Editing Propagator (section)

==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>.