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arXiv:2605.10725

Vacuum and thermal fluctuations of a scalar field with point interactions

Davide Fermi, Marco Gurgoglione

math-phcond-mat.mes-hallquant-phSubmitted May 11, 2026• Announced May 12, 2026• Revised Aug 26, 2026 (v2)
v1 · May 11, 2026v2 · Aug 26, 2026
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Abstract

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.

Comments
32 pages, 7 figures

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We study a dilute and ultracold Bose gas of interacting atoms by using an effective field theory which takes into account the finite-range effects of the interatomic potential. Within the formalism of functional integration from the grand canonical partition function, we derive beyond-mean-field analytical results which depend on both the scattering length and the effective range of the interaction. In particular, we calculate the equation of state of the bosonic system as a function of these interaction parameters both at zero and finite temperature including one-loop Gaussian fluctuation. In the case of zero-range effective interaction, we explicitly show that, due to quantum fluctuations, the bosonic system is thermodynamically stable only for very small values of the gas parameter. We find that a positive effective range above a critical threshold is necessary to remove the thermodynamical instability of the uniform configuration. Remarkably, also for relatively large values of the gas parameter, our finite-range results are in quite good agreement with recent zero-temperature Monte Carlo calculations obtained with hard-sphere bosons.
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.
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