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arXiv:2603.28379

The force of attraction between nucleons due to vacuum fluctuation

Anupam Ghosh

hep-phhep-thnucl-thquant-phSubmitted Mar 30, 2026• Announced Mar 31, 2026• Revised Jul 14, 2026 (v2)
v1 · Mar 30, 2026v2 · Jul 14, 2026
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Abstract

We investigate quantum vacuum interactions arising from the zero-point fluctuations of a spatially confined massive scalar field. Deriving analytical expressions for the planar interaction energy and vacuum pressure, we identify a fundamental transition from the canonical power-law scaling of massless fields to a distinct quantum saturation regime. We prove that in the macroscopic limit, where the boundary separation far exceeds the field's Compton wavelength ($a \gg λ_c$), the interaction energy does not vanish; instead, it asymptotes to a persistent constant, $-\hbar c / (24πλ_c^3)$. This reveals a cohesive zero-point energy reservoir inherent to massive vacua. Applying this formalism to the femtometer scale of the deuteron ($^2$H) nuclei, we demonstrate that confining massive pion fluctuations generates an attractive force between nucleons.

Comments
6 pages, 3 figures; expanded discussions and 2 new figures added

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We investigate quantum vacuum interactions arising from the zero-point fluctuations of a spatially confined massive scalar field. Deriving analytical expressions for the planar interaction energy and vacuum pressure, we identify a fundamental transition from the canonical power-law scaling of massless fields to a distinct quantum saturation regime. We prove that in the macroscopic limit, where the boundary separation far exceeds the field's Compton wavelength (a≫λca \gg λ_ca≫λc​), the interaction energy does not vanish; instead, it asymptotes to a persistent constant, −ℏc/(24πλc3)-\hbar c / (24πλ_c^3)−ℏc/(24πλc3​). This reveals a cohesive zero-point energy reservoir inherent to massive vacua. Applying this formalism to the femtometer scale of the deuteron (2^22H) nuclei, we demonstrate that confining massive pion fluctuations generates an attractive force between nucleons.
We investigate the vacuum and thermal fluctuations of a neutral massless scalar field living in Minkowski spacetime and interacting with a finite number of point-like obstacles, modelled by zero-range potentials. The system is described rigorously in terms of self-adjoint realizations of the Laplacian, under assumptions ensuring the absence of instabilities. Using the relative zeta-function technique, we determine the renormalized connected partition function and derive explicit expressions for the thermodynamic observables, characterizing both their low- and high-temperature behaviours. Furthermore, we derive of a convergent Born series expansion for the Casimir energy, which identifies multiple-scattering processes as the mechanism underlying vacuum forces. The latter decompose into pairwise contributions directed along the lines joining the obstacles, with intensities depending non-locally on the full configuration. We also present some numerical results for identical obstacles, indicating that the Casimir forces are always attractive in this context.
We show that quantum fluctuations of electromagnetic fields induce an extensive zero-point energy in insulators. The zero-point energy density is proportional to the quantum fluctuation of electric polarization in the many-body ground state, a fundamental quantum geometric property of solids known as the quantum weight. For fields produced by a superconducting LC circuit, the zero-point energy results in a repulsive force between the circuit and the material and an attractive force on the capacitor plates, which we call quantum geometric force. The force on the capacitor plates is proportional to the quantum weight, inversely proportional to the plate area, and independent of the plate distance. The proposed effects provide direct experimental access to the many-body quantum geometry and reveal a new quantum geometric effect in macroscopic solids induced by vacuum fluctuation.
Considering the Casimir effect due to phononic excitations of a weakly interacting dilute Bose–Einstein condensate (BEC), we derive a renormalized expression for the zero-temperature Casimir energy of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.
We investigate the Casimir effect for a massive real scalar field confined between two perfectly reflecting parallel plates in the presence of a position-dependent effective mass, a mechanism for coupling a scalar background to a scalar field. Exact normal modes are obtained by solving the corresponding Klein-Gordon equation, leading to a transverse energy spectrum that exhibits a characteristic Landau-like structure despite the absence of an external magnetic field. Upon quantization of the field, the vacuum energy is evaluated by means of generalized zeta-function regularization together with an appropriate renormalization procedure. The renormalized vacuum energy naturally separates into a Landau-like contribution and an additional term induced by the spatial dependence of the effective mass. We show analytically and numerically that both contributions are exponentially suppressed in the strong-coupling regime. In the opposite limit, the Landau-like contribution smoothly reproduces the standard vacuum energy for a massive scalar field confined between parallel plates, whereas the additional contribution becomes singular owing to the restricted domain of validity of the exact spectrum. Except in the vicinity of this singular limit, the vacuum energy is shown to be dominated by the Landau-like sector. Our results establish a direct connection between position-dependent effective masses and boundary-induced quantum vacuum phenomena, providing a new exactly solvable framework for the investigation of Casimir effects in spatially inhomogeneous relativistic systems.
\abstract{The Lamb shift, one of the most fundamental interactions in atomic physics, arises from the interaction of H atoms with the electromagnetic fluctuations of the quantum vacuum. The energy shift has been computed in a variety of ways. The energy shift, as Feynman and Power demonstrated, equals the change in the vacuum energy in the volume containing the H atoms due to the change in the index of refraction arising from the presence of the H atoms. By using this result and a group theoretical calculation of the contribution to the Lamb shift from each frequency of the vacuum fluctuations, we can obtain an expression for the size of the region of vacuum energy for each frequency \texorpdfstring{ωωω}{ω} around the H atom due to the Lamb shift. The ground state atom is surrounded by a region of positive vacuum energy that extends well beyond the atom for low frequencies. This region can be described as a steady state cloud of virtual quanta. For energies \texorpdfstring{E=ℏωE=\hbarωE=ℏω}{E=hbar omega} eV less than 1 eV, the radius of the positive energy region is approximately 14. 4/E Angstroms. For a vacuum fluctuation of wavelength \texorpdfstring{λλλ}{lambda} the radius is \texorpdfstring{(α/2π)λ(α/2π)λ(α/2π)λ}{(alpha/2pi) lambda}. Thus, for long wavelengths, the region has macroscopic dimensions. The energy-time Uncertainty Relation predicts a maximum possible radius that is larger than this by a factor of \texorpdfstring{1/4α1/ 4α1/4α}{1/(4 alpha)}.} \keyword{Bethe; radiative shift; shift spectral density; spectral volume; vacuum fluctuations; vacuum field; Lamb shift; QED; energy field, renormalization, zero point fluctuations; hydrogen atom}.
Heisenberg uncertainties lead to inevitable fluctuations in the measurement outcomes for quantum-mechanical observables. For quantum fields, these uncertainties result in random spatial structures in snapshots of a field, even when the field is in its ground (vacuum) state. Such `vacuum fluctuations' are at the heart of a wide range of phenomena, from spontaneous decay processes to the Casimir force and Hawking radiation. Their existence is a key manifestation of the quantumness of the physical world, but usually it is only their consequences that are directly observed. Here, we directly observe spatial vacuum fluctuations of a bosonic quantum field. Our experiments are based on a homogeneous planar atomic Bose--Einstein condensate. The condensate comprises two coherently coupled interacting components (spin states), and the quantum field describes its spin degrees of freedom. In the regime where the interactions dominate over the coherent coupling, our system emulates a (massive relativistic) sine-Gordon field. Images of the field reveal simultaneous fluctuations on different length scales, with scale-dependent amplitudes consistent with theoretical predictions for a vacuum state. Observing such fluctuations in the sine-Gordon limit opens many possibilities for laboratory simulations of relativistic fields in regimes that are presently not theoretically tractable.