Spacelike-Timelike Correspondence

• Spacelike-timelike correspondence is a framework connecting spatial kinematics with temporal phenomena using analytic continuation, dispersion relations, and controlled geometric mappings in QCD and gravity.
• It enables systematic translation between exclusive QCD amplitudes, hadronic form factors, and entanglement measures, revealing useful dualities and consistency tests in theoretical physics.
• The correspondence also underpins surface dualities, time-space splitting in relativity, and quantum-gravity formulations through precise analytic, geometric, and causal conditions.

Spacelike-timelike correspondence denotes a family of relations that connect objects defined in spacelike kinematics, spacelike separation, or spacelike geometry to counterparts in timelike regimes. In the literature surveyed here, the term does not refer to a single universal theorem. Rather, it names several technically distinct constructions: analytic continuation and crossing in exclusive QCD amplitudes, direct-versus-dispersive matching of hadronic form factors, reconstruction of timelike entanglement entropy from equal-time data, dualities between spacelike and timelike surfaces in Lorentzian geometry, and structural links between spacelike and timelike foliations, bundles, and compactness conditions in relativity and quantum gravity (Mueller et al., 2012, Guo et al., 2024, Truong, 2014). A persistent theme is that the correspondence is strongest when analyticity, factorization, or a controlled geometric structure is available, and weakest when additional singularities, operator exchanges, or global obstructions intervene.

1. Analyticity, crossing, and reconstruction principles

A central mechanism behind spacelike-timelike correspondence is analyticity. In hard exclusive QCD, the generalized Bjorken regime permits collinear factorization, so amplitudes decompose into perturbative coefficient functions convolved with GPDs or DAs. This analytic structure underlies explicit space-to-timelike relations at next-to-leading order, with logarithmic terms acquiring the standard continuation

$$\ln\frac{Q^2}{\mu^2}\rightarrow \ln\frac{Q^2}{\mu^2}-i\pi$$

when the hard scale is continued from spacelike to timelike kinematics (Mueller et al., 2012).

An analogous logic appears in resummation theory. For QCD observables with general leading behavior

$$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$

the timelike and spacelike quantities are related by dispersion-type transforms,

$$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$

and their modified Borel transforms satisfy

$$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$

(Ayala et al., 10 Sep 2025). This shows that the spacelike-timelike bridge can be formulated either at amplitude level or at the level of asymptotic series and renormalon structure.

In hadronic form-factor calculations, analyticity is often supplemented by dispersion relations between real and imaginary parts. In the light-front quark model treatment of $(\pi^0,\eta,\eta')\to\gamma^*\gamma$, the real and imaginary parts obey

$$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$

with the inverse relation for $\mathrm{Im}\,F(q^2)$, and these dispersive results agree with both direct timelike calculation and analytic continuation from the spacelike side (Choi et al., 2017).

2. Exclusive QCD amplitudes: DVCS, TCS, and conformal mappings

The most explicit spacelike-timelike correspondence in perturbative QCD is the relation between DVCS and TCS. At Born level, the two amplitudes are related by simple complex conjugation, reflecting crossing symmetry and the shared GPD content. At NLO, however, the correspondence is no longer exhausted by conjugation. The coefficient functions satisfy

$${}^{\mathrm{TCS}}T(x,\eta)=\pm\left({}^{\mathrm{DVCS}}T(x,\xi=\eta)+i\pi\,C_{\mathrm{coll}}(x,\xi=\eta)\right)^*,$$

with the plus sign for vector and the minus sign for axial-vector channels (Moutarde et al., 2013, Mueller et al., 2012).

This extra $i\pi C_{\mathrm{coll}}$ term is the signature of timelike kinematics at NLO. It produces new absorptive parts and, in detailed coefficient-function calculations, characteristic $\pi^2$ terms analogous to the large K-factor phenomenon in inclusive Drell–Yan analyses (Pire et al., 2011). The phenomenological implication is that spacelike and timelike amplitudes are complementary but not interchangeable beyond LO. In particular, the imaginary part in DVCS appears only in the DGLAP region, whereas in TCS it is present in both the ERBL and DGLAP regions (Pire et al., 2011).

The NLO corrections are not numerically marginal. Several analyses emphasize that gluonic contributions are large, often decisive, even in the medium-energy regime relevant to JLab12 and COMPASS-II. Quark-only NLO treatments can miss not only the normalization but also the sign of some observables, especially in the real parts of CFFs and in timelike observables (Moutarde et al., 2013, Moutarde et al., 2013). This is why the timelike-spacelike comparison is presented in these works as a stringent test of leading-twist factorization rather than a trivial crossing exercise.

A different QCD realization of the correspondence appears in the conformal mapping between the gluon distribution in a highly virtual timelike decay and the gluons in the lightcone wavefunction of a high-energy dipole. In an $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$0 gauge calculation, with vertex times kept fixed, the timelike angular variables and spacelike transverse coordinates are identified through

$$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$1

and the probability integrands match graph by graph before time integration (Mueller, 2018). The correspondence is exact at that level, but divergences appear after integrating over vertex times. A subtraction procedure, described as analogous to the Gell-Mann Low procedure in QED, is then used to define a conformal QCD through NLO by removing the self-energy pieces responsible for coupling renormalization (Mueller, 2018).

3. Hadronic form factors in spacelike and timelike regions

Transition and elastic form factors provide a second major setting in which spacelike-timelike correspondence is studied quantitatively. For $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$2 transition form factors, the light-front quark model analysis introduces a direct timelike calculation that does not rely solely on analytic continuation from the spacelike region. In the $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$3 frame with $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$4, the form factor develops a simple pole structure of the form $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$5, which makes the timelike singularity structure explicit and facilitates numerical evaluation (Choi et al., 2017).

The paper shows complete agreement among three methods: direct timelike LFQM calculation, analytic continuation $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$6 of the spacelike result, and the dispersion relation between $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$7 and $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$8 (Choi et al., 2017). This agreement is used to argue for internal consistency of the LFQM extension into the timelike domain. The same work also reports that the plus-component current for pseudoscalar mesons is free of light-front zero modes, so the transition form factor is valence dominated in the relevant formulation (Choi et al., 2017).

The phenomenology spans both low and high energy. In the low-timelike region, LFQM results are compared with Dalitz-decay data for $$\mathcal D(Q^2)=a(Q^2)^{\nu_0}+\mathcal O(a^{\nu_0+1}),\qquad \mathcal F(\sigma)=a(\sigma)^{\nu_0}+\mathcal O(a^{\nu_0+1}),$$9, including slope parameters at $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$0. In the high-energy regime, the LFQM is combined with QCD factorization to study the asymptotic behavior of TFFs in both spacelike and timelike domains, with generally good agreement except for the high-$$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$1 rise in the BaBar pion data, which is not reproduced (Choi et al., 2017).

A related but distinct implementation appears in the covariant spectator quark model for the $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$2. There, the four elastic form factors $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$3, $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$4, $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$5, and $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$6 are fitted to lattice-QCD data in the spacelike region and to $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$7 data in the timelike region (Ramalho, 2020). For large $$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$8, the timelike form factors are connected to their spacelike counterparts through asymptotic relations of the form

$$\mathcal{F}(\sigma)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}d\phi\,\mathcal{D}(\sigma e^{i\phi}), \qquad \mathcal{D}(Q^2)=Q^2\int_0^\infty \frac{d\sigma\,\mathcal{F}(\sigma)}{(\sigma+Q^2)^2},$$9

and the cross section is expressed through an effective form factor built from electric and magnetic multipoles (Ramalho, 2020). This suggests that, in baryon structure as in meson transitions, correspondence is most reliable in regimes where asymptotic analyticity and perturbative scaling are under control.

4. Timelike entanglement entropy from spacelike data

In quantum field theory, a different spacelike-timelike correspondence concerns entanglement entropy itself. For a broad class of states, timelike entanglement entropy is uniquely determined by a linear combination of the spacelike entanglement entropy and its first-order temporal derivative, all evaluated on a spacelike Cauchy surface such as $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$0 (Guo et al., 2024). The paper formulates this as an explicit integral expression involving the four lightcone-projected endpoints and first and second time derivatives of the equal-time entropy.

The result is verified for the vacuum, thermal, and Bañados states in two-dimensional CFT, and for holographic states including multiple intervals in large-$$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$1 CFTs admitting gravity duals (Guo et al., 2024). In these cases, the full complex timelike entropy is encoded in equal-time data and its local time evolution. For the thermal state, the entropy takes the explicit form

$$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$2

and the imaginary part arises entirely from the first-order time-derivative terms (Guo et al., 2024).

The replica-theoretic explanation is given in terms of twist operators. The Heisenberg evolution satisfies $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$3, and the imaginary part of timelike entanglement entropy originates from the non-commutativity between a twist operator and its first-order temporal derivative at coincident points (Guo et al., 2024). For the states considered, the commutator produces a delta function, which in turn yields the universal constant imaginary part.

The relation is not universal. It breaks down for generic excited states with nontrivial primary exchanges in the twist-operator OPE, and the authors anticipate that more general states would require higher-order time derivatives and additional local operator insertions (Guo et al., 2024). This makes the entanglement case an especially clear example of a spacelike-timelike correspondence that is exact only under identifiable dynamical restrictions.

5. Surface dualities in Lorentzian geometry

In differential geometry, spacelike-timelike correspondence often takes the form of a duality between surfaces of different causal type. A particularly general construction is the generalized Calabi correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in Lorentzian Killing submersions (Lee et al., 2013). If a graph has mean curvature $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$4 in $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$5, its twin graph lies in $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$6, is spacelike, and has mean curvature $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$7; the roles of mean curvature and bundle curvature are exchanged (Lee et al., 2013). The correspondence is involutive up to fiber translations and preserves conformal type.

The induced metrics of twin graphs are conformally equivalent, and the relation extends the classical Calabi correspondence between minimal graphs in $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$8 and maximal graphs in $$\widetilde{B}_{\mathcal{F}_1}(u;\kappa)=\frac{\sin(\pi u)}{\pi u}\,\widetilde{B}_{\mathcal{D}_1}(u;\kappa)$$9 (Lee et al., 2013). Applications include completeness results, nonexistence theorems controlled by the Cheeger constant, and the transfer of moduli-space information between Riemannian and Lorentzian settings.

Another correspondence, more specifically within $(\pi^0,\eta,\eta')\to\gamma^*\gamma$0, maps maximal surfaces with conelike singularities to timelike minimal surfaces by a linear transformation, with Wick rotation as a special case (Patel, 2019). The transformation preserves singularities and establishes a one-one correspondence between the two classes, while also preserving the one-one property of the Gauss map (Patel, 2019). This yields a Kobayashi-type theorem for timelike minimal surfaces: a complete timelike minimal surface with at least one conelike singularity and one-one Gauss map is congruent to

$(\pi^0,\eta,\eta')\to\gamma^*\gamma$1

for nonzero real $(\pi^0,\eta,\eta')\to\gamma^*\gamma$2 (Patel, 2019).

A more recent unification uses the Liouville equation

$(\pi^0,\eta,\eta')\to\gamma^*\gamma$3

with $(\pi^0,\eta,\eta')\to\gamma^*\gamma$4 for spacelike and $(\pi^0,\eta,\eta')\to\gamma^*\gamma$5 for timelike minimal surfaces in $(\pi^0,\eta,\eta')\to\gamma^*\gamma$6 (Cintra et al., 31 Dec 2025). Complex analysis handles the spacelike case, paracomplex analysis the timelike case, and Möbius-type transformations on the Gauss map encode the action of pseudo-isometries. In that formulation, local solutions of the Liouville equation determine the Gauss map and the Weierstrass data for both causal types (Cintra et al., 31 Dec 2025). This suggests a broad geometric pattern: the correspondence is most transparent when both causal classes can be embedded into a single analytic framework.

6. Time-space splitting, foliations, and quantum-geometric extensions

In general relativity, the separation of time and space is itself a geometric correspondence problem. For a spacetime $(\pi^0,\eta,\eta')\to\gamma^*\gamma$7, any separation of time and space is equivalent to introducing a possibly non-smooth Riemannian metric $(\pi^0,\eta,\eta')\to\gamma^*\gamma$8 on $(\pi^0,\eta,\eta')\to\gamma^*\gamma$9 (Truong, 2014). If $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$0 is smooth, it determines a smooth line bundle $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$1, the time bundle, whose fiber at each point is generated by a timelike vector. Time orientability is then equivalent to triviality of this bundle, characterized by the vanishing of the first Stiefel–Whitney class $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$2 (Truong, 2014). The quotient bundle $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$3 formalizes the complementary spacelike structure.

The canonical formulation of gravity yields another spacelike-timelike parallelism. The generalized Sen–Ashtekar–Barbero–Immirzi formalism exploits the isomorphism between $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$4 and $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$5 for spacelike foliation and the analogous isomorphism between $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$6 and $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$7 for timelike foliation (Perlov, 2020). This produces unified Gauss, diffeomorphism, and Hamiltonian constraints for both spacelike and timelike $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$8 foliations, with the Immirzi parameter shown to be covariant under foliation change and to disappear in the self-dual Ashtekar case, leaving the Hamiltonian constraint polynomial (Perlov, 2020).

Loop-quantum-gravity constructions similarly distinguish spacelike from timelike two-surfaces by the choice of normal vector in the linear simplicity constraints. With a spacelike normal, the reduced classical phase space is described by $$\mathrm{Re}\,F(q^2)=\frac{1}{\pi}\mathcal P\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,F(q'^2)}{q'^2-q^2}\,dq'^2,$$9, the reduced Hilbert space is spanned by $\mathrm{Im}\,F(q^2)$0 spin networks, and both spacelike and timelike faces acquire discrete area spectra (Rennert, 2016). The work is framed as a proposal for a quantum theory containing both spatial and temporal building blocks rather than only spacelike ones (Rennert, 2016).

7. Compactness, domain of validity, and common misconceptions

The relativistic theory of support makes clear that not every intuitive spacelike-timelike distinction is equivalent to every other. In a globally hyperbolic spacetime, a closed set $\mathrm{Im}\,F(q^2)$1 is spacelike compact iff it is contained in $\mathrm{Im}\,F(q^2)$2 for some compact $\mathrm{Im}\,F(q^2)$3, and this is equivalent to $\mathrm{Im}\,F(q^2)$4 being compact for every Cauchy surface $\mathrm{Im}\,F(q^2)$5 (Sanders, 2012). It is not enough to test compact intersection only on the Cauchy surfaces belonging to one chosen foliation; explicit counterexamples show that such a criterion can fail (Sanders, 2012).

Timelike compactness is different. A closed set is future or past compact iff it lies to the future or past of some Cauchy surface, equivalently iff its intersection with $\mathrm{Im}\,F(q^2)$6 is compact for every compact $\mathrm{Im}\,F(q^2)$7 (Sanders, 2012). The paper also shows that having compact intersection with all inextendible causal curves is not sufficient for timelike compactness (Sanders, 2012). These results are not metaphorical correspondences but exact logical equivalences, and they provide a rigorous vocabulary for support conditions in hyperbolic field theory.

Across the broader literature, the same caution recurs. In exclusive QCD, timelike amplitudes are not merely analytically continued spacelike amplitudes once NLO effects, absorptive parts, and gluon contributions are included (Pire et al., 2011, Moutarde et al., 2013). In entanglement theory, the linear reconstruction of timelike entropy from spacelike entropy fails for generic excited states (Guo et al., 2024). In geometric analysis, completeness or moduli-space statements transfer only under the explicit hypotheses of the corresponding duality (Lee et al., 2013, Patel, 2019). The general lesson is that spacelike-timelike correspondence is a precise technical relation, not a blanket identification of spacelike and timelike physics.

Taken together, these works show that the phrase “spacelike-timelike correspondence” names a recurrent structural idea rather than a single doctrine. Its implementations range from crossing and dispersion in QCD, to equal-time reconstruction in Lorentzian entanglement, to dualities of surfaces and foliations in geometry and gravity. What unifies them is not a universal formula, but the repeated appearance of controlled maps that transport information across the spacelike-timelike divide while preserving enough analytic, geometric, or causal structure to remain nontrivial and testable.