The Feynman Propagator

Derivation

The Feynman Propagator

D-379 Home PU-402 Threads fields · waves · matter Depends on Mode Expansion and Fock Space

Statement

For a free real scalar field \(\phi(x)\) of mass \(m\) satisfying the Klein–Gordon equation, the time-ordered vacuum two-point function \(D_F(x-y)\equiv\langle 0|T\,\phi(x)\phi(y)|0\rangle\) is a Green's function of the Klein–Gordon operator, \((\Box_x+m^2)\,D_F(x-y)=-\,i\,\delta^{(4)}(x-y)\), whose momentum-space representation \(\displaystyle D_F(x-y)=\int\!\frac{d^4p}{(2\pi)^4}\,\frac{i}{p^2-m^2+i\epsilon}\,e^{-ip\cdot(x-y)}\) is fixed uniquely by the \(+i\epsilon\) contour prescription, which encodes the time ordering.

Why it matters

The Feynman propagator is the single most-used object in perturbative quantum field theory: it is the amplitude for a particle to propagate between two spacetime points, the internal line of every Feynman diagram, and the building block from which all scattering amplitudes are assembled through Wick's theorem. Every tree-level cross section and every loop integral is written in terms of \(D_F\).

Its defining feature is not that it solves the Klein–Gordon equation — many Green's functions do — but which one it is. The retarded, advanced, and Feynman propagators share the same differential equation and differ only in how their poles are circumvented. The \(+i\epsilon\) prescription is the physics: it propagates positive-frequency (particle) modes forward in time and negative-frequency (antiparticle) modes backward, exactly reproducing the time-ordering \(T\). This is where causality and the particle/antiparticle structure of the vacuum enter the formalism.

Assumptions

The field is free and satisfies \((\Box+m^2)\phi=0\).With interactions the two-point function acquires a self-energy; \(D_F\) becomes the full propagator with a shifted pole and continuum cut, and is no longer a simple Green's function of the free operator. The state is the Poincaré-invariant vacuum \(|0\rangle\), annihilated by all \(a_{\vec p}\).In a thermal or non-vacuum state the two-point function gains occupation-number terms; the propagator splits into vacuum plus finite-temperature pieces (the Matsubara / thermal propagator). The equal-time canonical commutators hold: \([\phi(\vec x,t),\dot\phi(\vec y,t)]=i\,\delta^{(3)}(\vec x-\vec y)\), \([\phi,\phi]_{\rm ET}=0\).The contact term that makes \(D_F\) a Green's function comes entirely from these equal-time commutators; drop canonical quantization and the derivative of the time-ordering symbol produces no \(\delta^{(4)}\) source. Metric signature \((+,-,-,-)\) with \(\Box=\partial_\mu\partial^\mu=\partial_t^2-\nabla^2\), natural units \(\hbar=c=1\).Signs of \(p^2-m^2\) and the pole locations flip under \((-,+,+,+)\); the \(i\epsilon\) then attaches to \(m^2\) rather than to the whole denominator, and cross-checks against other texts fail if the convention is not tracked.

Derivation

\[ D_F(x-y)=\theta(x^0-y^0)\,\langle 0|\phi(x)\phi(y)|0\rangle+\theta(y^0-x^0)\,\langle 0|\phi(y)\phi(x)|0\rangle \]

Definition of the time-ordered product for bosonic fields: \(T\) places the later-time operator to the left, with the step function \(\theta\) selecting the ordering. A

\[ \langle 0|\phi(x)\phi(y)|0\rangle=\int\!\frac{d^3p}{(2\pi)^3}\,\frac{1}{2E_{\vec p}}\,e^{-ip\cdot(x-y)}\Big|_{p^0=E_{\vec p}},\qquad E_{\vec p}=\sqrt{\vec p^{\,2}+m^2} \]

Insert the mode expansion \(\phi=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\vec p}}}\big(a_{\vec p}e^{-ip\cdot x}+a_{\vec p}^\dagger e^{+ip\cdot x}\big)\) and use \(a_{\vec p}|0\rangle=0\), \([a_{\vec p},a^\dagger_{\vec q}]=(2\pi)^3\delta^{(3)}(\vec p-\vec q)\); only \(\langle 0|a\,a^\dagger|0\rangle\) survives (prior result: mode expansion, Fock space). A

\[ \partial_{x^0}D_F=\delta(x^0-y^0)\,\langle 0|[\phi(x),\phi(y)]|0\rangle\big|_{x^0=y^0}+T\big\langle\partial_{x^0}\phi(x)\,\phi(y)\big\rangle \]

Differentiate step 1 once. The derivative hits the two \(\theta\)-functions (giving \(\pm\delta(x^0-y^0)\), which combine into the equal-time commutator) and the fields inside. B

\[ \langle 0|[\phi(x),\phi(y)]|0\rangle\big|_{x^0=y^0}=0\quad\Longrightarrow\quad \partial_{x^0}D_F=T\big\langle\partial_{x^0}\phi(x)\,\phi(y)\big\rangle \]

The equal-time commutator of two fields vanishes (canonical quantization). The first term dies; only the time-ordered derivative survives. B

\[ \partial_{x^0}^2 D_F=\delta(x^0-y^0)\,\langle 0|[\dot\phi(x),\phi(y)]|0\rangle\big|_{x^0=y^0}+T\big\langle\partial_{x^0}^2\phi(x)\,\phi(y)\big\rangle \]

Differentiate step 4 again; the \(\theta\)-derivative now acts on \(\dot\phi\), producing the equal-time commutator of \(\dot\phi\) with \(\phi\). B

\[ [\dot\phi(\vec x,t),\phi(\vec y,t)]=-\,i\,\delta^{(3)}(\vec x-\vec y)\quad\Longrightarrow\quad \partial_{x^0}^2 D_F=-\,i\,\delta^{(4)}(x-y)+T\big\langle\ddot\phi(x)\,\phi(y)\big\rangle \]

The canonical momentum is \(\pi=\dot\phi\); the equal-time relation \([\phi,\pi]=i\delta^{(3)}\) gives \([\dot\phi,\phi]=-i\delta^{(3)}\). Combined with \(\delta(x^0-y^0)\,\delta^{(3)}(\vec x-\vec y)=\delta^{(4)}(x-y)\), this is the contact term — the seed of the source. C

\[ (\Box_x+m^2)\,D_F=(\partial_{x^0}^2-\nabla_x^2+m^2)\,D_F=-\,i\,\delta^{(4)}(x-y)+T\big\langle(\Box_x+m^2)\phi(x)\,\phi(y)\big\rangle \]

Add \(-\nabla_x^2+m^2\) acting on \(D_F\). Because \(-\nabla^2\) carries no time derivative it commutes through the \(\theta\)-functions, so it only acts on the fields inside \(T\), assembling the full Klein–Gordon operator there. B

\[ (\Box_x+m^2)\phi(x)=0\quad\Longrightarrow\quad \boxed{\,(\Box_x+m^2)\,D_F(x-y)=-\,i\,\delta^{(4)}(x-y)\,} \]

The free field obeys its equation of motion, killing the time-ordered remainder. \(D_F\) is a Green's function of the Klein–Gordon operator with source \(-i\delta^{(4)}\). A

\[ D_F(x-y)=\int\!\frac{d^4p}{(2\pi)^4}\,\tilde D_F(p)\,e^{-ip\cdot(x-y)},\qquad (-p^2+m^2)\,\tilde D_F(p)=-\,i \]

Fourier transform. Using \((\Box_x+m^2)e^{-ip\cdot(x-y)}=(-p^2+m^2)e^{-ip\cdot(x-y)}\) and \(\delta^{(4)}(x-y)=\int\frac{d^4p}{(2\pi)^4}e^{-ip\cdot(x-y)}\), the differential equation becomes algebraic. B

\[ \tilde D_F(p)=\frac{i}{p^2-m^2}\quad\Longrightarrow\quad D_F(x-y)=\int\!\frac{d^4p}{(2\pi)^4}\,\frac{i}{p^2-m^2}\,e^{-ip\cdot(x-y)} \]

Solve algebraically. The integrand is singular on the mass shell \(p^0=\pm E_{\vec p}\); the position-space integral is ambiguous until the contour past these poles is specified — this ambiguity is precisely the choice of Green's function. B

\[ \frac{i}{p^2-m^2+i\epsilon}=\frac{i}{(p^0)^2-E_{\vec p}^2+i\epsilon}=\frac{i}{\big(p^0-(E_{\vec p}-i\epsilon')\big)\big(p^0+(E_{\vec p}-i\epsilon')\big)} \]

Impose the Feynman prescription \(m^2\to m^2-i\epsilon\). The poles shift to \(p^0=+E_{\vec p}-i\epsilon'\) (lower half-plane) and \(p^0=-E_{\vec p}+i\epsilon'\) (upper half-plane), with \(\epsilon'=\epsilon/2E_{\vec p}>0\). C

\[ \int\!\frac{dp^0}{2\pi}\,\frac{i\,e^{-ip^0(x^0-y^0)}}{(p^0)^2-E_{\vec p}^2+i\epsilon}=\frac{1}{2E_{\vec p}}\Big[\theta(x^0-y^0)e^{-iE_{\vec p}(x^0-y^0)}+\theta(y^0-x^0)e^{+iE_{\vec p}(x^0-y^0)}\Big] \]

Close the \(p^0\) contour in the lower half-plane for \(x^0>y^0\) (so \(e^{-ip^0 t}\) decays), picking up the \(+E_{\vec p}\) pole; close above for \(x^0<y^0\), picking up \(-E_{\vec p}\). The residues reproduce exactly the two \(\theta\)-terms of step 1 with \(p^0=+E_{\vec p}\) on shell — confirming the \(+i\epsilon\) contour is the time-ordering. C

Result

\[ (\Box_x+m^2)\,D_F(x-y)=-\,i\,\delta^{(4)}(x-y),\qquad D_F(x-y)=\int\!\frac{d^4p}{(2\pi)^4}\,\frac{i\,e^{-ip\cdot(x-y)}}{p^2-m^2+i\epsilon} \]

Reading. The time-ordered vacuum expectation value \(\langle 0|T\phi(x)\phi(y)|0\rangle\) inverts the Klein–Gordon operator with a unit imaginary source at coincident points. Among all Green's functions of \((\Box+m^2)\), the \(+i\epsilon\) prescription selects the one that propagates positive-frequency modes forward in time and negative-frequency modes backward — the amplitude for the vacuum to create a quantum at the earlier point and absorb it at the later point (or the antiparticle in reverse). It is the internal line of every Feynman diagram.

Units check. In natural units a scalar field in four dimensions has mass dimension \([\phi]=1\), so \([D_F]=[\phi]^2=2\). In the momentum integral \([d^4p]=4\) and \([\,i/(p^2-m^2)\,]=-2\), giving \(4-2=2\). ✓ In the differential form \([\Box+m^2]=2\) and \([\delta^{(4)}(x-y)]=4\) (since \([d^4x]=-4\)), so \([(\Box+m^2)D_F]=2+2=4=[\delta^{(4)}]\). ✓

Limiting cases

Massless limit \(m\to 0\): \(\tilde D_F(p)=i/(p^2+i\epsilon)\); in position space \(D_F(x)=\dfrac{-1}{4\pi^2}\dfrac{1}{x^2-i\epsilon}\), the photon-like \(1/x^2\) propagator.
Static / equal-time limit: the spatial Fourier transform \(\int\!\frac{d^3p}{(2\pi)^3}\frac{e^{i\vec p\cdot\vec r}}{\vec p^2+m^2}=\dfrac{e^{-mr}}{4\pi r}\) is the Yukawa potential; the mass sets the range \(1/m\).
Heavy-mass / large \(m\): \(\tilde D_F\to -i/m^2\), a contact interaction; the particle cannot propagate and the exchange collapses to a point (basis of effective field theory).
Large spacelike separation \(r=\sqrt{-x^2}\gg 1/m\): \(D_F\sim e^{-mr}\), exponential decay — no signal, only virtual leakage over a Compton wavelength.
On the mass shell \(p^2\to m^2\): the propagator diverges as \(i/(p^2-m^2+i\epsilon)\); the imaginary part \(\to \pi\,\delta(p^2-m^2)\) is the physical single-particle pole.

Breaks when

Interactions are turned on. The free-field equation of motion (step 8) fails; \((\Box+m^2)\phi=-\lambda\phi^3/6\) etc. The two-point function becomes the full propagator \(\tilde D(p)=i/[p^2-m^2-\Sigma(p^2)+i\epsilon]\) with a self-energy \(\Sigma\) that shifts the pole (mass renormalization), adds a finite residue \(Z<1\), and grows a branch cut above the multiparticle threshold. \(D_F\) is then not a Green's function of the free operator.
The state is not the vacuum. At temperature \(T\) or finite density the ensemble average replaces \(\langle 0|\cdots|0\rangle\); \(\langle a^\dagger a\rangle=n_B(E_{\vec p})\ne 0\) adds \(2\pi\,\delta(p^2-m^2)\,n_B(|p^0|)\) to the propagator. Real-time thermal field theory requires the full \(2\times 2\) Schwinger–Keldysh matrix, not a single \(D_F\).
The contour is chosen differently. Replacing \(+i\epsilon\) by the retarded prescription \(p^0\to p^0+i\epsilon\) (both poles below the axis) gives the same differential equation but a propagator supported only in the forward light cone — a classical, not time-ordered, object. The Green's-function statement survives; the identification with \(\langle 0|T\phi\phi|0\rangle\) does not.
Spacetime is curved or a boundary is present. Translation invariance is lost, \(D_F(x-y)\) is no longer a function of \(x-y\) alone, the mode expansion of step 2 has no global positive-frequency split, and the very notion of "the vacuum" becomes observer-dependent (Unruh/Hawking).

Failure modes

Sign of the source. Writing \((\Box+m^2)D_F=+i\delta^{(4)}\) or \(=\delta^{(4)}\). The overall factor of \(i\) and its sign are fixed by the mode normalization and the equal-time commutator; getting them wrong flips every amplitude's phase.
Wrong pole for the wrong metric. Using \(i/(p^2-m^2+i\epsilon)\) with signature \((-,+,+,+)\). In that convention \(p^2=-{p^0}^2+\vec p^2\) and the correct Feynman denominator is \(-i/(p^2+m^2-i\epsilon)\); mixing conventions inverts the contour.
Dropping the \(i\epsilon\) as "a technicality." Without it the \(p^0\) integral is undefined; the \(i\epsilon\) is the entire content that distinguishes \(D_F\) from the retarded, advanced, or Wightman functions, all of which solve the same equation.
Forgetting that \(p^0\) is off shell. Treating the internal-line four-momentum as satisfying \(p^0=E_{\vec p}\). In the full four-dimensional integral \(p^2\ne m^2\) in general — that is what "virtual" means; only the poles are on shell.
Missing the second delta-term. When differentiating \(T\)-products, students often keep only the \(\delta(x^0-y^0)\) from one \(\theta\) and forget the second, losing the factor that combines into the commutator (steps 3, 5).

Discussion

The derivation exposes a clean division of labor. The Klein–Gordon operator \((\Box+m^2)\) is dictated by the free dynamics and is shared by every Green's function. The delta-function source is manufactured entirely by the equal-time canonical commutator \([\dot\phi,\phi]=-i\delta^{(3)}\) hitting the discontinuity of the time-ordering symbol — this is the one place where quantization, not classical field theory, enters. And the choice of which Green's function is made once and for all by the \(+i\epsilon\), which is nothing but the analytic encoding of "later operators to the left." Three independent ingredients — dynamics, canonical structure, boundary condition — meet in a single object.

The contour prescription is where the vacuum's particle content is written. Closing the \(p^0\) integral below for \(x^0>y^0\) picks the pole at \(+E_{\vec p}\): a positive-energy quantum moving forward in time. Closing above for \(x^0<y^0\) picks \(-E_{\vec p}\): interpreted as a positive-energy antiparticle moving forward, or equivalently a negative-energy solution moving backward (Feynman–Stückelberg). A single analytic function thus describes particle and antiparticle propagation as two faces of one amplitude — the reason \(D_F\) and not the retarded propagator is the natural object for a relativistic quantum theory, where pair creation forbids a strictly forward-in-time single-particle description.

There is a deeper, Euclidean face. Rotating \(p^0\to ip^0_E\) (Wick rotation) turns \(p^2-m^2+i\epsilon\) into \(-(p_E^2+m^2)\), a strictly negative-definite denominator with no poles on the real axis — the \(i\epsilon\) is exactly what makes this rotation legal by keeping the poles off the rotating contour. The Feynman propagator is the analytic continuation of the Euclidean two-point function \(1/(p_E^2+m^2)\), the covariance of a Gaussian measure. This is why the path integral with the \(e^{-S_E}\) weight computes time-ordered correlators: the \(T\)-ordering and the \(+i\epsilon\) are the Lorentzian shadow of Euclidean reflection positivity. The same \(i\epsilon\) that orders operators is the one that guarantees convergence of the functional integral.

Common misconceptions. The propagator is not "the wavefunction of the particle" and it is not a probability. It is a two-point correlation function / transition amplitude, complex-valued, with an imaginary part that (via the optical theorem) measures the rate of real particle production. Nor does the internal line "put a real particle on shell": the four-momentum flowing through \(\tilde D_F(p)\) is integrated over all of \(p^0\), off shell everywhere except at the poles. Finally, \(D_F\) is symmetric, \(D_F(x-y)=D_F(y-x)\), even though the definition uses time ordering — the two \(\theta\)-terms exchange under \(x\leftrightarrow y\).

Worked examples

Yukawa range from the static propagator — the pion-exchange nuclear force

\[ V(r)\;\propto\;\int\!\frac{d^3p}{(2\pi)^3}\,\frac{e^{i\vec p\cdot\vec r}}{\vec p^{\,2}+m^2}=\frac{e^{-mr}}{4\pi r} \]

Take the equal-time (static) limit of \(\tilde D_F\); the spatial transform of \(1/(\vec p^2+m^2)\) is the screened Coulomb / Yukawa form. Symbols first: the range is \(R=1/m\). B

\[ m_\pi=140~\text{MeV},\qquad R=\frac{1}{m_\pi}=\frac{\hbar c}{m_\pi c^2}=\frac{197.3~\text{MeV}\cdot\text{fm}}{140~\text{MeV}} \]

Restore \(\hbar c=197.3~\text{MeV·fm}\) to convert an inverse energy into a length. A

\[ R\approx 1.41~\text{fm};\qquad \frac{V(2R)}{V(R)}=\frac{e^{-2}/2R}{e^{-1}/R}=\tfrac{1}{2}e^{-1}\approx 0.184 \]

Numbers in. The potential falls by a factor \(\sim 5.4\) each time \(r\) increases by one range, plus the \(1/r\) prefactor. A

\[ R=\frac{\hbar c}{m_\pi c^2}\approx 1.4~\text{fm} \]

Reading. The Compton wavelength of the exchanged quantum sets the reach of the force. A 140 MeV pion gives a nuclear force of range \(\sim\!1.4\) fm — precisely the scale of the nucleon–nucleon interaction Yukawa predicted from exactly this propagator. A massless exchange (\(m\to0\)) gives infinite range: the Coulomb \(1/r\).

On-shell energy and pole location for a virtual particle

\[ \tilde D_F(p)=\frac{i}{p^2-m^2+i\epsilon},\qquad p^2=(p^0)^2-\vec p^{\,2},\qquad E_{\vec p}=\sqrt{\vec p^{\,2}+m^2} \]

Locate the poles of the propagator as a function of \(p^0\) at fixed spatial momentum. Symbols: poles at \(p^0=\pm(E_{\vec p}-i\epsilon')\). B

\[ m=140~\text{MeV},\qquad |\vec p|=300~\text{MeV}\ \Rightarrow\ E_{\vec p}=\sqrt{300^2+140^2}\ \text{MeV} \]

Insert numbers for a pion carrying 300 MeV of three-momentum. A

\[ E_{\vec p}=\sqrt{90000+19600}=\sqrt{109600}\approx 331~\text{MeV} \]

Evaluate the on-shell energy. The poles sit at \(p^0\approx+331~\text{MeV}\) (just below the real axis) and \(p^0\approx-331~\text{MeV}\) (just above). A

\[ \text{Off shell at }p^0=200~\text{MeV}:\quad p^2-m^2=200^2-300^2-140^2=-89600~\text{MeV}^2,\quad \tilde D_F=\frac{i}{-89600~\text{MeV}^2} \]

A concrete off-shell value: with \(p^0=200\) MeV the line is far from its pole, \(|\tilde D_F|\approx 1.1\times10^{-5}~\text{MeV}^{-2}\) — finite, virtual, suppressed. A

\[ p^0_{\text{pole}}=\pm\big(E_{\vec p}-i\epsilon'\big),\quad E_{\vec p}\approx 331~\text{MeV} \]

Reading. The propagator has poles only on the mass shell \(p^0=\pm E_{\vec p}\); the \(+i\epsilon\) nudges the \(+E_{\vec p}\) pole into the lower half-plane and the \(-E_{\vec p}\) pole into the upper half-plane — the analytic signature of time ordering. Away from the poles (here at \(p^0=200\) MeV) the internal line is off shell and its amplitude is finite and small, the hallmark of a virtual particle.

Problems

Green's-function sign. Verify directly in momentum space that \(\tilde D_F(p)=i/(p^2-m^2+i\epsilon)\) satisfies \((\Box_x+m^2)D_F=-i\delta^{(4)}(x-y)\), and state what \(\tilde D_F\) would be if the source were \(+i\delta^{(4)}\) instead.
Solution
Acting with \((\Box_x+m^2)\) on \(D_F=\int\frac{d^4p}{(2\pi)^4}\tilde D_F(p)e^{-ip\cdot(x-y)}\) brings down \((-p^2+m^2)\). Thus \((\Box+m^2)D_F=\int\frac{d^4p}{(2\pi)^4}(m^2-p^2)\frac{i}{p^2-m^2+i\epsilon}e^{-ip\cdot(x-y)}\). As \(\epsilon\to0\), \((m^2-p^2)/(p^2-m^2)=-1\), so the integrand \(\to -i\,e^{-ip\cdot(x-y)}\), giving \(-i\int\frac{d^4p}{(2\pi)^4}e^{-ip\cdot(x-y)}=-i\,\delta^{(4)}(x-y)\). ✓ For a \(+i\delta^{(4)}\) source the residue flips: \(\tilde D_F=-i/(p^2-m^2+i\epsilon)\).
Yukawa transform. Show \(\displaystyle\int\!\frac{d^3p}{(2\pi)^3}\frac{e^{i\vec p\cdot\vec r}}{\vec p^{\,2}+m^2}=\frac{e^{-mr}}{4\pi r}\), and evaluate the ratio of the potential at \(r=0.5\) fm to that at \(r=1.5\) fm for \(m=140\) MeV.
Solution
Do the angular integral: \(\int d\Omega\,e^{i\vec p\cdot\vec r}=4\pi\,\frac{\sin(pr)}{pr}\). Then \(I=\frac{1}{(2\pi)^3}\int_0^\infty dp\,p^2\frac{4\pi}{pr}\frac{\sin pr}{p^2+m^2}=\frac{1}{2\pi^2 r}\int_0^\infty\frac{p\sin(pr)}{p^2+m^2}dp\). The remaining integral is \(\frac{\pi}{2}e^{-mr}\) (standard contour result), giving \(I=\frac{e^{-mr}}{4\pi r}\). ✓ With \(1/m=1.41\) fm: \(\frac{V(0.5)}{V(1.5)}=\frac{e^{-0.5/1.41}/0.5}{e^{-1.5/1.41}/1.5}=3\,e^{(1.5-0.5)/1.41}=3\,e^{0.709}\approx 3\times2.03\approx 6.1\).
Contour and time ordering. Perform the \(p^0\) integral \(\int\frac{dp^0}{2\pi}\frac{i\,e^{-ip^0 t}}{(p^0)^2-E^2+i\epsilon}\) by residues for \(t>0\) and \(t<0\), and identify the two on-shell frequencies with the terms of \(T\phi\phi\).
Solution
Poles at \(p^0=\pm(E-i\epsilon')\). For \(t>0\), \(e^{-ip^0t}\) decays as \(\mathrm{Im}\,p^0\to-\infty\), close below, enclosing \(p^0=+E-i\epsilon'\) clockwise: residue gives \(\frac{i}{2\pi}\cdot(-2\pi i)\frac{e^{-iEt}}{2E}=\frac{e^{-iEt}}{2E}\). For \(t<0\), close above, enclosing \(p^0=-E+i\epsilon'\) counterclockwise: \(\frac{e^{+iEt}}{2E}\). Combined: \(\frac{1}{2E}[\theta(t)e^{-iEt}+\theta(-t)e^{+iEt}]\). The \(\theta(t)e^{-iEt}\) term is \(\langle\phi(x)\phi(y)\rangle\) (positive frequency, forward); the \(\theta(-t)e^{+iEt}\) term is \(\langle\phi(y)\phi(x)\rangle\) — exactly the time-ordered product.
Massless propagator. Take \(m\to0\) and use \(\int\frac{d^4p}{(2\pi)^4}\frac{-i}{p^2+i\epsilon}e^{-ip\cdot x}=\frac{1}{4\pi^2}\frac{1}{x^2-i\epsilon}\) (up to sign) to identify the light-cone singularity structure. What is the mass dimension of this \(D_F\), and does it match \([\phi]^2\)?
Solution
\(D_F(x)=\frac{-1}{4\pi^2}\frac{1}{x^2-i\epsilon}\) with \(x^2=t^2-\vec x^2\). It is singular on the light cone \(x^2=0\); the \(-i\epsilon\) tells how to circle that singularity (principal value plus a \(\delta(x^2)\) piece supported on the cone, i.e. real massless propagation). Dimension: \([1/x^2]=+2=[\phi]^2\) for a scalar in 4D. ✓ The massless field mediates a long-range \(1/r\) (Coulomb) static potential, the \(m\to0\) limit of Yukawa.
Symmetry of the propagator. Prove \(D_F(x-y)=D_F(y-x)\) for the free scalar, and explain why this is consistent with the manifestly ordering-dependent definition via \(T\).
Solution
From the definition, \(D_F(y-x)=\theta(y^0-x^0)\langle\phi(y)\phi(x)\rangle+\theta(x^0-y^0)\langle\phi(x)\phi(y)\rangle\), which is the same two terms as \(D_F(x-y)\) with the roles of the \(\theta\)'s and correlators swapped in tandem — i.e. identical. Equivalently, in momentum space \(\tilde D_F(p)=i/(p^2-m^2+i\epsilon)\) is even in \(p\), so its Fourier transform is even in \(x-y\). The \(T\)-symbol is symmetric under exchanging both the operators and their time labels together, so no asymmetry survives. Physically: exchanging the endpoints swaps "emit then absorb" for "absorb then emit," and for the vacuum two-point function these give the same amplitude.