Editing Propagator

{{Short description|Function in quantum field theory showing probability amplitudes of moving particles}}
{{about|time evolution in [[quantum field theory]]|propagation of plants|Plant propagation}}
{{Use American English|date=January 2019}}{{Quantum field theory}}
In [[quantum mechanics]] and [[quantum field theory]], the '''propagator''' is a function that specifies the [[probability amplitude]] for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In [[Feynman diagram]]s, which serve to calculate the rate of collisions in [[quantum field theory]], [[virtual particle]]s contribute their propagator to the rate of the [[scattering]] event described by the respective diagram. Propagators may also be viewed as the [[inverse operation|inverse]] of the [[wave operator]] appropriate to the particle, and are, therefore, often called ''(causal) [[Green's function (many-body theory)|Green's functions]]'' (called "''causal''" to distinguish it from the elliptic Laplacian Green's function).<ref>[http://www.mathtube.org/sites/default/files/lecture-notes/Lamoureux_Michael.pdf The mathematics of PDEs and the wave equation], p 32., Michael P. Lamoureux, University of Calgary, Seismic Imaging Summer School,  August 7–11, 2006, Calgary.</ref><ref>[http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part4.pdf Ch.: 9 Green's functions], p 6., J Peacock, FOURIER ANALYSIS LECTURE COURSE: LECTURE 15.</ref>

==Non-relativistic propagators==
In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a [[Elementary particle|particle]] to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).

The [[Green's function]] G for the [[Schrödinger equation]] is a function
<math display="block">G(x, t; x', t') = \frac{1}{i\hbar} \Theta(t - t') K(x, t; x', t')</math>
satisfying
<math display="block">\left( i\hbar \frac{\partial}{\partial t} - H_x \right) G(x, t; x', t') = \delta(x - x') \delta(t - t'),</math>
where {{math|''H''}} denotes the [[Hamiltonian (quantum mechanics)|Hamiltonian]], {{math|''δ''(''x'')}} denotes the [[Dirac delta-function]] and {{math|Θ(''t'')}} is the [[Heaviside step function]]. The [[Integral transform|kernel]] of the above Schrödinger differential operator in the big parentheses is denoted by {{math|''K''(''x'', ''t'' ;''x′'', ''t′'')}} and called the '''propagator'''.<ref group=nb> While the term propagator sometimes refers to {{mvar|G}} as well, this article will use the term to refer to {{mvar|K}}.</ref>

This propagator may also be written as the transition amplitude
<math display="block">K(x, t; x', t') = \big\langle x \big| U(t, t') \big| x' \big\rangle,</math>
where {{math|''U''(''t'', ''t′'')}} is the [[unitary operator|unitary]] time-evolution operator for the system taking states at time {{mvar|t′}} to states at time {{mvar|t}}.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=314,337}} Note the initial condition enforced by
<math display="block">\lim_{t \to t'} K(x, t; x', t') = \delta(x - x').</math>
The propagator may also be found by using a [[Path_integral_formulation#Path_integral_in_quantum_mechanics|path integral]]:
: <math>K(x, t; x', t') = \int \exp \left[\frac{i}{\hbar} \int_{t'}^{t} L(\dot{q}, q, t) \, dt\right] D[q(t)],</math>
where {{mvar|L}} denotes the [[Lagrangian mechanics|Lagrangian]] and the boundary conditions are given by {{math|''q''(''t'') {{=}} ''x'', ''q''(''t′'') {{=}} ''x′''}}. The paths that are summed over move only forwards in time and are integrated with the differential <math>D[q(t)]</math> following the path in time.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|p=2273}}

The propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is given by
: <math>\psi(x, t) = \int_{-\infty}^\infty \psi(x', t') K(x, t; x', t') \, dx'.</math>
If {{math|''K''(''x'', ''t''; ''x''&prime;, ''t''&prime;)}} only depends on the difference {{math|''x'' − ''x′''}}, this is a [[convolution]] of the initial wave function and the propagator.

===Examples===
{{see also|Path integral formulation#Simple harmonic oscillator| Heat equation#Fundamental solutions}}
For a time-translationally invariant system, the propagator only depends on the time difference {{math|''t'' − ''t''′}}, so it may be rewritten as
<math display="block">K(x, t; x', t') = K(x, x'; t - t').</math>

The [[Wave packet#Free propagator|propagator of a one-dimensional free particle]], obtainable from, e.g., the [[Path integral formulation#Free particle|path integral]], is then
{{Equation box 1
|indent = :
|equation = <math>K(x, x'; t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} dk\, e^{ik(x-x')} e^{-\frac{i\hbar k^2 t}{2m}} = \left(\frac{m}{2\pi i\hbar t}\right)^{\frac{1}{2}} e^{-\frac{m(x-x')^2}{2i\hbar t}}.</math>
|border colour = #0073CF
|bgcolor = #F9FFF7}}

Similarly, the propagator of a one-dimensional [[Quantum harmonic oscillator#Natural length and energy scales|quantum harmonic oscillator]] is the [[Mehler kernel]],<ref>E. U. Condon, [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf "Immersion of the Fourier transform in a continuous group of functional transformations"], ''Proc. Natl. Acad. Sci. USA'' '''23''', (1937) 158–164.</ref><ref>[[Wolfgang Pauli]], ''Wave Mechanics: Volume 5 of Pauli Lectures on Physics'' (Dover Books on Physics, 2000) {{ISBN|0486414620}}. Section 44.</ref> 
{{Equation box 1
|indent = :
|equation = <math>K(x, x'; t) = \left(\frac{m\omega}{2\pi i\hbar \sin \omega t}\right)^{\frac{1}{2}} \exp\left(-\frac{m\omega\big((x^2 + x'^2) \cos\omega t - 2xx'\big)}{2i\hbar \sin\omega t}\right).</math>
|border colour = #0073CF
|bgcolor = #F9FFF7}}
The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,<ref>Kolsrud, M. (1956). Exact quantum dynamical solutions for oscillator-like systems, ''Physical Review'' '''104'''(4), 1186.</ref>
<math display="block">\begin{align}
&\exp \left( -\frac{it}{\hbar} \left( \frac{1}{2m} \mathsf{p}^2 + \frac{1}{2} m\omega^2 \mathsf{x}^2 \right) \right) \\
&= \exp \left( -\frac{im\omega}{2\hbar} \mathsf{x}^2\tan\frac{\omega t}{2} \right) \exp \left( -\frac{i}{2m\omega \hbar}\mathsf{p}^2 \sin(\omega t) \right) \exp \left( -\frac{im\omega }{2\hbar} \mathsf{x}^2 \tan\frac{\omega t}{2} \right),
\end{align}</math>
valid for operators <math>\mathsf{x}</math> and <math>\mathsf{p}</math> satisfying the [[Canonical_commutation_relation|Heisenberg relation]] <math>[\mathsf{x},\mathsf{p}] = i\hbar</math>.

For the {{mvar|N}}-dimensional case, the propagator can be simply obtained by the product
<math display="block">K(\vec{x}, \vec{x}'; t) = \prod_{q=1}^N K(x_q, x_q'; t).</math>

==Relativistic propagators==
In [[relativistic quantum mechanics]] and [[quantum field theory]] the propagators are [[Lorentz-invariant]]. They give the amplitude for a [[Elementary particle|particle]] to travel between two [[spacetime]] events.

===Scalar propagator===
In quantum field theory, the theory of a free (or non-interacting) [[scalar field]] is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes [[Spin (physics)|spin]]-zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.

=== Position space ===
The position space propagators are [[Green's function]]s for the [[Klein–Gordon equation]]. This means that they are functions {{math|''G''(''x'', ''y'')}} satisfying
<math display="block">\left(\square_x + m^2\right) G(x, y) = -\delta(x - y),</math>
where
* {{mvar|x, y}} are two points in [[Minkowski spacetime]],
* <math>\square_x = \tfrac{\partial^2}{\partial t^2} - \nabla^2</math> is the [[d'Alembertian]] operator acting on the {{mvar|x}} coordinates,
* {{math|''δ''(''x'' − ''y'')}} is the [[Dirac delta function]].

(As typical in [[special relativity|relativistic]] quantum field theory calculations, we use units where the [[speed of light]] {{mvar|c}} and the [[reduced Planck constant]] {{mvar|ħ}} are set to unity.)

We shall restrict attention to 4-dimensional [[Minkowski spacetime]]. We can perform a [[Fourier transform]] of the equation for the propagator, obtaining
<math display="block">\left(-p^2 + m^2\right) G(p) = -1.</math>

This equation can be inverted in the sense of [[Distribution (mathematics)|distributions]], noting that the equation {{math|1=''xf''(''x'') = 1}} has the solution (see [[Sokhotski–Plemelj theorem]])
<math display="block">f(x) = \frac{1}{x \pm i\varepsilon} = \frac{1}{x} \mp i\pi\delta(x),</math>
with {{mvar|ε}} implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.

The solution is
{{Equation box 1
|indent = :
|equation = <math>G(x, y) = \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{p^2 - m^2 \pm i\varepsilon},</math>
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where 
<math display="block">p(x - y) := p_0(x^0 - y^0) - \vec{p} \cdot (\vec{x} - \vec{y})</math> 
is the [[4-vector]] inner product.

The different choices for how to deform the [[Methods of contour integration|integration contour]] in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the <math>p_0</math> integral.

The integrand then has two poles at 
<math display="block">p_0 = \pm \sqrt{\vec{p}^2 + m^2},</math> 
so different choices of how to avoid these lead to different propagators.

=== 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>

===Momentum space propagator===
The [[Fourier transform]] of the position space propagators can be thought of as propagators in [[momentum space]]. These take a much simpler form than the position space propagators.

They are often written with an explicit {{mvar|ε}} term although this is understood to be a reminder about which integration contour is appropriate (see above). This {{mvar|ε}} term is included to incorporate boundary conditions and [[causality]] (see below).

For a [[4-momentum]] {{mvar|p}} the causal and Feynman propagators in momentum space are:

:<math>\tilde{G}_\text{ret}(p) = \frac{1}{(p_0+i\varepsilon)^2 - \vec{p}^2 - m^2}</math>
:<math>\tilde{G}_\text{adv}(p) = \frac{1}{(p_0-i\varepsilon)^2 - \vec{p}^2 - m^2}</math>
:<math>\tilde{G}_F(p) = \frac{1}{p^2 -  m^2 + i\varepsilon}. </math>

For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of {{mvar|i}} (conventions vary).

===Faster than light?===
{{More citations needed section|date=November 2022}}
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is ''nonzero'' outside of the [[light cone]], though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle travelling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?

The answer is no: while in [[classical mechanics]] the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is [[commutator]]s that determine which operators can affect one another.

So what ''does'' the spacelike part of the propagator represent? In QFT the [[vacuum]] is an active participant, and [[particle number]]s and field values are related by an [[uncertainty principle]]; field values are uncertain even for particle number ''zero''. There is a nonzero [[probability amplitude]] to find a significant fluctuation in the vacuum value of the field {{math|Φ(''x'')}} if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an [[EPR paradox|EPR correlation]].  Indeed, the propagator is often called a ''two-point correlation function'' for the [[free field]].

Since, by the postulates of quantum field theory, all [[observable]] operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.

Regarding virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-[[antiparticle]] pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum.  In [[Richard Feynman|Feynman]]'s language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone.  However, no signaling back in time is allowed.

====Explanation using limits====
This can be made clearer by writing the propagator in the following form for a massless particle:
<math display="block">G^\varepsilon_F(x, y) = \frac{\varepsilon}{(x - y)^2 + i \varepsilon^2}.</math>

This is the usual definition but normalised by a factor of <math>\varepsilon</math>. Then the rule is that one only takes the limit <math>\varepsilon \to 0</math> at the end of a calculation.

One sees that 
<math display="block">G^\varepsilon_F(x, y) = \frac{1}{\varepsilon} \quad\text{if}~~~ (x - y)^2 = 0,</math>
and
<math display="block">\lim_{\varepsilon \to 0} G^\varepsilon_F(x, y) = 0 \quad\text{if}~~~ (x - y)^2 \neq 0.</math>
Hence this means that a single massless particle will always stay on the light cone. It is also shown that the total probability for a photon at any time must be normalised by the reciprocal of the following factor:
<math display="block">
\lim_{\varepsilon \to 0} \int |G^\varepsilon_F(0, x)|^2 \, dx^3 
 = \lim_{\varepsilon \to 0} \int \frac{\varepsilon^2}{(\mathbf{x}^2 - t^2)^2 + \varepsilon^4} \, dx^3
 = 2 \pi^2 |t|.
</math>
We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.

===Propagators in Feynman diagrams===
The most common use of the propagator is in calculating [[probability amplitude]]s for particle interactions using [[Feynman diagram]]s.  These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every ''internal line'', that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's [[Lagrangian (field theory)|Lagrangian]] for every internal vertex where lines meet.  These prescriptions are known as ''Feynman rules''.

Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be [[off shell]]. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on shell.

The energy carried by the particle in the propagator can even be ''negative''. This can be interpreted simply as the case in which, instead of a particle going one way, its [[antiparticle]] is going the ''other'' way, and therefore carrying an opposing flow of positive energy.  The propagator encompasses both possibilities.  It does mean that one has to be careful about minus signs for the case of [[fermions]], whose propagators are not [[even function]]s in the energy and momentum (see below).

Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed ''loop'', the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of [[renormalization]].

===Other theories===
==== Spin {{frac|1|2}} ====
If the particle possesses [[Spin (physics)|spin]] then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The differential equation satisfied by the propagator for a spin {{frac|1|2}} particle is given by<ref>{{harvnb|Greiner|Reinhardt|2008|loc=Ch.2}}</ref>

:<math>(i\not\nabla' - m)S_F(x', x) = I_4\delta^4(x'-x),</math>

where {{math|''I''<sub>4</sub>}} is the unit matrix in four dimensions, and employing the [[Feynman slash notation]]. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation,
<math display="block">S_F(x', x) = \int\frac{d^4p}{(2\pi)^4}\exp{\left[-ip \cdot(x'-x)\right]}\tilde S_F(p),</math>
the equation becomes

: <math>
\begin{align}
& (i \not \nabla' - m)\int\frac{d^4p}{(2\pi)^4}\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt]
= {} & \int\frac{d^4p}{(2\pi)^4}(\not p - m)\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt]
= {} & \int\frac{d^4p}{(2\pi)^4}I_4\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt]
= {} & I_4\delta^4(x'-x),
\end{align}
</math>

where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus

:<math>(\not p - m I_4)\tilde S_F(p) = I_4.</math>

By multiplying from the left with
<math display="block">(\not p + m)</math>
(dropping unit matrices from the notation) and using properties of the [[gamma matrices]],
<math display="block">\begin{align}
\not p \not p & = \tfrac{1}{2}(\not p \not p + \not p \not p) \\[6pt]
& = \tfrac{1}{2}(\gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) \\[6pt]
& = \tfrac{1}{2}(\gamma_\mu  \gamma_\nu + \gamma_\nu\gamma_\mu)p^\mu p^\nu \\[6pt]
& = g_{\mu\nu}p^\mu p^\nu = p_\nu p^\nu  = p^2,
\end{align}</math>

the momentum-space propagator used in Feynman diagrams for a [[Dirac equation|Dirac]] field representing the [[electron]] in [[quantum electrodynamics]] is found to have form

:<math> \tilde{S}_F(p) = \frac{(\not p + m)}{p^2 - m^2 + i \varepsilon} = \frac{(\gamma^\mu p_\mu + m)}{p^2 - m^2 + i \varepsilon}.</math>

The {{math|''iε''}} downstairs is a prescription for how to handle the poles in the complex {{math|''p''<sub>0</sub>}}-plane. It automatically yields the [[Feynman propagator|Feynman contour of integration]] by shifting the poles appropriately. It is sometimes written

:<math>\tilde{S}_F(p) = {1 \over \gamma^\mu p_\mu - m + i\varepsilon} = {1 \over \not p - m + i\varepsilon} </math>

for short. It should be remembered that this expression is just shorthand notation for {{math|(''γ''<sub>''μ''</sub>''p''<sup>''μ''</sup> − ''m'')<sup>−1</sup>}}. "One over matrix" is otherwise nonsensical. In position space one has
<math display="block">S_F(x-y) = \int \frac{d^4 p}{(2\pi)^4} \, e^{-i p \cdot (x-y)} \frac{\gamma^\mu p_\mu + m}{p^2 - m^2 + i \varepsilon} = \left( \frac{\gamma^\mu (x-y)_\mu}{|x-y|^5} + \frac{m}{|x-y|^3} \right) J_1(m |x-y|).</math>

This is related to the Feynman propagator by

:<math>S_F(x-y) = (i \not \partial + m) G_F(x-y)</math>

where <math>\not \partial := \gamma^\mu \partial_\mu</math>.

==== Spin 1 ====
The propagator for a [[gauge boson]] in a [[gauge theory]] depends on the choice of convention to fix the gauge. For the gauge used by Feynman and [[Ernst Stueckelberg|Stueckelberg]], the propagator for a [[photon]] is
:<math>{-i g^{\mu\nu} \over p^2 + i\varepsilon }.</math>

The general form with gauge parameter {{math|''λ''}}, up to overall sign and the factor of <math>i</math>, reads
:<math> -i\frac{g^{\mu\nu} + \left(1-\frac{1}{\lambda}\right)\frac{p^\mu p^\nu}{p^2}}{p^2+i\varepsilon}.</math>

The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter {{math|''λ''}}, up to overall sign and the factor of <math>i</math>, reads
:<math> \frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}}{k^2-m^2+i\varepsilon}+\frac{\frac{k_\mu k_\nu}{m^2}}{k^2-\frac{m^2}{\lambda}+i\varepsilon}.</math>

With these general forms one obtains the propagators in unitary gauge for {{math|''λ'' {{=}} 0}}, the propagator in Feynman or 't Hooft gauge for {{math|''λ'' {{=}} 1}} and in Landau or Lorenz gauge for {{math|''λ'' {{=}} ∞}}. There are also other notations where the gauge parameter is the inverse of {{mvar|λ}}, usually denoted {{mvar|ξ}} (see [[Gauge fixing#Rξ gauges|{{math|''R''<sub>ξ</sub>}} gauges]]). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.

Unitary gauge:
:<math>\frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}}{k^2-m^2+i\varepsilon}.</math>

Feynman ('t Hooft) gauge:
:<math>\frac{g_{\mu\nu}}{k^2-m^2+i\varepsilon}.</math>

Landau (Lorenz) gauge:
:<math>\frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}}{k^2-m^2+i\varepsilon}.</math>

===Graviton propagator===
The graviton propagator for [[Minkowski space]] in [[general relativity]] is <ref>[https://dspace.library.uu.nl/bitstream/handle/1874/4837/Quantum_theory_of_gravitation.pdf?sequence=2&isAllowed=y Quantum theory of gravitation] library.uu.nl</ref>
<math display="block">G_{\alpha\beta~\mu\nu} = \frac{\mathcal{P}^2_{\alpha\beta~\mu\nu}}{k^2} - \frac{\mathcal{P}^0_s{}_{\alpha\beta~\mu\nu}}{2k^2} = \frac{g_{\alpha\mu} g_{\beta\nu}+ g_{\beta\mu}g_{\alpha\nu}- \frac{2}{D-2} g_{\mu\nu}g_{\alpha\beta}}{k^2},</math>
where <math>D</math> is the number of spacetime dimensions, <math>\mathcal{P}^2</math> is the transverse and traceless [[Spin (physics)#Spin projection quantum number and multiplicity|spin-2 projection operator]] and <math>\mathcal{P}^0_s</math> is a spin-0 scalar [[multiplet]]. 
The graviton propagator for [[Anti-de Sitter space|(Anti) de Sitter space]] is 
<math display="block">G = \frac{\mathcal{P}^2}{2H^2-\Box} + \frac{\mathcal{P}^0_s}{2(\Box+4H^2)},</math>
where <math>H</math> is the [[Hubble's law|Hubble constant]]. Note that upon taking the limit <math>H \to 0</math> and <math>\Box \to -k^2</math>, the AdS propagator reduces to the Minkowski propagator.<ref>{{cite web| url=https://cds.cern.ch/record/378516/files/9902042.pdf |title=Graviton and gauge boson propagators in AdSd+1}}</ref>

==Related singular functions==
{{further|Green's function (many-body theory)|Correlation function (quantum field theory)}}
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in [[quantum field theory]].  These functions are most simply defined in terms of the [[vacuum expectation value]] of products of field operators.

===Solutions to the Klein–Gordon equation===
====Pauli&ndash;Jordan function====
The commutator of two scalar field operators defines the [[Wolfgang Pauli|Pauli]]&ndash;[[Pascual Jordan|Jordan]] function <math>\Delta(x-y)</math> by<ref>{{Cite journal |last1=Pauli |first1=Wolfgang |last2=Jordan |first2=Pascual |year=1928 |title=Zur Quantenelektrodynamik ladungsfreier Felder |journal=Zeitschrift für Physik |volume=47 |issue=3–4 |pages=151–173|doi=10.1007/BF02055793 |bibcode=1928ZPhy...47..151J |s2cid=120536476 }}</ref><ref name="BD">{{Cite book |last1=Bjorken |first1=James D. |title=Relativistic Quantum Fields |last2=Drell |first2=Sidney David |publisher=[[McGraw-Hill]] |year=1964 |isbn=978-0070054943 |series=International series in pure and applied physics |location=New York, NY |chapter=Appendix C}}</ref>

:<math>\langle 0 | \left[ \Phi(x),\Phi(y) \right] | 0 \rangle = i \, \Delta(x-y)</math>
with
:<math>\,\Delta(x-y) = G_\text{ret} (x-y) - G_\text{adv}(x-y)</math>

This satisfies 
:<math>\Delta(x-y) = -\Delta(y-x)</math> 
and is zero if <math>(x-y)^2 < 0</math>.

====Positive and negative frequency parts (cut propagators)====
We can define the positive and negative frequency parts of <math>\Delta(x-y)</math>, sometimes called cut propagators,  in a relativistically invariant way.

This allows us to define the positive frequency part:
:<math>\Delta_+(x-y) = \langle 0 | \Phi(x) \Phi(y) |0 \rangle, </math>

and the negative frequency part:
:<math>\Delta_-(x-y) = \langle 0 | \Phi(y) \Phi(x) |0 \rangle. </math>

These satisfy<ref name="BD"/> 
:<math>\,i \Delta = \Delta_+ - \Delta_-</math>

and
:<math>(\Box_x + m^2) \Delta_{\pm}(x-y) = 0.</math>

====Auxiliary function====
The anti-commutator of two scalar field operators defines <math>\Delta_1(x-y)</math> function by
:<math>\langle 0 | \left\{ \Phi(x),\Phi(y) \right\} | 0 \rangle = \Delta_1(x-y)</math>
with
:<math>\,\Delta_1(x-y) = \Delta_+ (x-y) + \Delta_-(x-y).</math>

This satisfies <math>\,\Delta_1(x-y) = \Delta_1(y-x).</math>

===Green's functions for the Klein–Gordon equation===
The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein–Gordon equation.

They are related to the singular functions by<ref name="BD"/>
:<math>G_\text{ret}(x-y) = \Delta(x-y) \Theta(x^0-y^0) </math>
:<math>G_\text{adv}(x-y) = -\Delta(x-y) \Theta(y^0-x^0) </math>
:<math>2 G_F(x-y) = -i \,\Delta_1(x-y) + \varepsilon(x^0 - y^0) \,\Delta(x-y) </math>
where <math>\varepsilon(x^0-y^0)</math> is the sign of <math>x^0-y^0</math>.

== See also ==

* [[Source field]]
* [[LSZ reduction formula]]

==Notes==
{{reflist|group=nb}}
{{reflist}}

==References==
*{{cite book|last1=Bjorken|first1=J.|author-link1=James Bjorken|last2=Drell|first2=S.|author-link2=Sidney Drell|title=Relativistic Quantum Fields|url=https://archive.org/details/relativisticquan0000bjor_c5q0|url-access=registration|location=New York|publisher=[[McGraw-Hill]]|year=1965|isbn=0-07-005494-0}} (Appendix C.)
*{{cite book|last1=Bogoliubov|first1=N.|author-link1=Nikolay Bogolyubov|last2=Shirkov|first2=D. V.|author-link2=Dmitry Shirkov|title=Introduction to the theory of quantized fields|publisher=[[Wiley-Interscience]]|isbn=0-470-08613-0|year=1959}} (Especially pp.&nbsp;136–156 and Appendix A)
* {{cite book | last=Cohen-Tannoudji | first=Claude | last2=Diu | first2=Bernard | last3=Laloë | first3=Franck | title=Quantum Mechanics, Volume 1 | publisher=John Wiley & Sons | publication-place=Weinheim | date=2019| isbn=978-3-527-34553-3}}
*{{cite book|editor-last1=DeWitt-Morette|editor-first1=C.|editor-link1=Cécile DeWitt-Morette|editor-last2=DeWitt|editor-first2=B.|editor-link2=Bryce DeWitt|title=Relativity, Groups and Topology|publisher=[[Blackie and Son]]|location=Glasgow|isbn=0-444-86858-5}} (section Dynamical Theory of Groups & Fields, Especially pp.&nbsp;615–624)
*{{cite book|title=Quantum Electrodynamics|first1=W.|last1=Greiner|author-link1=Walter Greiner|first2=J.|last2=Reinhardt|edition=4th|year=2008|isbn=9783540875604|publisher=[[Springer Verlag]]|url=https://books.google.com/books?id=5Kd3dBL8a64C&q=walter+greiner+Quantum+electrodynamics}}
*{{cite book|title=Field Quantization|first1=W.|last1=Greiner|author-link1=Walter Greiner|first2=J.|last2=Reinhardt|year=1996|isbn=9783540591795|publisher=Springer Verlag|url=https://archive.org/details/fieldquantizatio0000grei|url-access=registration}}
*{{cite book|last=Griffiths|first=D. J.|title=Introduction to Elementary Particles|location=New York|publisher=[[John Wiley & Sons]]|year=1987|isbn=0-471-60386-4}}
*{{cite book|last=Griffiths|first=D. J.|title=Introduction to Quantum Mechanics|location=Upper Saddle River|publisher=[[Prentice Hall]]|year=2004|isbn=0-131-11892-7}}
*{{citation|last1=Halliwell|first1=J.J.|last2=Orwitz|first2=M.|title=Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology|journal=Physical Review D|volume=48|issue=2|pages=748–768|arxiv=gr-qc/9211004|bibcode = 1993PhRvD..48..748H |doi = 10.1103/PhysRevD.48.748 |pmid=10016304|year=1993|s2cid=16381314}}
*{{cite book|last=Huang|first=Kerson|author-link=Kerson Huang|title=Quantum Field Theory: From Operators to Path Integrals|location=New York|publisher=John Wiley & Sons|year=1998|isbn=0-471-14120-8}}
*{{cite book|last1=Itzykson|first1=C.|author-link1=Claude Itzykson|last2=Zuber|first2=J-B.|author-link2=Jean-Bernard Zuber|title=Quantum Field Theory|url=https://archive.org/details/quantumfieldtheo0000itzy|url-access=registration|location=New York|publisher=McGraw-Hill|year=1980|isbn=0-07-032071-3}}
*{{cite book|last=Pokorski|first=S.|title=Gauge Field Theories|location=Cambridge|publisher=[[Cambridge University Press]]|year=1987|isbn=0-521-36846-4}}  ''(Has useful appendices of Feynman diagram rules, including propagators, in the back.)''
*{{cite book|last=Schulman|first=L. S.|title=Techniques & Applications of Path Integration|publisher=John Wiley & Sons|location=New York|year=1981|isbn=0-471-76450-7}}
*Scharf, G. (1995). ''Finite Quantum Electrodynamics, The Causal Approach.'' Springer. {{ISBN|978-3-642-63345-4}}.

== External links ==
* [https://arxiv.org/abs/quant-ph/0205085 Three Methods for Computing the Feynman Propagator]

[[Category:Quantum mechanics]]
[[Category:Quantum field theory]]
[[Category:Mathematical physics]]