Microscopic Wormhole States in Quantum Gravity

• Microscopic wormhole states are precise characterizations of quantum-induced spacetime bridges emerging from entangled fields and modified gravity frameworks.
• The analysis employs quadratic Palatini gravity, quantum foam dynamics, and noncommutative geometry to elucidate state doubling and throat dynamics.
• Implications include resolving classical singularities, deepening our grasp of quantum gravity, and substantiating the ER=EPR correspondence in emergent spacetime.

A microscopic description of wormhole states addresses the precise physical structures, dynamics, and statistical underpinnings by which nontrivial spacetime topology—specifically, wormholes—arises from fundamental or quantum constituents. This topic connects geometric aspects of gravity with quantum entanglement, nonperturbative quantum field theory, and quantum gravity scenarios, and is central to linking gravitational phenomena such as ER=EPR, quantum foam, black hole interiors, and emergent geometry to their underlying microphysics.

1. Wormhole Geometry and Emergence from Quantum Entanglement

A key insight of the modern understanding is the identification of wormhole solutions with quantum entanglement structures. In quadratic Palatini gravity, the Lagrangian

$$\mathcal{L}_G = \frac{1}{2\kappa^2} f(R, Q), \quad \text{with } f(R, Q) = R + l_P^2 (a R + Q)$$

($l_P$ is the Planck length), admits charged solutions in which a wormhole geometry emerges for all charge configurations (Lobo et al., 2014). The line element

$$ds^2 = -A(x,v)e^{2\psi(x,v)} dv^2 + 2 e^{\psi(x,v)}dv\, dx + r^2(x,v)d\Omega^2$$

features an area radius function $r(x, v)$ that achieves a nonzero minimum at the wormhole throat $x=0$—a bounce that prevents curvature singularities and connects two asymptotically flat regions.

This geometric feature has a direct interpretation in terms of entanglement: it gives a spacetime realization of the ER=EPR conjecture, where maximally entangled states of two subsystems (as in EPR pairs or black hole microstates) correspond geometrically to a non-traversable Einstein–Rosen (ER) bridge. The area-minimizing surface at the throat encodes the entanglement entropy, bridging quantum information theory with gravitational geometry.

2. Spacetime Microstructure and Quantum Foam

Quadratic Palatini gravity leads naturally to a spacetime microstructure with foam-like attributes. The fundamental degree of freedom is not a smooth manifold, but an ensemble of transient, microscopic wormholes dynamically generated by quantum vacuum fluctuations. As described in (Lobo et al., 2014), the spontaneous creation and annihilation of entangled particle–antiparticle pairs is always accompanied by the emergence of Planckian, non-traversable wormholes connecting these particles, supporting the view that quantum vacuum “bubbles” pervade spacetime at small scales. This is closely connected to the concept of Wheeler's quantum foam.

This scenario postulates that the vacuum structure is granular: on Planckian distances, wormhole states flicker into and out of existence as the topology of spacetime fluctuates, such that spacetime itself should be regarded as an emergent phenomenon arising from quantum entanglement and topology change at the smallest scales.

3. Microscopic Field Theories and State Doubling

Explicit field theoretic analysis, such as for a real massive scalar field in an Ellis wormhole background (Smolyakov, 26 Jun 2025), demonstrates a doubling of quantum states. For every asymptotic momentum, there are two linearly independent sets of solutions—symmetric and antisymmetric under $r \mapsto -r$—which, after appropriate linear combinations, become localized in either “universe” connected by the wormhole. These solutions have the form

$$\psi_{l,s/a}(k,r) \sim \sin(k|r|+\kappa_{l,s/a}(k))$$

and the normalized “physical” states in isotropic coordinates are

$$\phi_+(\vec{k}, X) = \frac{1}{\sqrt{2}}[\phi_s(\vec{k}, X) + \phi_a(\vec{k}, X)], \ \phi_-(\vec{k}, X) = \frac{1}{\sqrt{2}}[\phi_s(\vec{k}, X) - \phi_a(\vec{k}, X)],$$

which inhabit distinct asymptotic regions. The Hamiltonian correspondingly exhibits a doubled structure,

$$H = \int d^3k \sqrt{\vec{k}^2 + M^2}[a_+^\dagger(\vec{k})a_+(\vec{k}) + a_-^\dagger(\vec{k})a_-(\vec{k})],$$

reflecting the nontrivial topology of the wormhole (with $R^2 \times S^2$ topology) and enabling localized, non-overlapping quantum states in each “universe.”

Similar degeneracy is observed in Schwarzschild spacetime's maximal extension, suggesting that the doubling is a generic feature associated with nontrivial spacetime topology rather than properties of the matter content or specific field.

4. Statistical and Quantum Gravity Mechanisms

Microscopic wormhole states require matter sources that violate the null energy condition. For Planck-scale throats, these can arise from fields with negative kinetic terms (phantom fields), Casimir energy, or energy-momentum distributions produced by quantum effects and noncommutative geometry (Culetu, 2014, Kuhfittig, 8 Aug 2025).

In the Planckian regime:

• Massless scalar fields with negative kinetic terms generate negative energy density and pressure, yielding energy densities independent of Newton's constant, analogous to Casimir vacua.
• Noncommutative geometry replaces Dirac delta distributions with Lorentzian or Gaussian profiles, yielding smeared, quantum-spread energy densities,

$$\rho(r) = \frac{m \sqrt{\gamma}}{\pi^2 (r^2 + \gamma)^2}$$

for a mass $m$ and noncommutative parameter $\sqrt{\gamma}$.

• The Casimir effect provides another quantum-route to negative energy densities, though of insufficient magnitude for astrophysically significant wormholes unless further mechanisms (such as modified gravity $f(Q)$ theories) amplify the effective gravitational response (Kuhfittig, 8 Aug 2025).

Modifications to Einstein gravity, for example, through $f(Q)$ terms in the action (where $Q$ is the non-metricity scalar), introduce further degrees of freedom that can enhance the effective mass and allow even wormholes with low energy density to become macroscopic. The field equations then acquire explicit dependence on the function $f(Q)$ and its derivatives; with appropriate choices of $f(Q)$, the shape function $b(r)$ can be amplified and the integrated mass increased, rendering a macroscopic wormhole possible from microscopic quantum matter sources.

5. Energetics, Junctions, and Dynamics of Microscopic Wormholes

Microscopic analysis using junction formalism reveals that the energy content of the wormhole spacetime differs for various quasi-local energy measures:

• The Komar energy, integrating the contributions of negative energy density and radial pressure, can vanish due to cancellation,

$W_K = 0,$

• The ADM energy is nonzero and negative for these configurations, typically equal in magnitude to the Planck energy,

$W_{ADM} = -b,$

where $b$ is the throat radius ($\sim$ Planck length) (Culetu, 2014).

Dynamical analyses of the wormhole throat, especially when modeled as a thin shell, yield hyperbolic equations of motion of the form

$\ddot{R} R + \dot{R}^2 - 1 = 0,$

with solutions $R(t) = \sqrt{t^2 + b^2}$, corresponding to Lorentz-invariant expanding throats. The acceleration is set by the surface energy density, $a_n = -2\pi|\sigma|$, paralleling the domain wall and bubble nucleation physics of early-universe cosmology.

The microscopic origin of the throat's energy and pressure is thus directly traceable to quantum fields, noncommutative geometry, or quantum-gravitational corrections, rather than hypothesized exotic “fluids.”

6. Relation to the ER=EPR Conjecture and Quantum Information

An important bridge between microscopic wormhole structure and quantum information emerges in the realization that spontaneous vacuum fluctuations not only create maximally entangled EPR pairs but that each such pair is geometrically connected by a nontraversable bridge (a Planck-scale Einstein–Rosen wormhole) (Lobo et al., 2014). The nonlocal, yet non-causal, nature of the bridge directly encodes the quantum entanglement: the wormhole is interpreted as the geometric avatar of the entangled state.

In this framework, spacetime “foaminess,” the ubiquitous quantum creation and annihilation of entangled pairs, and nontrivial wormhole topology are all aspects of the same underlying microphysical quantum entanglement structure.

7. Implications for Quantum Gravity and Emergent Spacetime

The conclusion that spacetime, at the smallest scales, is an emergent, foam-like medium whose connectivity and geometric features are statistical manifestations of quantum entanglement and vacuum fluctuations points to deep implications for the nature of quantum gravity:

• Singularities in classical gravity become resolved by quantum-induced wormhole throats.
• Spacetime connectivity is fundamentally encoded in the entanglement properties of quantum fields.
• Insights from holography and modified gravity suggest that geometry and topology can, in principle, be engineered by suitable (quantum) matter content and interactions.

Collectively, these results support the view that wormhole states, both microscopic and macroscopic, arise as inevitable consequences of quantum gravitational effects, potentially unifying aspects of gravitational, quantum field, and quantum information theory in the description of the fabric of spacetime.