Lorentzian Boundary Conditions and the iε Prescription

The symbol $i\epsilon$ is not decoration on an algebraic denominator. It records a boundary approach that turns a singular expression into a distribution and places its energy poles. For the massive free scalar in the selected Minkowski vacuum, Feynman boundary data give vacuum time ordering and in–out semantics, while retarded and advanced data give future- and past-supported response. These kernels agree away from the mass shell but are different distributions on it. A Feynman correlator is also only an ingredient in a transition-amplitude calculation: it does not specify the external states or perform an LSZ reduction.

Required background. Scalar Propagators, Ordered Correlators, and Sources supplies the correlator taxonomy, raw-correlator versus Green-inverse normalization, and free-scalar mode conventions.

Helpful background. Contour Deformation, Pinches, and Causal Prescriptions supplies legal contour deformation and pinch criteria; Hyperbolic Equations and Causal Propagators supplies support-selected inverses.

Boundary values are part of the Green function

Work in $d$-dimensional Minkowski space with a real free scalar of mass $m>0$. Set

$$P=\Box+m^2, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}, \qquad s=p^2-m^2=(p^0)^2-E_{\mathbf p}^2.$$

The differential expression $P$ does not have a unique inverse until its domain and boundary data are fixed. In momentum space, all the inverses below look like $-1/s$ when $s\neq0$. Their difference is concentrated on the characteristic set $s=0$, exactly where the naïve algebraic quotient is undefined.

There is one normalization distinction to retain throughout. The vacuum time-ordered correlator is

$$D_F(x-y) = \langle0|\mathrm T\{\phi(x)\phi(y)\}|0\rangle,$$

and it obeys

$$P D_F=-i\delta^{(d)}.$$

The corresponding delta-normalized inverse is

$$G_F=iD_F, \qquad P G_F=\delta^{(d)}.$$

Retarded and advanced kernels below use this $G$ normalization as well. Thus all three can be compared as inverses of the same $P$ without silently moving a factor of $i$ between columns.

The elementary boundary-value identity is

$$\lim_{\epsilon\downarrow0} \frac{1}{u\pm i\epsilon} = \operatorname{PV}\frac1u \mp i\pi\delta(u),$$

or, equivalently,

$$\frac1{u+i0}-\frac1{u-i0} =-2\pi i\,\delta(u).$$

These are equalities of tempered distributions after smearing, not pointwise assignments at $u=0$. In particular,

$$\delta(s) = \frac{\delta(p^0-E_{\mathbf p}) +\delta(p^0+E_{\mathbf p})} {2E_{\mathbf p}}.$$

Changing a boundary approach therefore changes an on-shell term even when every off-shell value is unchanged. Weinberg gives the principal-value and delta decomposition, together with its asymptotic meaning, in Weinberg 1995, § 3.1, pp. 112–113.

Two limits must not be conflated:

• for fixed $\epsilon>0$, a deformed finite-dimensional or regulated Gaussian may be absolutely damped and its poles sit a finite distance from the real axis;
• $i0$ denotes the boundary distribution obtained only after performing the relevant integrations or smearings and then taking $\epsilon\downarrow0$.

The first can be a regulator. The second is part of the definition of the Lorentzian distribution. Neither makes $\epsilon$ a physical decay width.

The Feynman limit and vacuum time ordering

For the selected free vacuum,

$$\widetilde D_F(p) = \frac{i}{s+i0}, \qquad \widetilde G_F(p) = -\frac1{s+i0}.$$

As a boundary distribution in the energy variable,

$$\frac1{s+i0} = \frac1{2E_{\mathbf p}} \left( \frac1{p^0-E_{\mathbf p}+i0} - \frac1{p^0+E_{\mathbf p}-i0} \right).$$

Thus the positive-energy pole lies at $+E_{\mathbf p}-i0$, below the real axis, and the negative-energy pole lies at $-E_{\mathbf p}+i0$, above it. With the inherited inverse Fourier factor $e^{-ip^0t}$,

$$\begin{aligned} D_F(t,\mathbf p) &= \int_{-\infty}^{\infty} \frac{\mathrm dp^0}{2\pi} \frac{i e^{-ip^0t}}{s+i0} \\ &= \frac{ \theta(t)e^{-iE_{\mathbf p}t} +\theta(-t)e^{+iE_{\mathbf p}t}} {2E_{\mathbf p}} \\ &= \frac{e^{-iE_{\mathbf p}|t|}}{2E_{\mathbf p}}. \end{aligned}$$

For $t>0$, the large semicircle closes below and is traversed clockwise; the enclosed positive-energy pole produces $e^{-iE_{\mathbf p}t}$. For $t<0$, the contour closes above counterclockwise; the negative-energy pole produces $e^{+iE_{\mathbf p}t}$. Positive frequency therefore propagates toward later time and negative frequency toward earlier time. This is the frequency boundary condition behind vacuum time ordering. It is not future-cone support.

The operator derivation, pole placement, residue calculation, and contact equation appear in Schwartz 2014, § 6.2, pp. 75–77. Schwartz writes the opposite Fourier exponential; translating $k=-p$ reverses the drawn contour orientation while leaving the pole set and resulting time-domain kernel invariant. Weinberg gives the structural step-function form and its positive/negative-frequency asymptotics in Weinberg 1995, § 6.2, pp. 274–277, with his mostly-plus metric and propagator prefactor translated to the site convention.

The opposite boundary value must also carry the conjugated raw-correlator prefactor. The anti-time-ordered vacuum correlator is

$$\widetilde D_{\bar F}(p) = -\frac{i}{s-i0}, \qquad P D_{\bar F}=+i\delta^{(d)}.$$

A bare expression $i/(s-i0)$ is therefore not the anti-time-ordered correlator in this normalization.

Feynman, retarded, and advanced poles in the energy plane

Use delta-normalized inverses for a direct comparison:

$$\begin{aligned} \widetilde G_F(p) &= -\frac1{(p^0)^2-E_{\mathbf p}^2+i0}, \\ \widetilde G_{\mathrm{ret}}(p) &= -\lim_{\epsilon\downarrow0} \frac1{(p^0+i\epsilon)^2-E_{\mathbf p}^2}, \\ \widetilde G_{\mathrm{adv}}(p) &= -\lim_{\epsilon\downarrow0} \frac1{(p^0-i\epsilon)^2-E_{\mathbf p}^2}. \end{aligned}$$

The finite-$\epsilon$ Feynman roots are the roots of $(p^0)^2-E_{\mathbf p}^2+i\epsilon$; their imaginary displacements approach the locations listed below. Retarded and advanced denominators are written differently because their infinitesimal sign depends on the energy branch. One may abbreviate them as $-1/(s\pm i0\,p^0)$ only as boundary values; this is not an algebraic identity between the displayed finite-$\epsilon$ regulators.

The same point is especially transparent in the on-shell decompositions

$$\begin{aligned} \widetilde G_F(p) &= -\operatorname{PV}\frac1s +i\pi\delta(s), \\ \widetilde G_{\mathrm{ret}}(p) &= -\operatorname{PV}\frac1s +i\pi\operatorname{sgn}(p^0)\delta(s), \\ \widetilde G_{\mathrm{adv}}(p) &= -\operatorname{PV}\frac1s -i\pi\operatorname{sgn}(p^0)\delta(s). \end{aligned}$$

They share the same principal-value inverse and differ only in the mass-shell distribution. The raw correlator instead has

$$\widetilde D_F(p) = i\operatorname{PV}\frac1s +\pi\delta(s),$$

which again exposes the normalization factor that distinguishes $D_F$ from $G_F$.

Inverse	Energy-plane poles	Defining meaning
$G_F$	$+E_{\mathbf p}-i0$ and $-E_{\mathbf p}+i0$	selected-vacuum time ordering; positive frequency forward and negative frequency backward
$G_{\mathrm{ret}}$	$+E_{\mathbf p}-i0$ and $-E_{\mathbf p}-i0$	future-supported response
$G_{\mathrm{adv}}$	$+E_{\mathbf p}+i0$ and $-E_{\mathbf p}+i0$	past-supported response

Closing the contours with the same $e^{-ip^0t}$ convention gives

$$\begin{aligned} G_F(t,\mathbf p) &= \frac{i e^{-iE_{\mathbf p}|t|}}{2E_{\mathbf p}}, \\ G_{\mathrm{ret}}(t,\mathbf p) &= \theta(t)\frac{\sin(E_{\mathbf p}t)}{E_{\mathbf p}}, \\ G_{\mathrm{adv}}(t,\mathbf p) &= -\theta(-t)\frac{\sin(E_{\mathbf p}t)}{E_{\mathbf p}}. \end{aligned}$$

The pole placement now makes the temporal support statement visible. Both retarded poles lie below, so $t<0$ encloses no pole; both advanced poles lie above, so $t>0$ encloses no pole. The energy contour proves this time-support statement; finite propagation for the Klein–Gordon equation, equivalently the free-commutator calculation on the prerequisite page, sharpens it to the causal cones. In spacetime,

$$\operatorname{supp}G_{\mathrm{ret}}\subseteq J^+, \qquad \operatorname{supp}G_{\mathrm{adv}}\subseteq J^-.$$

By contrast, $G_F$ has one pole on each side and is not supported in either causal cone. In terms of the vacuum commutator $C(x-y)=\langle0|[\phi(x),\phi(y)]|0\rangle$,

$$G_{\mathrm{ret}}=i\theta(t)C, \qquad G_{\mathrm{adv}}=-i\theta(-t)C, \qquad G_{\mathrm{ret}}-G_{\mathrm{adv}}=iC.$$

Altland and Simons compare time-ordered, retarded, and advanced functions and their half-plane analyticity in Altland and Simons 2023, § 7.3, pp. 394–397. Their response convention contains $-i\theta$ because the perturbing Hamiltonian is written $H_{\mathrm{int}}=+F\phi$; the site source $+J\phi$ in the action means $F=-J$, producing the displayed $+i\theta$ response.

There is also a local normalization check. For fixed $\mathbf p$, each $G$ satisfies

$$(\partial_t^2+E_{\mathbf p}^2)G(t,\mathbf p)=\delta(t).$$

For $G_{\mathrm{ret}}$, the kernel vanishes for $t<0$, is continuous at zero, and has derivative jump

$$\partial_tG_{\mathrm{ret}}(0^+,\mathbf p) - \partial_tG_{\mathrm{ret}}(0^-,\mathbf p) =1.$$

The advanced and Feynman kernels have the same unit derivative jump. This independently verifies

$$P G_F=P G_{\mathrm{ret}}=P G_{\mathrm{adv}}=\delta^{(d)},$$

while the raw correlator retains $P D_F=-i\delta^{(d)}$.

Convergence deformations and vacuum selection

Several related operations are often compressed into the same $i\epsilon$ notation. They should be kept logically separate:

Operation	What it does	What it does not do alone
distributional limit $\epsilon\downarrow0$	defines a boundary value in $\mathcal S'$	provide a pointwise value on shell
energy-pole displacement	fixes frequency boundary data or causal support	choose arbitrary external states
finite positive damping	controls a regulated oscillatory Gaussian or large-time limit	define a general interacting continuum measure
asymptotic vacuum projection	suppresses higher-energy components under stated spectral hypotheses	guarantee a unique vacuum in every theory

Let $R$ label a finite regulator and let $\mathcal C_R$ be its chosen real integration cycle. One may write the free Lorentzian Gaussian schematically as

$$\mathcal Z_{F,\epsilon}[J] = \int_{\mathcal C_R}\mathrm d\mu_R(\phi) \exp\!\left[ -\frac i2\phi\mathbin{\cdot}(P_R-i\epsilon B_R)\mathbin{\cdot}\phi +iJ\mathbin{\cdot}\phi \right],$$

where $B_R$ is positive on the retained real modes and $\epsilon>0$. For $B_R=I$,

$$-\frac i2\phi\mathbin{\cdot}(P_R-i\epsilon I)\mathbin{\cdot}\phi = -\frac i2\phi\mathbin{\cdot}P_R\mathbin{\cdot}\phi -\frac\epsilon2\phi\mathbin{\cdot}\phi.$$

The last term supplies genuine finite-dimensional damping, while $(P_R-i\epsilon I)^{-1}$ approaches the Feynman inverse when the regulator and boundary limits are controlled. This calculation does not establish an absolute interacting continuum path integral.

Vacuum selection can be seen independently in Hamiltonian language. At a finite regulator, or whenever a discrete spectral resolution is appropriate, shift the ground energy to $E_0=0$ and expand a boundary state as $|\psi\rangle=\sum_n c_n|n\rangle$. Then

$$e^{-i(1-i\epsilon)HT}|\psi\rangle = \sum_n c_n e^{-iE_nT}e^{-\epsilon E_nT}|n\rangle.$$

Taking $T\to\infty$ at fixed $\epsilon>0$ suppresses components with $E_n>0$. More generally, for a fixed self-adjoint Hamiltonian bounded below, the spectral theorem makes $e^{-\epsilon HT}$ converge strongly to the spectral projection onto the bottom eigenspace even without a positive gap. After normalization by the corresponding vacuum amplitude, and only then taking $\epsilon\downarrow0$, the limit selects the normalized projection of $|\psi\rangle$ into that eigenspace when the projection is nonzero. A degenerate ground space therefore need not select a unique vector. If the spectral bottom is not a normalizable eigenstate, or the boundary state has zero projection onto it, this argument selects no vacuum vector. Removing the volume or regulator, or identifying this fixed-$H$ projection with adiabatic continuation from another Hamiltonian, can fail in infrared sectors, at level crossings, or for unstable vacua.

Srednicki carries out the ground-state projection and oscillator contour check in Srednicki 2007, §§ 6–7, pp. 60–65. Free-field vacuum boundary wave functionals lead to the same Feynman deformation in Schwartz 2014, § 14.4, pp. 264–267 and Weinberg 1995, § 9.2, pp. 385–389. These arguments relate damping, endpoint vacuum data, and the $+i0$ boundary value in the controlled free setting; they do not make every adiabatic-switching regulator identical at finite $\epsilon$.

From vacuum correlators to in–out data

The normalized in–out source functional with Feynman boundary data generates selected-vacuum time-ordered correlators. Schematically,

$$Z_F[J] = \frac{ \langle0,\mathrm{out}|\, \mathrm T\exp\!\left(i\int\mathrm d^dx\,J(x)\phi(x)\right) |0,\mathrm{in}\rangle} {\langle0,\mathrm{out}|0,\mathrm{in}\rangle},$$

so ordinary source derivatives inherit the vacuum endpoints and Feynman ordering already built into $Z_F$.

This statement does not identify a transition amplitude from its internal denominator. A nonvacuum amplitude additionally requires the chosen in/out wave functionals or asymptotic states. A scattering calculation further requires stable external one-particle poles, state and residue normalization, existence of the relevant asymptotic limits, and infrared control. Those conditions are not consequences of writing $+i0$.

The distinction can be summarized as follows:

Datum	Selects	Still missing
Feynman pole boundary value	vacuum time-ordered internal correlator	external states and reduction
retarded support	response to past source data	an in–out transition amplitude
advanced support	response fixed from future data	a preparation protocol for ordinary causal evolution
endpoint wave functional	a specified boundary state	proof that asymptotic scattering states exist
density operator at finite initial time	an initial-value expectation problem	closed-time-path branch grammar

Nonvacuum density matrices and finite-time expectation values are developed later in this chapter. They are not obtained by changing a single sign in the Feynman denominator.

Distributional checks and failure modes

Four checks diagnose most prescription mistakes.

1. Smearing check. Pair the proposed kernel with a Schwartz test function before taking $\epsilon\downarrow0$. If the argument instead substitutes $\epsilon=0$ at the pole, no distribution has been defined.
2. Pole–support check. Under $e^{-ip^0t}$, both poles below must give future support and both above must give past support. One pole on each side gives neither.
3. Contact check. Apply $P$ and recover $\delta^{(d)}$ for a delta-normalized inverse, or $-i\delta^{(d)}$ for the raw $D_F$ correlator.
4. State-and-order check. Name the state, ordering, or support condition that the pole prescription implements. The algebraic denominator alone is not an answer.

Several tempting statements fail these checks.

“Feynman means retarded causality.” Feynman ordering has one energy pole in each half-plane and is not future-supported. Causal response uses the retarded commutator kernel.

“Retarded means $s+i0$.” A uniform invariant $+i0$ is Feynman. Retarded displacement depends on the sign of $p^0$ and places both energy poles below.

“A small $\epsilon$ is a small width.” The prescription defines a boundary distribution and is removed at the end. A physical width comes from dynamical analytic structure, typically a continued complex pole, and remains a physical parameter.

“The same finite regulator proves the same vacuum.” Distinct damping or switching procedures can approach the same boundary value without being identical away from the limit. Vacuum projection also needs overlap and spectral hypotheses, and may fail or become nonunique.

“The prescription solves every contour problem.” Multiple denominators can pinch an integration contour, thresholds introduce cuts, gauge choices can add auxiliary poles, and massless zero modes can invalidate the massive free benchmark. Those are separate analyses.

Check your understanding

1. Recover the three time kernels

Using $e^{-ip^0t}$, close each energy contour for $t>0$ and $t<0$. Which poles contribute for Feynman, retarded, and advanced boundary data?

Answer

For $t>0$, close below clockwise. Feynman encloses only $+E-i0$; retarded encloses both poles; advanced encloses none. For $t<0$, close above counterclockwise. Feynman encloses only $-E+i0$; advanced encloses both poles; retarded encloses none. The residues give $G_F=i e^{-iE|t|}/(2E)$, $G_{\mathrm{ret}}=\theta(t)\sin(Et)/E$, and $G_{\mathrm{adv}}=-\theta(-t)\sin(Et)/E$.

2. Locate the on-shell difference

Why can two prescriptions agree for every $s\neq0$ yet define different Green functions?

Answer

Their difference is a distribution supported at $s=0$. The identity $(s+i0)^{-1}-(s-i0)^{-1}=-2\pi i\delta(s)$ makes the missing on-shell term explicit. Off-shell algebra cannot recover it.

3. Test a vacuum-projection claim

What goes wrong if $c_0=0$ in the damped Hamiltonian limit, or if the ground state is degenerate?

Answer

If $c_0=0$, damping cannot create a missing vacuum component; it selects the lowest-energy component that is actually present, if the limit exists. With a degenerate ground energy, it selects the normalized projection into the overlapping ground subspace rather than a unique vacuum.

4. Separate a correlator from an amplitude

Does $i/(p^2-m^2+i0)$ specify a scattering amplitude?

Answer

No. It specifies the free vacuum time-ordered two-point boundary value. A scattering amplitude also needs external in/out states, stable poles and residue normalization, asymptotic limits, interaction data, and infrared control.

Where the prescription hands off

• Wick Rotation and Analytic Continuation asks whether a declared Lorentzian boundary value can be reached from a Euclidean contour without crossing singularities.
• Retarded, Advanced, and Spectral Correlators develops causal response and its spectral representation.
• Contour Deformation, Pinches, and Causal Prescriptions treats legal deformations and pinch obstructions beyond this one-denominator benchmark.
• Thermal Propagators and Spectral Representations supplies the state, periodicity, and contour data needed at finite temperature.
• A corresponding computational contour comparison would require a separate implementation with declared inputs and numerical checks; the static pole, residue, support, and contact checks above do not depend on one.

References

• Altland, Alexander, and Ben Simons. Condensed Matter Field Theory. Third ed. Cambridge: Cambridge University Press, 2023. DOI.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.

• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.

• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. First ed. Cambridge: Cambridge University Press, 1995. DOI.