Annihilation-to-Nothing: Quantum Gravity & Beyond

• Annihilation-to-Nothing is a process where fields, spacetime geometries, or particle systems transition into a complete state of nullity through deterministic or probabilistic dynamics.
• Its realization spans quantum gravity, cosmology, and statistical mechanics, illustrated by bubble of nothing decays, destructive wave interference, and kinetic particle annihilation.
• The phenomenon provides insights into singularity avoidance, vacuum instability, and the limitations of effective theories, highlighting the need for UV-complete models.

The annihilation-to-nothing scenario refers to a class of physical, geometric, or quantum processes in which some field, state, spacetime geometry, or material system transitions—deterministically or probabilistically—into an absence of excitations, geometry, or even spacetime itself. This concept arises in quantum gravity, cosmology, kinetic theory, and mathematical models of low-dimensional universes. In high-energy and gravitational frameworks, annihilation-to-nothing typically denotes a vacuum instability leading to "nothingness" construed as the endpoint of tunneling processes—often realized as topology-changing transitions such as the "bubble of nothing." In kinetic particle systems and statistical mechanics, the concept connotes complete mass extinction or mass decay to zero, driven by annihilating dynamics. The term is also deployed, in a more formal sense, as a quantum-gravitational boundary condition ensuring singularity avoidance in black-hole or cosmological wavefunctions.

1. Annihilation-to-Nothing in Quantum Cosmology and Gravity

In semiclassical quantum gravity, particularly in the Wheeler–DeWitt framework for black hole and cosmological interiors, annihilation-to-nothing appears as a dynamical enforcement of the DeWitt boundary condition, $\Psi = 0$, at the would-be classical singularity $r=0$. Canonically quantized metrics (e.g., the Schwarzschild interior in Kantowski–Sachs variables $(X,Y)$) yield a quantum-mechanical equation

$$\left(\partial_X^2 - \partial_Y^2 + 4r_s^2 e^{2Y}\right) \Psi(X,Y) = 0$$

whose general solution consists of superposed wave packets representing "ingoing" and "outgoing" classical geometries. Properly constructed, these wave packets overlap with a relative $\pi$ phase near $X\approx 0$ (i.e., the transition region between classical interior and singularity), resulting in destructive interference and a vanishing total wavefunction: both branches "annihilate to nothing" at the quantum transition surface (Bouhmadi-López et al., 2019, Brahma et al., 2021).

This scenario is robust in the minisuperspace WDW context, where reflection symmetry and path separation in superspace naturally produce cancellation at the singularity, offering a non-singular quantum evolution and, under some interpretations, a quantum resolution of the black hole information paradox (Bouhmadi-López et al., 2019, Chien et al., 2023). The mechanism can be mapped onto matter models such as collapsing null shells, where time-reversal symmetry guarantees the presence of an anti-shell, and the shell and anti-shell branches destructively interfere at the classical singularity, enforcing the DeWitt boundary condition (Brahma et al., 2021).

2. Spacetime Annihilation via Bubble of Nothing Decays

In higher-dimensional quantum gravity and string theory contexts, annihilation-to-nothing is realized by the nucleation of a "bubble of nothing," a topologically non-trivial decay of a compactified spacetime into a region where spacetime ceases to exist (Brown et al., 2011, Blanco-Pillado et al., 2023). The archetypal example is Witten's decay of ${\mathbb M}_4 \times S^1$ via an O(4)-symmetric instanton, in which the $S^1$ radius shrinks smoothly to zero at the bubble wall: $$ds^2 = \frac{dr^2}{1 - R^2/r^2} + r^2 d\Omega_3^2 + (1 - R^2/r^2) R^2 d\phi^2$$ for $r \geq R$, such that at $r = R$ the $S^1$ "pinches off," excising a hole in spacetime—this boundary is "nothing" as the volume (and all fields) shrink to zero size, and the interior's curvature length approaches zero, i.e., nothing is identified with the limit of AdS curvature length $\ell \to 0$ (Brown et al., 2011).

These transitions fit into the Coleman–De Luccia (CDL) framework as a limiting process: the instanton interpolates between a "false vacuum" (e.g., stabilized compactification) and a true vacuum in which certain geometric moduli go to a limit that removes spacetime entirely. The corresponding Euclidean action difference $\Delta S_E$ determines the nucleation rate,

$\Gamma \sim \exp(-\Delta S_E/\hbar),$

and as the bubble approaches the nothing limit, the interior potential $V_4(R)$ diverges to $-\infty$ as $R \to 0$ (Brown et al., 2011, Blanco-Pillado et al., 2023).

A systematic classification of "bubbles of nothing" (BoNs) based on the asymptotic structure of the modulus potential $V(\phi)$ reveals multiple types (type 0, type $-$, type $+$, type $-*$), distinguished by their scaling behavior and higher-dimensional interpretation. The existence and quenching of BoN decays are controlled by both geometric and dynamical parameters, such as the size of compactification radii and the specifics of the modulus potential. Not all compactifications admit BoN instability—a critical radius must be exceeded for annihilation-to-nothing to proceed (Blanco-Pillado et al., 2023).

3. Annihilation-to-Nothing in Kinetic and Statistical Systems

The annihilation-to-nothing scenario arises in low-dimensional systems of randomly distributed annihilating particles. In the one-dimensional ballistic annihilation model, particles start at Poisson points on $\mathbb{R}$ or $\mathbb{R}_+$ and move at constant (possibly zero) velocities until they collide and annihilate pairwise (Sidoravicius et al., 2016, Biswas et al., 2021).

For symmetric velocity distributions on $\mathbb{R}_+$, all positive-velocity particles are almost surely annihilated—complete extinction of one sector occurs, i.e., "annihilation to nothing." On the full line with a "three-speed" distribution (e.g., velocities $\{-1,0,+1\}$ with probability $p$ for $0$), a phase transition is observed: for $p > 1/3$, zero-velocity particles can survive forever, while for lower $p$, the system typically experiences mass extinction (Sidoravicius et al., 2016).

The kinetic picture is formalized in the Kac-model framework for annihilating particle systems, where propagation of chaos holds for bounded kernels, and the macroscopic evolution is described by a nonlinear Boltzmann hierarchy with annihilation. In the homogeneous limit, the total density $n(t) = \int f(t,v) dv$ strictly decreases, and $\lim_{t \to \infty} n(t) = 0$, giving literal mass annihilation-to-nothing (Lods et al., 2019).

4. Boundary Condition Realizations and Quantum Decoherence

The imposition of vanishing (DeWitt) boundary conditions at classically singular or otherwise inaccessible boundaries embodies annihilation-to-nothing as a quantum mechanism. In models of black hole interiors (with or without cosmological horizons), numeric and analytic solutions of the WDW equation demonstrate that properly initialized wave functionals (with oppositely signed Gaussian branches) destroy each other via interference at a designated transition surface ($X=0$), enforcing $\Psi = 0$ and confining the quantum state to the non-singular region (Bouhmadi-López et al., 2019, Chien et al., 2023).

This annihilation has significant interpretive consequences: it provides an intrinsic, boundary-driven mechanism of quantum-to-classical transition (classicalization), acts as a dynamical realization of gravitational decoherence, and underpins a natural "no-boundary" outcome in the quantum gravitational path integral. The mechanism generalizes to settings with cosmological horizons, where a similar boundary condition at a finite coordinate radius cuts off quantum coherence and prevents leakage of information or norm to infinity (Chien et al., 2023).

5. Annihilation-to-Nothing in Effective Theories and the Limits of GR

The robustness of the annihilation-to-nothing scenario in quantum gravity critically depends on the underlying theory of gravity. Explicit calculations in modified Wheeler–DeWitt equations incorporating higher curvature corrections (such as $R^2$, $R_{\mu\nu}R^{\mu\nu}$, and $Riemann^3$ terms) show that the cancellation at the transition surface $X=0$ is destroyed by these corrections (Kanai, 11 Dec 2025). Perturbative expansions around classical GR reveal that the would-be vanishing wavefunction $\Psi(X=0,Y)$ picks up non-vanishing, cutoff-dependent contributions as $\gamma$ or $\eta$ (coefficients of higher curvature operators) become nonzero, resulting in ill-defined, non-normalizable solutions at the singularity.

Therefore, singularity resolution via annihilation-to-nothing—at least in EFT extensions of GR—is not viable, and UV-complete quantum gravity, with fundamentally new degrees of freedom or nonlocal effects, is required to restore the boundary condition $\Psi = 0$ at the quantum “end of spacetime” (Kanai, 11 Dec 2025). This limitation highlights that the quantum gravitational resolution of spacetime singularities through annihilation-to-nothing is essentially a statement about physics beyond low-energy EFT.

6. Geometric and Model-Analogy Realizations

Toy models such as the "Lineland" construction—where particles are intersections of a moving circle with a line—provide instructive geometric realization of annihilation-to-nothing. Here, particle pairs are created at coincident points, separate under mutual "interaction," and annihilate when they rejoin, with the system's macroscopic laws inferred directly from kinematic geometry (Platini et al., 2014). The analytic tractability of such analogies illuminates the universal features of annihilation and boundary-driven extinction across diverse physical regimes.

In cosmology, dark matter models involving microhalo formation and subsequent gravothermal collapse demonstrate a scenario where annihilation proceeds to near-completion: heavy dark matter particles, after forming dense structures, can annihilate completely, returning the universe to a radiation-dominated phase in time for successful nucleosynthesis (Chuzhoy, 2010). This process is explicitly constructed as "annihilation to nothing" in the context of early universe relic composition.

7. Generalized Implications and Theoretical Significance

The annihilation-to-nothing scenario constitutes a unifying paradigm for interpreting singularity avoidance, complete extinction, or spacetime decay across gravitational, quantum, and statistical frameworks. Its mathematical realization relies on the interplay of symmetries, topological changes, and interference effects in both microscopic and macroscopic systems.

In gravitational physics, annihilation-to-nothing codifies the notion that spacetime itself can be dynamically removed via nonperturbative tunneling events, or suppressed in quantum evolution via boundary conditions or interference. In kinetic and statistical mechanics, it embodies the physical extinction of all excitations or modes—often via probabilistic, random, or thermodynamic processes. Its full realization, however, may depend critically on the UV completion of the underlying theory and the precise degrees of freedom available near classical singularities or phase boundaries (Brown et al., 2011, Blanco-Pillado et al., 2023, Kanai, 11 Dec 2025).