Position Space Analytic Continuations

• Position space analytic continuations are rigorous extensions of spatial functions into the complex domain, revealing hidden analytic properties and regularity in various physical systems.
• This technique employs complexification and contour deformation to establish real-analyticity of solutions in PDEs and temper quantum field correlators via analytic regularization.
• Applications span from proving unique continuation in nonlocal parabolic equations to probing deep holographic aspects of black hole interiors and matching position-momentum formulations.

Position space analytic continuations refer to the mathematically precise extension of functions defined on real spatial coordinates (and, in physical contexts, often spacetime coordinates) into the complex domain. This technique underpins both foundational results in analytic regularity theory for partial differential equations (PDEs) and modern analytic approaches to quantum field theory, gravitational holography, and the structure of correlation functions. Analytic continuation in position space can reveal hidden properties, regularity, and physical regimes not accessible by real coordinates alone, and is tightly connected with the interplay between position- and momentum-space formulations in both mathematics and theoretical physics.

1. Analytic Continuation in the Theory of Partial Differential Equations

The analytic continuation of solutions to PDEs in spatial variables is pivotal for understanding regularity properties and unique continuation. In the context of nonlocal parabolic equations, solutions to fractional powers of parabolic operators, such as $$L^s = (\partial_t - \mathrm{div}_x(B(x)\nabla_x))^s$$ for $0 < s < 1$, typically lack an obvious local structure in the $x$-variables. The extension method, as described in (Banerjee et al., 2022), circumvents this by embedding the $n$-dimensional problem into a higher-dimensional space through the introduction of an auxiliary extension variable $y>0$, yielding a degenerate, yet locally parabolic, PDE: $$y^a \partial_t U - \mathrm{div}_{x,y}\left(y^a\widetilde{B}(x)\nabla_{x,y}U\right) = 0,$$ where $a=1-2s\in(-1,1)$ and $\widetilde{B}(x)$ is a uniformly elliptic $(n+1)\times(n+1)$ matrix. The function $u(t,x)$ of the original problem is recovered as the trace $U(t,x,0)$. A key result is that, under analyticity assumptions on $B(x)$, every weak solution $U$ even in $y$ is real-analytic in the $x$-variables, for each fixed $(t,y)$. The proof leverages the complexification of $x$, establishing the holomorphic extension of the Green's function kernel and deriving explicit Cauchy-type bounds, such as

$$\sup_{|x-x_0|<\rho} |\partial_x^\alpha U| \leq C M \rho^{-|\alpha|}|\alpha|!,$$

which controls the Taylor expansion's convergence radius in $x$ and underpins unique continuation statements for the original operator (Banerjee et al., 2022).

2. Analytic Continuations in Position Space Correlators

Position space analytic continuations extend to the field of quantum field theory (QFT) and conformal field theory (CFT), where correlation functions are naturally defined on real spacetime but exhibit richer analytic structure in the complexified domain. In momentum-space and conformal algebraic contexts, such as in the two-dimensional Galilean Conformal Algebra (GCA), position-space correlators with certain parameters may fall outside the space of tempered distributions, precluding straightforward Fourier transforms.

The specific case addressed in (Chetia et al., 25 Dec 2025) involves two- and three-point GCA correlators: $$G^{(2)}(x_{1}, t_{1}; x_{2}, t_{2}) = C^{(2)} \delta_{\Delta_1,\Delta_2}\delta_{\xi_1,\xi_2}(t_{12})^{-2\Delta_1} \exp\!\left( \frac{2\xi_1 x_{12}}{t_{12}} \right),$$ where the boost eigenvalues $\xi_i$ are real. However, the unbounded exponentials hinder the existence of the Fourier transform. By analytically continuing to imaginary boost eigenvalues $\xi_i \to -i\widetilde{\xi}_i$, the exponentials become oscillatory,

$\exp\left(-2i\widetilde{\xi}\,\frac{x}{t}\right),$

rendering the correlators tempered and their Fourier transforms well-defined. Notably, the transformed correlators are found to coincide exactly with the directly constructed momentum-space correlation functions from the Ward identities, after this analytic continuation. This establishes analytic continuation as a technical necessity for consistent position-momentum correspondence in certain field-theoretic settings (Chetia et al., 25 Dec 2025).

3. Position Space Analytic Continuations and Black Hole Interiors

Analytic continuation in position space plays a fundamental role in holographic duality, particularly in the extraction of bulk black hole interior physics from the boundary quantum theory. In the large-mass (WKB/geodesic) approximation, the boundary two-point function in an asymptotically AdS spacetime is represented as a Fourier integral over momentum-space Wightman functions,

$$G^+(\Delta\tau, \Delta x) \simeq \int dw\,dk\, e^{i w \Delta\tau + i k \Delta x} e^{m Z(w,k)},$$

whose saddle-point evaluation identifies the classical bulk spacelike geodesic connecting the boundary points. However, analytic continuation of the insertion points (complexification of $(\tau, x)$ variables) can deform the integration contour in $(w, k)$ and force it to pass around singularities or branch points in the $Z(w,k)$ plane. Specifically, rotating the separation variable, such as $(\tau-ix)\to e^{-2\pi i}(\tau-ix)$, causes the contour to encircle new branch cuts corresponding to additional, typically timelike, geodesics that traverse deep into the black hole interior, potentially reaching or crossing the inner (Cauchy) horizon (Ahmad et al., 5 Jan 2026).

This methodology provides a direct boundary signature of regions causally inaccessible from the exterior. For instance, in the rotating BTZ black hole, a $2\pi$ rotation of the branch point corresponds to shifting operator insertions into the second asymptotic universe and threading the geodesic through the black hole interior, as reflected in the boundary two-point function's monodromy structure and periodicity in complexified coordinates.

4. Technical Methods and Nonperturbative Effects

Position space analytic continuation is intimately linked to the analytic structure of the relevant Green or heat kernels, the behavior of their complex extensions, and the convergence of Fourier or Laplace integrals under complex deformation of parameters or insertion points. Complexification of the kernel variables enables the derivation of sharp bounds and holomorphicity domains, as exemplified by the extension of the heat kernel $p(x,x',t)$ in (Banerjee et al., 2022) and the construction of holomorphic product kernels in both spatial and extension variables.

Nonperturbative effects are rendered accessible through position-space analytic continuation. When continued sufficiently far in the imaginary direction (e.g., in imaginary time or cross ratios), the analytic structure (Stokes phenomena) induces contributions from infinitely many "image" saddles in the geodesic approximation. Each image corresponds to a distinct class of geodesic trajectories, capturing timelike windings and phase jumps as they traverse nontrivial topological or causal structures, such as black hole interiors or universes connected via analytic continuation (Ahmad et al., 5 Jan 2026). These nonprincipal contributions are summable as image sums and are sensitive to global geometric features and quantum corrections.

5. Applications and Physical Implications

The ability to perform analytic continuation in position space has significant implications for both rigorous analysis and physical interpretation. In mathematical analysis, it underpins strong unique continuation principles for nonlocal parabolic and related operators, as the real-analyticity forced by such continuation implies that vanishing on an open set entails global triviality of solutions (Banerjee et al., 2022). In theoretical physics, analytic continuation of correlators supplies a calculational tool to match position-space and momentum-space constructions, regulate divergent expressions, and unearth the physical influence of analytic structures on observables.

Notably, in holographic settings, such continuation provides a calculable boundary probe of interior phenomena, including the instability of Cauchy horizons (mass inflation, stress tensor instability) and can reveal quantum or semiclassical effects that arise when boundary correlators are continued beyond naive domain of holomorphy (Ahmad et al., 5 Jan 2026). This establishes analytic continuation as a core tool in both mathematical analysis and the study of quantum gravitational systems.

6. Summary Table: Key Examples of Position Space Analytic Continuation

Context	Analytic Continuation	Outcome
Nonlocal parabolic PDEs	Complexification of $x$	Real-analyticity; unique continuation (Banerjee et al., 2022)
2D GCA correlators	$\xi\to -i\widetilde{\xi}$	Tempered correlators; Fourier transform matches Ward-identity solution (Chetia et al., 25 Dec 2025)
Holography/Black holes	Complex $({\tau},x)$ insertions	Boundary access to interior geodesics; image sums for nonperturbative contributions (Ahmad et al., 5 Jan 2026)

A plausible implication is that, with further development, analytic continuation techniques in position space may offer broader insights into the analytic topology of solution spaces for nonlocal operators, as well as concrete computational handles on quantum effects associated with deep bulk or topologically nontrivial regions in gravitational duals.