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  • Open Access

Area scaling of dynamical degrees of freedom in regularized scalar field theory

Oliver Friedrich1,2, Kristina Giesel3, and Varun Kushwaha1,*

  • *Contact author: varun.kushwaha@lmu.de
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Phys. Rev. D 114, 045020 – Published 24 August, 2026

DOI: https://doi.org/10.1103/tjc4-221z

Abstract

How many canonical degrees of freedom are dynamically needed to reproduce the Hamiltonian evolution of a regulated field theory? We address this question for a UV/IR-regularized classical scalar field by identifying the minimal symplectic dimension required to reproduce a single trajectory by an autonomous Hamiltonian system. Using symplectic model order reduction as a structure-preserving diagnostic, we show that, for the free scalar field, this dimension is controlled not by the volume-extensive number of regulated field modes but by the smaller number of distinct normal-mode frequencies below the ultraviolet cutoff. In a flat cubic box, this gives an area-type scaling with the size of the region, up to slowly varying corrections. For geodesic balls in maximally symmetric curved spaces, positive curvature produces a superarea enhancement that remains subextensive, while negative curvature suppresses the scaling, with the flat result recovered smoothly in the small-curvature limit. Numerical experiments indicate that the same frequency-based mechanism persists in weakly interacting λϕ4 theory on quasi-integrable timescales. Beyond the counting result, the reduced dynamics has a characteristic algebraic structure: it decomposes into independent oscillator blocks, while reconstructed field modes are linear combinations of these blocks and have Poisson brackets governed by a projector rather than the identity. In this precise classical sense, overlapping degrees of freedom arise dynamically, without modifying the canonical structure by hand. The results provide a controlled setting in which area-type dynamical scaling and overlap structures can be studied before quantization and help separate ordinary Hamiltonian compression effects from genuinely gravitational mechanisms.

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Phys. Rev. D 114, 045020– Published 24 August, 2026
  • Received 7 April 2026
  • Accepted 24 July 2026
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