Area Scaling of Dynamical Degrees of Freedom in Regularised Scalar Field Theory

TL;DR

The paper investigates how many canonical directions are dynamically explored by a UV/IR-regularised classical scalar field during Hamiltonian evolution. It introduces Symplectic Model Order Reduction (SMOR) as a structure-preserving diagnostic to define the minimal symplectic dimension $d_{ m min}$ needed to reproduce a single trajectory, linking this dimension to the count of distinct normal-mode frequencies below the UV cutoff. For a free field, the authors show $d_{ m min}=4 n_ Omega$, leading to area-type scaling in flat space and curvature-dependent corrections in curved geometries; weak interactions preserve this scaling on pre-resonant timescales. They uncover an intrinsic overlap structure: unreduced modes become dependent on a shared set of reduced canonical variables, with Poisson brackets governed by a finite-rank projector, providing a purely classical mechanism for overlapping degrees of freedom. The work offers a controlled classical setting to study area-like scaling and overlaps prior to quantisation and discusses connections to holographic ideas, while outlining clear directions for extending the framework to other fields and quantum contexts.

Abstract

How many canonical degrees of freedom does a quantum field theory actually use during its Hamiltonian evolution? For a UV/IR-regularised classical scalar field, we address this question directly at the level of phase-space dynamics 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 minimal dimension is controlled not by the volume-extensive number of discretised field variables, but by the much smaller number of distinct normal-mode frequencies below the ultraviolet cutoff. In flat space, this leads to an area-type scaling with the size of the region, up to slowly varying corrections. On geodesic balls in maximally symmetric curved spaces, positive curvature induces mild super-area growth, while negative curvature suppresses the scaling, with the flat result recovered smoothly in the small-curvature limit. Numerical experiments further indicate that this behaviour persists in weakly interacting $λφ^4$ theory over quasi-integrable time scales. Beyond counting, the reduced dynamics exhibits a distinctive internal structure: it decomposes into independent oscillator blocks, while linear combinations of these blocks generate a larger family of apparent field modes whose Poisson brackets are governed by a projector rather than the identity. This reveals a purely classical and dynamical mechanism by which overlapping degrees of freedom arise, without modifying canonical structures by hand. Our results provide a controlled field-theoretic setting in which area-type scaling and overlap phenomena can be studied prior to quantisation, helping to identify which aspects of such structures--often discussed in holographic contexts--can already arise from classical Hamiltonian dynamics.

Figures (5)

• Figure 1: Relative projection error versus reduced dimension for the scalar field, extracted from trajectories sampled over a finite observation window $T_{\rm obs}$. (a) Free theory ($\lambda=0$): the error drops sharply at the threshold $d_{\min}=4\,n_\Omega \propto (L_{\rm IR}\Lambda_{\rm UV})^2$, independent of the (generic) Gaussian random initial condition. (b) Weakly interacting $\lambda\phi^4$ theory: initial amplitudes are fixed at the ground--state scale of the corresponding harmonic oscillators. The sharp jump is smoothed into a crossover due to finite--time frequency resolution, but the knee remains close to the free--field threshold for small $\lambda$.
• Figure 2: Verification of the three--square counting estimate. Top: numerical count $|\mathcal{R}(X)|$ of integers $n\le X$ representable as a sum of three squares (red points), compared with the analytic asymptotic \ref{['eq:R-asymptotic']} (black dashed line). Bottom: relative error between the numerical count and the analytic estimate, which remains at the level of $\mathcal{O}(10^{-8})$ over the range of $X$ relevant for our simulations.
• Figure 3: Validation of the analytic construction of the symplectic basis $\bm{V}^\star$. Panel (a) shows the principal angles between the analytic subspace spanned by $\mathbb{U}_{\mathrm{an}}$ and the numerical left–singular subspace obtained by direct SVD of the complex snapshot matrix $\mathbb{X}_c$. All angles are smaller than $10^{-15}\,\mathrm{rad}$, demonstrating that the analytic and numerical subspaces are indistinguishable to machine precision. Panel (b) compares the singular values $\sigma_{s,\pm}$ obtained analytically from Eq. \ref{['eq:sv-2x2']} with those computed numerically by SVD. The two sets coincide to within relative differences below $10^{-15}$ in all cases. Together, these results confirm that the analytic expressions for the singular vectors and singular values---derived solely from the initial amplitudes and mode frequencies---exactly reproduce the numerical outcome. The symplectic basis $\bm{V}^\star$ can thus be obtained fully analytically, without any numerical decomposition.
• Figure 4: Nonlinearity budget $\varepsilon(t)$ as defined in \ref{['eq:eps-appendix']} for several $\lambda>0$. Curves remain well below unity over the entire time interval, indicating that the quartic term acts as a small perturbation of the quadratic dynamics.
• Figure 5: Fractional frequency shift $(\omega_{\mathrm{peak}}-\omega_{\mathrm{free}})/\omega_{\mathrm{free}}$ versus $\omega_{\mathrm{free}}$ for a representative set of modes and several $\lambda$. The shaded band is the finite–time resolution $\sim 2\pi/T_{\mathrm{obs}}$ expressed as a relative tolerance. Points within the band are unresolved at this time baseline; small excursions at larger $\lambda$ reflect mild, expected frequency renormalisation.