynman diagrams

In perturbation theory, propagators appear as internal lines in Feynman diagrams.^[4]

Each internal line contributes a propagator factor, while vertices represent interactions. The full amplitude of a process is obtained by combining propagators and interaction terms according to specific rules.

Types of propagators

Different fields have different propagators:

Scalar propagator
Fermion propagator
Gauge boson propagator

For example, the fermion propagator is: $S_{F} (p) = \frac{i (γ^{μ} p_{μ} + m)}{p^{2} - m^{2} + i ϵ}$

These reflect the spin and internal structure of the corresponding particles.^[2]

Green's function interpretation

Mathematically, propagators are Green's functions of the field equations. For a scalar field: $(◻ + m^{2}) D_{F} (x - y) = - i δ^{(4)} (x - y)$

This shows that the propagator acts as the fundamental solution to the field equation.^[3]

Causality and time ordering

The time-ordering operator ensures that causality is preserved in relativistic quantum theory. Events are ordered such that operators at later times act first in the correlation function.

This structure ensures consistency with relativistic causality and quantum mechanics.^[1]

Conceptual importance

Propagators connect the abstract field formalism to measurable quantities. They provide the link between:

field operators
particle propagation
observable scattering processes

They are therefore one of the central computational and conceptual tools in quantum field theory.

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Physics:Quantum Propagators in quantum field theory

Definition

Momentum-space representation

Physical interpretation

Role in Feynman diagrams

Types of propagators

Green's function interpretation

Causality and time ordering

Conceptual importance

See also

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