Editing Propagator (section)

=== Causal propagators ===

==== Retarded propagator ====
[[Image:CausalRetardedPropagatorPath.svg]]

A contour going clockwise over both poles gives the '''causal retarded propagator'''. This is zero if {{mvar|x-y}} is spacelike or {{mvar|y}} is to the future of {{mvar|x}}, so it is zero if {{math|''x'' ⁰< ''y'' ⁰}}.

This choice of contour is equivalent to calculating the [[Limit (mathematics)|limit]],
<math display="block">G_\text{ret}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0+i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(x^0 - y^0)}{2\pi} \delta(\tau_{xy}^2) + \Theta(x^0 - y^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}.</math>

Here 
<math display="block">\Theta (x) := \begin{cases}
1 & x \ge 0 \\
0 & x < 0
\end{cases}</math>
is the [[Heaviside step function]],
<math display="block">\tau_{xy}:= \sqrt{ (x^0 - y^0)^2 - (\vec{x} - \vec{y})^2}</math>
is the [[proper time]] from {{mvar|x}} to {{mvar|y}}, and <math>J_1</math> is a [[Bessel function of the first kind]]. The propagator is non-zero only if <math>y \prec x</math>, i.e., {{mvar|y}} [[causal structure|causally precedes]] {{mvar|x}}, which, for Minkowski spacetime, means
:<math>y^0 \leq x^0</math> and <math>\tau_{xy}^2 \geq 0 ~.</math>

This expression can be related to the [[vacuum expectation value]] of the [[commutator]] of the free scalar field operator,
<math display="block">G_\text{ret}(x,y) = -i \langle 0| \left[ \Phi(x), \Phi(y) \right] |0\rangle \Theta(x^0 - y^0),</math>
where 
<math display="block">\left[\Phi(x), \Phi(y) \right] := \Phi(x) \Phi(y) - \Phi(y) \Phi(x).</math>

==== Advanced propagator ====
[[Image:CausalAdvancedPropagatorPath.svg]]

A contour going anti-clockwise under both poles gives the '''causal advanced propagator'''. This is zero if {{mvar|x-y}} is spacelike or if {{mvar|y}} is to the past of {{mvar|x}}, so it is zero if {{math|''x'' ⁰> ''y'' ⁰}}.

This choice of contour is equivalent to calculating the limit<ref>{{cite book |last1=Scharf |first1=Günter |title=Finite Quantum Electrodynamics, The Causal Approach |date=13 November 2012 |publisher=Springer |isbn=978-3-642-63345-4 |pages=89}}</ref>
<math display="block">
G_\text{adv}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2\pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0 - i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(y^0-x^0)}{2\pi}\delta(\tau_{xy}^2) + \Theta(y^0-x^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}.
</math>

This expression can also be expressed in terms of the [[vacuum expectation value]] of the [[commutator]] of the free scalar field.
In this case,
<math display="block">G_\text{adv}(x,y) = i \langle 0|\left[ \Phi(x), \Phi(y) \right]|0\rangle \Theta(y^0 - x^0)~.</math>

====Feynman propagator====
[[Image:FeynmanPropagatorPath.svg]]

A contour going under the left pole and over the right pole gives the '''Feynman propagator''', introduced by [[Richard Feynman]] in 1948.<ref>{{Citation |last=Feynman |first=R. P. |title=Space-Time Approach to Non-Relativistic Quantum Mechanics |url=http://www.worldscientific.com/doi/abs/10.1142/9789812567635_0002 |work=Feynman's Thesis — A New Approach to Quantum Theory |year=2005 |pages=71–109 |publisher=WORLD SCIENTIFIC |language=en |doi=10.1142/9789812567635_0002 |bibcode=2005ftna.book...71F |isbn=978-981-256-366-8 |access-date=2022-08-17}}</ref>

This choice of contour is equivalent to calculating the limit<ref>{{cite book |last=Huang |first=Kerson |title=Quantum Field Theory: From Operators to Path Integrals |publisher=John Wiley & Sons |year=1998 |isbn=0-471-14120-8 |location=New York |page=30 |author-link=Kerson Huang}}</ref>   
<math display="block">G_F(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{p^2 -  m^2 + i\varepsilon} = \begin{cases}
-\frac{1}{4 \pi} \delta(\tau_{xy}^2) + \frac{m}{8 \pi \tau_{xy}} H_1^{(1)}(m \tau_{xy}) & \tau_{xy}^2 \geq 0 \\ -\frac{i m}{ 4 \pi^2 \sqrt{-\tau_{xy}^2}} K_1(m \sqrt{-\tau_{xy}^2}) & \tau_{xy}^2 < 0.
\end{cases} </math>

Here, {{math|''H''<sub>1</sub><sup>(1)</sup>}} is a [[Bessel function#Hankel functions|Hankel function]] and {{math|''K''<sub>1</sub>}} is a [[Bessel function#Modified Bessel functions: I.CE.B1.2C K.CE.B1|modified Bessel function]].

This expression can be derived directly from the field theory as the [[vacuum expectation value]] of the ''[[time-ordered]] product'' of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same,
<math display="block">
\begin{align}
G_F(x-y) & = -i \lang 0|T(\Phi(x) \Phi(y))|0 \rang \\[4pt]
& = -i \left \lang 0| \left [\Theta(x^0 - y^0) \Phi(x)\Phi(y) + \Theta(y^0 - x^0) \Phi(y)\Phi(x) \right] |0 \right \rang.
\end{align}</math>

This expression is [[Lorentz invariant]], as long as the field operators commute with one another when the points {{mvar|x}} and {{mvar|y}} are separated by a [[spacelike]] interval.

The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, and then to show that the {{math|Θ}} functions providing the causal time ordering may be obtained by a [[line integral|contour integral]] along the energy axis, if the integrand is as above (hence the infinitesimal imaginary part), to move the pole off the real line.

The propagator may also be derived using the [[path integral formulation]] of quantum theory.

==== Dirac propagator ====
Introduced by [[Paul Dirac]] in 1938.<ref>{{Cite journal |date=1938-08-05 |title=Classical theory of radiating electrons |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1938.0124 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |language=en |volume=167 |issue=929 |pages=148–169 |doi=10.1098/rspa.1938.0124 |s2cid=122020006 |issn=0080-4630|url-access=subscription }}</ref><ref>{{Cite web |title=Dirac propagator in nLab |url=https://ncatlab.org/nlab/show/Dirac+propagator |access-date=2023-11-08 |website=ncatlab.org}}</ref>