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Inquantum mechanics andquantum field theory, thepropagator is a function that specifies theprobability 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. InFeynman diagrams, which serve to calculate the rate of collisions inquantum field theory,virtual particles contribute their propagator to the rate of thescattering event described by the respective diagram. Propagators may also be viewed as theinverse of thewave operator appropriate to the particle, and are, therefore, often called(causal)Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).^[1][2]

In non-relativistic quantum mechanics, the propagator gives the probability amplitude for aparticle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).

TheGreen's function G for theSchrödinger equation is a function$${\displaystyle G(x,t;x^{\prime },t^{\prime })=\frac{1}{i\hslash }\mathrm{\Theta }(t-t^{\prime })K(x,t;x^{\prime },t^{\prime })}$$satisfying$${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-H_x\right)G(x,t;x^{\prime },t^{\prime })=\delta (x-x^{\prime })\delta (t-t^{\prime }),}$$whereH denotes theHamiltonian,δ(x) denotes theDirac delta-function andΘ(t) is theHeaviside step function. Thekernel of the above Schrödinger differential operator in the big parentheses is denoted byK(x,t ;x′,t′) and called thepropagator.^{[nb 1]}

This propagator may also be written as the transition amplitude$${\displaystyle K(x,t;x^{\prime },t^{\prime })=⟨x|U(t,t^{\prime })|x^{\prime }⟩,}$$whereU(t,t′) is theunitary time-evolution operator for the system taking states at timet′ to states at timet.^[3] Note the initial condition enforced by$${\displaystyle \underset{t\to t^{\prime }}{lim}K(x,t;x^{\prime },t^{\prime })=\delta (x-x^{\prime }).}$$The propagator may also be found by using apath integral:

${\displaystyle K(x,t;x^{\prime },t^{\prime })=\int \exp \left[\frac{i}{\hslash }{\int }_{t^{\prime }}^{t}L(\dot{q},q,t)dt\right]D[q(t)],}$

whereL denotes theLagrangian and the boundary conditions are given byq(t) =x,q(t′) =x′. The paths that are summed over move only forwards in time and are integrated with the differential${\displaystyle D[q(t)]}$ following the path in time.^[4]

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

${\displaystyle \psi (x,t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\psi (x^{\prime },t^{\prime })K(x,t;x^{\prime },t^{\prime })d{x}^{\prime }.}$

IfK(x,t;x′,t′) only depends on the differencex −x′, this is aconvolution of the initial wave function and the propagator.

For a time-translationally invariant system, the propagator only depends on the time differencet −t′, so it may be rewritten as$${\displaystyle K(x,t;x^{\prime },t^{\prime })=K(x,x^{\prime };t-t^{\prime }).}$$

Thepropagator of a one-dimensional free particle, obtainable from, e.g., thepath integral, is then

Similarly, the propagator of a one-dimensionalquantum harmonic oscillator is theMehler kernel,^[5][6]

The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,^[7]valid for operators${\displaystyle \mathsf{x}}$ and${\displaystyle \mathsf{p}}$ satisfying theHeisenberg relation${\displaystyle [\mathsf{x},\mathsf{p}]=i\hslash }$.

For theN-dimensional case, the propagator can be simply obtained by the product$${\displaystyle K(\overrightarrow{x},{\overrightarrow{x}}^{\prime };t)=\prod _{q=1}^{N}K(x_q,x_q^{\prime };t).}$$

Inrelativistic quantum mechanics andquantum field theory the propagators areLorentz-invariant. They give the amplitude for aparticle to travel between twospacetime events.

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 describesspin-zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.

The position space propagators areGreen's functions for theKlein–Gordon equation. This means that they are functionsG(x,y) satisfying$${\displaystyle \left({◻}_x+m^2\right)G(x,y)=-\delta (x-y),}$$where

• x, y are two points inMinkowski spacetime,
• ${\displaystyle {◻}_x=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2}$ is thed'Alembertian operator acting on thex coordinates,
• δ(x −y) is theDirac delta function.

(As typical inrelativistic quantum field theory calculations, we use units where thespeed of lightc and thereduced Planck constantħ are set to unity.)

We shall restrict attention to 4-dimensionalMinkowski spacetime. We can perform aFourier transform of the equation for the propagator, obtaining$${\displaystyle \left(-p^2+m^2\right)G(p)=-1.}$$

This equation can be inverted in the sense ofdistributions, noting that the equationxf(x) = 1 has the solution (seeSokhotski–Plemelj theorem)$${\displaystyle f(x)=\frac{1}{x\pm i\epsilon }=\frac{1}{x}\mp i\pi \delta (x),}$$withε implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.

The solution is

where$${\displaystyle p(x-y):=p_0(x^0-y^0)-\overrightarrow{p}\cdot (\overrightarrow{x}-\overrightarrow{y})}$$ is the4-vector inner product.

The different choices for how to deform theintegration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the${\displaystyle p_0}$ integral.

The integrand then has two poles at$${\displaystyle p_0=\pm \sqrt{{\overrightarrow{p}}^2+m^2},}$$ so different choices of how to avoid these lead to different propagators.

A contour going clockwise over both poles gives thecausal retarded propagator. This is zero ifx-y is spacelike ory is to the future ofx, so it is zero ifx ⁰<y ⁰.

This choice of contour is equivalent to calculating thelimit,$${\displaystyle G_{\text{ret}}(x,y)=\underset{\epsilon \to 0}{lim}\frac{1}{(2\pi {)}^4}\int d^{4}p\frac{e^{-ip(x-y)}}{(p_0+i\epsilon {)}^2-{\overrightarrow{p}}^2-m^2}=-\frac{\mathrm{\Theta }(x^0-y^0)}{2\pi }\delta ({\tau }_{xy}^2)+\mathrm{\Theta }(x^0-y^0)\mathrm{\Theta }({\tau }_{xy}^2)\frac{m{J}_1(m{\tau }_{xy})}{4\pi {\tau }_{xy}}.}$$

Hereis theHeaviside step function,$${\displaystyle {\tau }_{xy}:=\sqrt{(x^0-y^0{)}^2-(\overrightarrow{x}-\overrightarrow{y}{)}^2}}$$is theproper time fromx toy, and${\displaystyle J_1}$ is aBessel function of the first kind. The propagator is non-zero only if${\displaystyle y\prec x}$, i.e.,ycausally precedesx, which, for Minkowski spacetime, means

${\displaystyle y^0\le x^0}$ and${\displaystyle {\tau }_{xy}^2\ge 0.}$

This expression can be related to thevacuum expectation value of thecommutator of the free scalar field operator,$${\displaystyle G_{\text{ret}}(x,y)=-i⟨0|\left[\mathrm{\Phi }(x),\mathrm{\Phi }(y)\right]|0⟩\mathrm{\Theta }(x^0-y^0),}$$where$${\displaystyle \left[\mathrm{\Phi }(x),\mathrm{\Phi }(y)\right]:=\mathrm{\Phi }(x)\mathrm{\Phi }(y)-\mathrm{\Phi }(y)\mathrm{\Phi }(x).}$$

A contour going anti-clockwise under both poles gives thecausal advanced propagator. This is zero ifx-y is spacelike or ify is to the past ofx, so it is zero ifx ⁰>y ⁰.

This choice of contour is equivalent to calculating the limit^[8]$${\displaystyle G_{\text{adv}}(x,y)=\underset{\epsilon \to 0}{lim}\frac{1}{(2\pi {)}^4}\int d^{4}p\frac{e^{-ip(x-y)}}{(p_0-i\epsilon {)}^2-{\overrightarrow{p}}^2-m^2}=-\frac{\mathrm{\Theta }(y^0-x^0)}{2\pi }\delta ({\tau }_{xy}^2)+\mathrm{\Theta }(y^0-x^0)\mathrm{\Theta }({\tau }_{xy}^2)\frac{m{J}_1(m{\tau }_{xy})}{4\pi {\tau }_{xy}}.}$$

This expression can also be expressed in terms of thevacuum expectation value of thecommutator of the free scalar field.In this case,$${\displaystyle G_{\text{adv}}(x,y)=i⟨0|\left[\mathrm{\Phi }(x),\mathrm{\Phi }(y)\right]|0⟩\mathrm{\Theta }(y^0-x^0).}$$

A contour going under the left pole and over the right pole gives theFeynman propagator, introduced byRichard Feynman in 1948.^[9]

This choice of contour is equivalent to calculating the limit^[10]

Here,H₁⁽¹⁾ is aHankel function andK₁ is amodified Bessel function.

This expression can be derived directly from the field theory as thevacuum expectation value of thetime-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,

This expression isLorentz invariant, as long as the field operators commute with one another when the pointsx andy are separated by aspacelike 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 twoΘ functions, one for the particle and one for its anti-particle, providing the causal time ordering may be obtained by acontour 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 thepath integral formulation of quantum theory.

Introduced byPaul Dirac in 1938.^[11][12]

TheFourier transform of the position space propagators can be thought of as propagators inmomentum space. These take a much simpler form than the position space propagators.

They are often written with an explicitε term although this is understood to be a reminder about which integration contour is appropriate (see above). Thisε term is included to incorporate boundary conditions andcausality (see below).

For a4-momentump the causal and Feynman propagators in momentum space are:

${\displaystyle {\tilde{G}}_{\text{ret}}(p)=\frac{1}{(p_0+i\epsilon {)}^2-{\overrightarrow{p}}^2-m^2}}$

${\displaystyle {\tilde{G}}_{\text{adv}}(p)=\frac{1}{(p_0-i\epsilon {)}^2-{\overrightarrow{p}}^2-m^2}}$

${\displaystyle {\tilde{G}}_F(p)=\frac{1}{p^2-m^2+i\epsilon }.}$

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

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The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator isnonzero outside of thelight 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 inclassical 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 iscommutators that determine which operators can affect one another.

So whatdoes the spacelike part of the propagator represent? In QFT thevacuum is an active participant, andparticle numbers and field values are related by anuncertainty principle; field values are uncertain even for particle numberzero. There is a nonzeroprobability amplitude to find a significant fluctuation in the vacuum value of the fieldΦ(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 anEPR correlation. Indeed, the propagator is often called atwo-point correlation function for thefree field.

Since, by the postulates of quantum field theory, allobservable 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. InFeynman'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.

This can be made clearer by writing the propagator in the following form for a massless particle:$${\displaystyle G_F^{\epsilon }(x,y)=\frac{\epsilon }{(x-y{)}^2+i{\epsilon }^2}.}$$

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

One sees that$${\displaystyle G_F^{\epsilon }(x,y)=\frac{1}{\epsilon }\text{if}(x-y{)}^2=0,}$$and$${\displaystyle \underset{\epsilon \to 0}{lim}{G}_F^{\epsilon }(x,y)=0\text{if}(x-y{)}^2\ne 0.}$$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:$${\displaystyle \underset{\epsilon \to 0}{lim}\int |G_F^{\epsilon }(0,x){|}^{2}d{x}^3=\underset{\epsilon \to 0}{lim}\int \frac{{\epsilon }^2}{({\mathbf{x}}^2-t^2{)}^2+{\epsilon }^4}d{x}^3=2{\pi }^2|t|.}$$We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.

The most common use of the propagator is in calculatingprobability amplitudes for particle interactions usingFeynman diagrams. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for everyinternal 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'sLagrangian for every internal vertex where lines meet. These prescriptions are known asFeynman 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 beoff 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 benegative. This can be interpreted simply as the case in which, instead of a particle going one way, itsantiparticle is going theother 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 offermions, whose propagators are noteven functions 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 closedloop, 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 ofrenormalization.

If the particle possessesspin 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 spin1⁄2 particle is given by^[13]

${\displaystyle (i{\mathrm{\nabla ̸}}^{\prime }-m)S_F(x^{\prime },x)=I_4{\delta }^4(x^{\prime }-x),}$

whereI₄ is the unit matrix in four dimensions, and employing theFeynman slash notation. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation,$${\displaystyle S_F(x^{\prime },x)=\int \frac{d^{4}p}{(2\pi {)}^4}\exp \left[-ip\cdot (x^{\prime }-x)\right]{\tilde{S}}_F(p),}$$the equation becomes

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

${\displaystyle (\mathrm{p̸}-m{I}_4){\tilde{S}}_F(p)=I_4.}$

By multiplying from the left with$${\displaystyle (\mathrm{p̸}+m)}$$(dropping unit matrices from the notation) and using properties of thegamma matrices,

the momentum-space propagator used in Feynman diagrams for aDirac field representing theelectron inquantum electrodynamics is found to have form

${\displaystyle {\tilde{S}}_F(p)=\frac{(\mathrm{p̸}+m)}{p^2-m^2+i\epsilon }=\frac{({\gamma }^{\mu }{p}_{\mu }+m)}{p^2-m^2+i\epsilon }.}$

Theiε downstairs is a prescription for how to handle the poles in the complexp₀-plane. It automatically yields theFeynman contour of integration by shifting the poles appropriately. It is sometimes written

${\displaystyle {\tilde{S}}_F(p)=\frac{1}{{\gamma }^{\mu }{p}_{\mu }-m+i\epsilon }=\frac{1}{\mathrm{p̸}-m+i\epsilon }}$

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$${\displaystyle S_F(x-y)=\int \frac{d^{4}p}{(2\pi {)}^4}{e}^{-ip\cdot (x-y)}\frac{{\gamma }^{\mu }{p}_{\mu }+m}{p^2-m^2+i\epsilon }=\left(\frac{{\gamma }^{\mu }(x-y{)}_{\mu }}{|x-y{|}^5}+\frac{m}{|x-y{|}^3}\right)J_1(m|x-y|).}$$

This is related to the Feynman propagator by

${\displaystyle S_F(x-y)=(i\mathrm{\partial ̸}+m)G_F(x-y)}$

where${\displaystyle \mathrm{\partial ̸}:={\gamma }^{\mu }{\mathrm{\partial }}_{\mu }}$.

The propagator for agauge boson in agauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman andStueckelberg, the propagator for aphoton is

${\displaystyle \frac{-i{g}^{\mu \nu }}{p^2+i\epsilon }.}$

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

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

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

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

With these general forms one obtains the propagators in unitary gauge forλ = 0, the propagator in Feynman or 't Hooft gauge forλ = 1 and in Landau or Lorenz gauge forλ = ∞. There are also other notations where the gauge parameter is the inverse ofλ, usually denotedξ (seeR_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:

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

Feynman ('t Hooft) gauge:

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

Landau (Lorenz) gauge:

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

The graviton propagator forMinkowski space ingeneral relativity is^[14]$${\displaystyle G_{\alpha \beta \mu \nu }=\frac{{\mathcal{P}}_{\alpha \beta \mu \nu }^2}{k^2}-\frac{{\mathcal{P}}_s^0{}_{\alpha \beta \mu \nu }}{2k^2}=\frac{g_{\alpha \mu }{g}_{\beta \nu }+g_{\beta \mu }{g}_{\alpha \nu }-\frac{2}{D-2}{g}_{\mu \nu }{g}_{\alpha \beta }}{k^2},}$$where${\displaystyle D}$ is the number of spacetime dimensions,${\displaystyle {\mathcal{P}}^2}$ is the transverse and tracelessspin-2 projection operator and${\displaystyle {\mathcal{P}}_s^0}$ is a spin-0 scalarmultiplet. The graviton propagator for(Anti) de Sitter space is$${\displaystyle G=\frac{{\mathcal{P}}^2}{2H^2-◻}+\frac{{\mathcal{P}}_s^0}{2(◻+4H^2)},}$$where${\displaystyle H}$ is theHubble constant. Note that upon taking the limit${\displaystyle H\to 0}$ and${\displaystyle ◻\to -k^2}$, the AdS propagator reduces to the Minkowski propagator.^[15]

The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important inquantum field theory. These functions are most simply defined in terms of thevacuum expectation value of products of field operators.

The commutator of two scalar field operators defines thePauli–Jordan function${\displaystyle \mathrm{\Delta }(x-y)}$ by^[16][17]

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

with

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

This satisfies

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

and is zero if.

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

This allows us to define the positive frequency part:

${\displaystyle {\mathrm{\Delta }}_{+}(x-y)=⟨0|\mathrm{\Phi }(x)\mathrm{\Phi }(y)|0⟩,}$

and the negative frequency part:

${\displaystyle {\mathrm{\Delta }}_{-}(x-y)=⟨0|\mathrm{\Phi }(y)\mathrm{\Phi }(x)|0⟩.}$

These satisfy^[17]

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

and

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

The anti-commutator of two scalar field operators defines${\displaystyle {\mathrm{\Delta }}_1(x-y)}$ function by

${\displaystyle ⟨0|\left\{\mathrm{\Phi }(x),\mathrm{\Phi }(y)\right\}|0⟩={\mathrm{\Delta }}_1(x-y)}$

with

${\displaystyle {\mathrm{\Delta }}_1(x-y)={\mathrm{\Delta }}_{+}(x-y)+{\mathrm{\Delta }}_{-}(x-y).}$

This satisfies${\displaystyle {\mathrm{\Delta }}_1(x-y)={\mathrm{\Delta }}_1(y-x).}$

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^[17]

${\displaystyle G_{\text{ret}}(x-y)=\mathrm{\Delta }(x-y)\mathrm{\Theta }(x^0-y^0)}$

${\displaystyle G_{\text{adv}}(x-y)=-\mathrm{\Delta }(x-y)\mathrm{\Theta }(y^0-x^0)}$

${\displaystyle 2G_F(x-y)=-i{\mathrm{\Delta }}_1(x-y)+\epsilon (x^0-y^0)\mathrm{\Delta }(x-y)}$

where${\displaystyle \epsilon (x^0-y^0)}$ is the sign of${\displaystyle x^0-y^0}$.

• Source field
• LSZ reduction formula

• Bjorken, J.;Drell, S. (1965).Relativistic Quantum Fields. New York:McGraw-Hill.ISBN 0-07-005494-0. (Appendix C.)
• Bogoliubov, N.;Shirkov, D. V. (1959).Introduction to the theory of quantized fields.Wiley-Interscience.ISBN 0-470-08613-0.{{cite book}}:ISBN / Date incompatibility (help) (Especially pp. 136–156 and Appendix A)
• Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019).Quantum Mechanics, Volume 1. Weinheim: John Wiley & Sons.ISBN 978-3-527-34553-3.
• DeWitt-Morette, C.;DeWitt, B. (eds.).Relativity, Groups and Topology. Glasgow:Blackie and Son.ISBN 0-444-86858-5. (section Dynamical Theory of Groups & Fields, Especially pp. 615–624)
• Greiner, W.; Reinhardt, J. (2008).Quantum Electrodynamics (4th ed.).Springer Verlag.ISBN 9783540875604.
• Greiner, W.; Reinhardt, J. (1996).Field Quantization. Springer Verlag.ISBN 9783540591795.
• Griffiths, D. J. (1987).Introduction to Elementary Particles. New York:John Wiley & Sons.ISBN 0-471-60386-4.
• Griffiths, D. J. (2004).Introduction to Quantum Mechanics. Upper Saddle River:Prentice Hall.ISBN 0-131-11892-7.
• Halliwell, J.J.; Orwitz, M. (1993), "Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology",Physical Review D,48 (2):748–768,arXiv:gr-qc/9211004,Bibcode:1993PhRvD..48..748H,doi:10.1103/PhysRevD.48.748,PMID 10016304,S2CID 16381314
• Huang, Kerson (1998).Quantum Field Theory: From Operators to Path Integrals. New York: John Wiley & Sons.ISBN 0-471-14120-8.
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• Pokorski, S. (1987).Gauge Field Theories. Cambridge:Cambridge University Press.ISBN 0-521-36846-4.(Has useful appendices of Feynman diagram rules, including propagators, in the back.)
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• Scharf, G. (1995).Finite Quantum Electrodynamics, The Causal Approach. Springer.ISBN 978-3-642-63345-4.

• Three Methods for Computing the Feynman Propagator

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• Mathematical physics

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