Retarded propagator

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

This choice of contour is equivalent to calculating the limit,

$${\displaystyle G_{\text{ret}}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{(p_{0}+i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}=-{\frac {\Theta (x^{0}-y^{0})}{2\pi }}\delta (\tau _{xy}^{2})+\Theta (x^{0}-y^{0})\Theta (\tau _{xy}^{2}){\frac {mJ_{1}(m\tau _{xy})}{4\pi \tau _{xy}}}.}$$

Here

$${\displaystyle \Theta (x):={\begin{cases}1&x\geq 0\\0&x<0\end{cases}}}$$

is the Heaviside step function,

$${\displaystyle \tau _{xy}:={\sqrt {(x^{0}-y^{0})^{2}-({\vec {x}}-{\vec {y}})^{2}}}}$$

is the proper time from x to y, and ${\displaystyle J_{1}}$ is a Bessel function of the first kind. The propagator is non-zero only if ${\displaystyle y\prec x}$, i.e., y causally precedes x, which, for Minkowski spacetime, means

${\displaystyle y^{0}\leq x^{0}}$ and ${\displaystyle \tau _{xy}^{2}\geq 0~.}$

This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator,

$${\displaystyle G_{\text{ret}}(x,y)=-i\langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle \Theta (x^{0}-y^{0})}$$

where

$${\displaystyle \left[\Phi (x),\Phi (y)\right]:=\Phi (x)\Phi (y)-\Phi (y)\Phi (x)}$$

is the commutator.

Advanced propagator

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

This choice of contour is equivalent to calculating the limit^[6]

$${\displaystyle G_{\text{adv}}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{(p_{0}-i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}=-{\frac {\Theta (y^{0}-x^{0})}{2\pi }}\delta (\tau _{xy}^{2})+\Theta (y^{0}-x^{0})\Theta (\tau _{xy}^{2}){\frac {mJ_{1}(m\tau _{xy})}{4\pi \tau _{xy}}}.}$$

This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field. In this case,

$${\displaystyle G_{\text{adv}}(x,y)=i\langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle \Theta (y^{0}-x^{0})~.}$$

Feynman propagator

A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948.^[7]

This choice of contour is equivalent to calculating the limit^[8]

$${\displaystyle G_{F}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{p^{2}-m^{2}+i\varepsilon }}={\begin{cases}-{\frac {1}{4\pi }}\delta (\tau _{xy}^{2})+{\frac {m}{8\pi \tau _{xy}}}H_{1}^{(1)}(m\tau _{xy})&\tau _{xy}^{2}\geq 0\\-{\frac {im}{4\pi ^{2}{\sqrt {-\tau _{xy}^{2}}}}}K_{1}(m{\sqrt {-\tau _{xy}^{2}}})&\tau _{xy}^{2}<0.\end{cases}}}$$

Here, H₁⁽¹⁾ is a Hankel function and K₁ is a 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,

$${\displaystyle {\begin{aligned}G_{F}(x-y)&=-i\langle 0|T(\Phi (x)\Phi (y))|0\rangle \\[4pt]&=-i\left\langle 0|\left[\Theta (x^{0}-y^{0})\Phi (x)\Phi (y)+\Theta (y^{0}-x^{0})\Phi (y)\Phi (x)\right]|0\right\rangle .\end{aligned}}}$$

This expression is Lorentz invariant, as long as the field operators commute with one another when the points x and 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 Θ functions providing the causal time ordering may be obtained by a 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.^[9][10]

Explanation using limits

This can be made clearer by writing the propagator in the following form for a massless particle:

$${\displaystyle G_{F}^{\varepsilon }(x,y)={\frac {\varepsilon }{(x-y)^{2}+i\varepsilon ^{2}}}.}$$

This is the usual definition but normalised by a factor of ${\displaystyle \varepsilon }$. Then the rule is that one only takes the limit ${\displaystyle \varepsilon \to 0}$ at the end of a calculation.

One sees that

$${\displaystyle G_{F}^{\varepsilon }(x,y)={\frac {1}{\varepsilon }}\quad {\text{if}}~~~(x-y)^{2}=0,}$$

and

$${\displaystyle \lim _{\varepsilon \to 0}G_{F}^{\varepsilon }(x,y)=0\quad {\text{if}}~~~(x-y)^{2}\neq 0.}$$

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:

$${\displaystyle \lim _{\varepsilon \to 0}\int |G_{F}^{\varepsilon }(0,x)|^{2}\,dx^{3}=\lim _{\varepsilon \to 0}\int {\frac {\varepsilon ^{2}}{(\mathbf {x} ^{2}-t^{2})^{2}+\varepsilon ^{4}}}\,dx^{3}=2\pi ^{2}|t|.}$$

We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.

Spin 1⁄2

If the particle possesses 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 1⁄2 particle is given by^[11]

${\displaystyle (i\not \nabla '-m)S_{F}(x',x)=I_{4}\delta ^{4}(x'-x),}$

where I₄ 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,

$${\displaystyle S_{F}(x',x)=\int {\frac {d^{4}p}{(2\pi )^{4}}}\exp {\left[-ip\cdot (x'-x)\right]}{\tilde {S}}_{F}(p),}$$

the equation becomes

${\displaystyle {\begin{aligned}&(i\not \nabla '-m)\int {\frac {d^{4}p}{(2\pi )^{4}}}{\tilde {S}}_{F}(p)\exp {\left[-ip\cdot (x'-x)\right]}\\[6pt]={}&\int {\frac {d^{4}p}{(2\pi )^{4}}}(\not p-m){\tilde {S}}_{F}(p)\exp {\left[-ip\cdot (x'-x)\right]}\\[6pt]={}&\int {\frac {d^{4}p}{(2\pi )^{4}}}I_{4}\exp {\left[-ip\cdot (x'-x)\right]}\\[6pt]={}&I_{4}\delta ^{4}(x'-x),\end{aligned}}}$

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

${\displaystyle (\not p-mI_{4}){\tilde {S}}_{F}(p)=I_{4}.}$

By multiplying from the left with

$${\displaystyle (\not p+m)}$$

(dropping unit matrices from the notation) and using properties of the gamma matrices,

$${\displaystyle {\begin{aligned}\not p\not p&={\tfrac {1}{2}}(\not p\not p+\not p\not p)\\[6pt]&={\tfrac {1}{2}}(\gamma _{\mu }p^{\mu }\gamma _{\nu }p^{\nu }+\gamma _{\nu }p^{\nu }\gamma _{\mu }p^{\mu })\\[6pt]&={\tfrac {1}{2}}(\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu })p^{\mu }p^{\nu }\\[6pt]&=g_{\mu \nu }p^{\mu }p^{\nu }=p_{\nu }p^{\nu }=p^{2},\end{aligned}}}$$

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

${\displaystyle {\tilde {S}}_{F}(p)={\frac {(\not p+m)}{p^{2}-m^{2}+i\varepsilon }}={\frac {(\gamma ^{\mu }p_{\mu }+m)}{p^{2}-m^{2}+i\varepsilon }}.}$

The iε downstairs is a prescription for how to handle the poles in the complex p₀-plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written

${\displaystyle {\tilde {S}}_{F}(p)={1 \over \gamma ^{\mu }p_{\mu }-m+i\varepsilon }={1 \over \not p-m+i\varepsilon }}$

for short. It should be remembered that this expression is just shorthand notation for (γ_μp^μ − m)⁻¹. "One over matrix" is otherwise nonsensical. In position space one has

$${\displaystyle S_{F}(x-y)=\int {\frac {d^{4}p}{(2\pi )^{4}}}\,e^{-ip\cdot (x-y)}{\frac {\gamma ^{\mu }p_{\mu }+m}{p^{2}-m^{2}+i\varepsilon }}=\left({\frac {\gamma ^{\mu }(x-y)_{\mu }}{|x-y|^{5}}}+{\frac {m}{|x-y|^{3}}}\right)J_{1}(m|x-y|).}$$

This is related to the Feynman propagator by

${\displaystyle S_{F}(x-y)=(i\not \partial +m)G_{F}(x-y)}$

where ${\displaystyle \not \partial$ :=\gamma ^{\mu }\partial _{\mu }} .

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 Stueckelberg, the propagator for a photon is

${\displaystyle {-ig^{\mu \nu } \over p^{2}+i\varepsilon }.}$

The general form with gauge parameter λ, up to overall sign and the factor of ${\displaystyle i}$, reads

${\displaystyle -i{\frac {g^{\mu \nu }+\left(1-{\frac {1}{\lambda }}\right){\frac {p^{\mu }p^{\nu }}{p^{2}}}}{p^{2}+i\varepsilon }}.}$

The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter λ, up to overall sign and the factor of ${\displaystyle i}$, reads

${\displaystyle {\frac {g_{\mu \nu }-{\frac {k_{\mu }k_{\nu }}{m^{2}}}}{k^{2}-m^{2}+i\varepsilon }}+{\frac {\frac {k_{\mu }k_{\nu }}{m^{2}}}{k^{2}-{\frac {m^{2}}{\lambda }}+i\varepsilon }}.}$

With these general forms one obtains the propagators in unitary gauge for λ = 0, the propagator in Feynman or 't Hooft gauge for λ = 1 and in Landau or Lorenz gauge for λ = ∞. There are also other notations where the gauge parameter is the inverse of λ, usually denoted ξ (see R_ξ gauges). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.

Unitary gauge:

${\displaystyle {\frac {g_{\mu \nu }-{\frac {k_{\mu }k_{\nu }}{m^{2}}}}{k^{2}-m^{2}+i\varepsilon }}.}$

Feynman ('t Hooft) gauge:

${\displaystyle {\frac {g_{\mu \nu }}{k^{2}-m^{2}+i\varepsilon }}.}$

Landau (Lorenz) gauge:

${\displaystyle {\frac {g_{\mu \nu }-{\frac {k_{\mu }k_{\nu }}{k^{2}}}}{k^{2}-m^{2}+i\varepsilon }}.}$

Pauli–Jordan function

The commutator of two scalar field operators defines the Pauli–Jordan function ${\displaystyle \Delta (x-y)}$ by^[16][14]

${\displaystyle \langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle =i\,\Delta (x-y)}$

with

${\displaystyle \,\Delta (x-y)=G_{\text{ret}}(x-y)-G_{\text{adv}}(x-y)}$

This satisfies

${\displaystyle \Delta (x-y)=-\Delta (y-x)}$

and is zero if ${\displaystyle (x-y)^{2}<0}$.

Positive and negative frequency parts (cut propagators)

We can define the positive and negative frequency parts of ${\displaystyle \Delta (x-y)}$, sometimes called cut propagators, in a relativistically invariant way.

This allows us to define the positive frequency part:

${\displaystyle \Delta _{+}(x-y)=\langle 0|\Phi (x)\Phi (y)|0\rangle ,}$

and the negative frequency part:

${\displaystyle \Delta _{-}(x-y)=\langle 0|\Phi (y)\Phi (x)|0\rangle .}$

These satisfy^[14]

${\displaystyle \,i\Delta =\Delta _{+}-\Delta _{-}}$

and

${\displaystyle (\Box _{x}+m^{2})\Delta _{\pm }(x-y)=0.}$

Auxiliary function

The anti-commutator of two scalar field operators defines ${\displaystyle \Delta _{1}(x-y)}$ function by

${\displaystyle \langle 0|\left\{\Phi (x),\Phi (y)\right\}|0\rangle =\Delta _{1}(x-y)}$

with

${\displaystyle \,\Delta _{1}(x-y)=\Delta _{+}(x-y)+\Delta _{-}(x-y).}$

This satisfies ${\displaystyle \,\Delta _{1}(x-y)=\Delta _{1}(y-x).}$