Lorentzian Correlators and Causal Orderings

A Euclidean correlator does not determine a usable Lorentzian formula until one specifies how every insertion approaches real time. The infinitesimal imaginary times fix the Wightman ordering, the continuation path fixes the sheet in cross-ratio space, and the two boundary values of a cut determine the commutator or double discontinuity used by analytic bootstrap methods.

Required background. Cross ratios and four-point kinematics fix $u$, $v$, $z$, and $\bar z$. Lorentzian boundary conditions and the $i\epsilon$ prescription explain how operator order is encoded by complex time.

Helpful background. Wick rotation and analytic continuation reviews the relation between Euclidean and real-time boundary values.

Lorentzian boundary values of a Euclidean correlator

Use the site’s $(+---)$ Lorentzian metric. Starting from Euclidean times $\tau_i$, continue

$$\tau_i=\epsilon_i+i t_i, \qquad \epsilon_1>\epsilon_2>\cdots>\epsilon_n>0.$$

Then the Euclidean ordering becomes the Wightman product

$$\lim_{\epsilon_1>\cdots>\epsilon_n\to0^+} \langle O_1(\epsilon_1+i t_1,\mathbf x_1)\cdots O_n(\epsilon_n+i t_n,\mathbf x_n)\rangle_E =\langle O_1(x_1)\cdots O_n(x_n)\rangle_W .$$

For a scalar two-point denominator,

$$x_{ij,E}^2 =\lvert\mathbf x_{ij}\rvert^2+(\epsilon_{ij}+it_{ij})^2 =\lvert\mathbf x_{ij}\rvert^2-(t_{ij}-i\epsilon_{ij})^2.$$

Exchanging two operators reverses the sign of the corresponding $\epsilon_{ij}$. Their difference is a commutator boundary value. Spacelike separated local operators commute, so the two continuations agree there; timelike separation can place them on different sides of a branch cut. This tube-domain viewpoint, and not the shorthand instruction “send $t_E$ to $it$,” is what keeps the ordering unambiguous Caron-Huot 2017, §2.2, pp. 6–9.

Time ordering is a sum of Wightman boundary values selected by step functions. Retarded correlators add a commutator and causal support. These are different distributions; one cannot change from one to another by relabeling a single analytic expression after the $i\epsilon$ hierarchy has been discarded.

Cross-ratio sheets and causal orderings

For identical real scalars, use

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(z,\bar z)}{(x_{12}^2x_{34}^2)^{\Delta_\phi}}, \qquad u=z\bar z, \qquad v=(1-z)(1-\bar z).$$

In a Euclidean configuration $\bar z=z^*$, and the correlator is single-valued even though individual conformal blocks need not be. The OPE converges in finite radial domains with an exponentially controlled high-dimension tail Pappadopulo et al. 2012, §§3–5. In Lorentzian signature $z$ and $\bar z$ are independent. The branch points $0$, $1$, and $\infty$ correspond to pairwise null configurations in the three OPE channels. A path winding $1-z$ or $1-\bar z$ changes the sheet; a path winding $z$ or $\bar z$ changes the sheet associated with a different channel.

It is therefore insufficient to state only a final pair of real numbers $(z,\bar z)$. Two paths can end at the same point and represent different operator orderings. A complete specification gives:

1. the initial Euclidean point;
2. the hierarchy of $\epsilon_i$;
3. the path of $z$ and $\bar z$ relative to $0$, $1$, and $\infty$;
4. the final boundary value, such as $\bar z\to\bar z\pm i0$;
5. the OPE channel and causal region in which a subsequent expansion is used.

The figure below is a visual guide to these data. Inspect the distinction between a permutation, which changes the cross-ratio coordinates, and a monodromy, which changes the analytic sheet.

Cross ratios, OPE channels, Euclidean regions, Lorentzian orderings, and analytic sheets. The diagram separates $\operatorname{Disc}\mathcal G=\mathcal G_+-\mathcal G_-$ from the phase-weighted clockwise/counterclockwise combination defining $\operatorname{dDisc}\mathcal G$. It is schematic: arrows represent specified homotopy classes, not continuous motion through an everywhere-analytic region.

The same content can be read without the figure:

Operation	Cross-ratio effect	Physical datum that must remain fixed
Permute points $1\leftrightarrow3$	$(u,v)\mapsto(v,u)$	Relabeled external operators and prefactor
Continue around $\bar z=1$	$1-\bar z\mapsto e^{\pm2\pi i}(1-\bar z)$	Wightman ordering and direction of winding
Approach a cut	$\bar z\to\bar z\pm i0$	Upper or lower boundary value
Take a lightcone limit	$z\to0$ with a declared behavior of $\bar z$	Order of limits, sheet, and OPE channel
Take a Regge limit	Continue to a Regge sheet, then scale both variables	Boost parameter, impact parameter, and smearing

The six basic permutations generated by exchanging four points act on $z$ by

$$z, \quad 1-z, \quad \frac{1}{z}, \quad \frac{1}{1-z}, \quad \frac{z}{z-1}, \quad \frac{z-1}{z},$$

and similarly on $\bar z$. The prefactor in the full correlator transforms too. Checking only the reduced function without its prefactor can therefore create a false sign or power mismatch.

Discontinuities and double discontinuities

For a cut in $\bar z$, define the oriented discontinuity

$$\operatorname{Disc}_{\bar z} f =f(\bar z+i0)-f(\bar z-i0).$$

Reversing this definition reverses every later commutator sign, so the convention must be carried into inversion and dispersion formulas. For identical external scalars, the double discontinuity around $\bar z=1$ is

$$\operatorname{dDisc}_{\bar z=1}\mathcal G =\mathcal G -\frac12\mathcal G^{\circlearrowleft} -\frac12\mathcal G^{\circlearrowright},$$

where the two superscripts denote counterclockwise and clockwise continuation around $\bar z=1$ while $z$ is held on its declared branch. For unequal external dimensions the two continued terms carry phases; omitting those phases changes the inversion kernel’s input. In the reflection-positive identical-scalar configuration used by the Lorentzian inversion formula, the double discontinuity is related to a product of commutators and has a positivity property Caron-Huot 2017, eqs. (2.14)–(2.16), pp. 8–9. The spacetime derivation makes the causal diamonds and commutator ordering explicit Simmons-Duffin, Stanford, and Witten 2018, §§2–3. That positivity statement does not extend automatically to mixed, charged, or nonunitary correlators.

Solvable check: generalized free field

For

$$\mathcal G_{\mathrm{GFF}}(u,v) =1+u^{\Delta_\phi}+\left(\frac{u}{v}\right)^{\Delta_\phi},$$

the crossed contraction contains $(1-\bar z)^{-\Delta_\phi}$. Away from contact distributions, its two monodromies are

$$(1-\bar z)^{-\Delta_\phi} \longmapsto e^{\mp2\pi i\Delta_\phi}(1-\bar z)^{-\Delta_\phi}.$$

Consequently,

$$\operatorname{dDisc}(1-\bar z)^{-\Delta_\phi} =2\sin^2(\pi\Delta_\phi)(1-\bar z)^{-\Delta_\phi}.$$

This check fixes the factor of two and the direction-independent double discontinuity. If $\Delta_\phi$ is an integer, the pointwise expression vanishes, but endpoint or contact distributions may still matter. A pointwise zero must not be used to discard such terms before the integration prescription is specified.

Failure modes

Endpoint without path. Reporting only real $z$ and $\bar z$ loses the homotopy class of the continuation. Record which branch point was wound and in which direction.

Euclidean OPE on a Lorentzian sheet. Euclidean convergence does not follow a correlator through a cut. Re-establish the applicable expansion or use a Lorentzian representation designed for that sheet.

Commutator sign by memory. The sign depends on the definition of Disc and on the $i\epsilon$ ordering. Derive it from the two boundary values.

Positivity outside its domain. Double-discontinuity positivity uses unitarity, reflection positivity, operator reality, and a specific ordering. Mixed or nonunitary systems require a separate argument.

Exercises

For $f(\bar z)=(1-\bar z)^\alpha$, compute its discontinuity and double discontinuity around $\bar z=1$.

Solution

The two monodromies are $f^{\circlearrowleft}=e^{2\pi i\alpha}f$ and $f^{\circlearrowright}=e^{-2\pi i\alpha}f$. Hence

$$\operatorname{dDisc} f =\bigl[1-\cos(2\pi\alpha)\bigr]f =2\sin^2(\pi\alpha)f.$$

On the principal branch and with the declared upper-minus-lower convention,

$$\operatorname{Disc}f =\left(e^{-i\pi\alpha}-e^{i\pi\alpha}\right) \lvert1-\bar z\rvert^\alpha =-2i\sin(\pi\alpha)\lvert1-\bar z\rvert^\alpha.$$

Reversing the discontinuity convention reverses this sign.

References

• Caron-Huot, Simon. “Analyticity in Spin in Conformal Theories.” Journal of High Energy Physics 2017, 078 (2017). doi:10.1007/JHEP09(2017)078.
• Pappadopulo, Duccio, Slava Rychkov, Johnny Espin, and Riccardo Rattazzi. “OPE Convergence in Conformal Field Theory.” Physical Review D 86, 105043 (2012). doi:10.1103/PhysRevD.86.105043.
• Simmons-Duffin, David, Douglas Stanford, and Edward Witten. “A Spacetime Derivation of the Lorentzian OPE Inversion Formula.” Journal of High Energy Physics 2018, 085 (2018). doi:10.1007/JHEP07(2018)085.