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

{{Quantum book backlink|Quantum field theory}}
'''Propagators in quantum field theory''' describe how quantum fields and their excitations propagate between different points in space-time.<ref name="peskin">Peskin, M. E.; Schroeder, D. V. ''An Introduction to Quantum Field Theory'' (1995).</ref> They are central to the calculation of physical processes, particularly in perturbation theory and Feynman diagrams.

<div style="float:right; border:1px solid #ccc; padding:4px; background:#fff8dc; margin:0 0 1em 1em; width:420px;">
[[File:Quantum_field_propagator.jpg|400px]]
<div style="font-size:90%;">Propagation of a quantum field excitation between two space-time points</div>
</div>

== 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:
<math>
D_F(x - y) = \langle 0 | T\{\phi(x)\phi(y)\} | 0 \rangle
</math>

where:
* <math>T</math> denotes time ordering
* <math>|0\rangle</math> is the vacuum state

This function encodes how disturbances in the field travel through space-time.<ref name="schwartz">Schwartz, M. D. ''Quantum Field Theory and the Standard Model'' (2014).</ref>

== Momentum-space representation ==
It is often useful to express propagators in momentum space:
<math>
D_F(p) = \frac{i}{p^2 - m^2 + i\epsilon}
</math>

where:
* <math>p^2 = p_\mu p^\mu</math>
* <math>m</math> is the mass of the particle
* <math>i\epsilon</math> ensures proper boundary conditions

This form is widely used in calculations involving scattering amplitudes.<ref name="peskin"/>

== 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.<ref name="weinberg">Weinberg, S. ''The Quantum Theory of Fields'' (1995).</ref>

== Role in Feynman diagrams ==
In perturbation theory, propagators appear as internal lines in Feynman diagrams.<ref name="feynman">Feynman, R. P. (1949). Space-time approach to quantum electrodynamics.</ref>

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:
<math>
S_F(p) = \frac{i(\gamma^\mu p_\mu + m)}{p^2 - m^2 + i\epsilon}
</math>

These reflect the spin and internal structure of the corresponding particles.<ref name="schwartz"/>

== Green's function interpretation ==
Mathematically, propagators are Green's functions of the field equations. For a scalar field:
<math>
(\Box + m^2) D_F(x - y) = -i \delta^{(4)}(x - y)
</math>

This shows that the propagator acts as the fundamental solution to the field equation.<ref name="weinberg"/>

== 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.<ref name="peskin"/>

== 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.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum field theory (QFT) core|1}}

Physics:Quantum Propagators in quantum field theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template