正则标量场理论中动力学自由度的面积标度
Area Scaling of Dynamical Degrees of Freedom in Regularised Scalar Field Theory
AI总结：
研究正则标量场理论中动力学自由度的面积标度，通过相空间动力学确定最小辛维度，揭示曲率对面积标度的影响及经典动力学中叠加自由度的机制。
摘要：
量子场论在哈密顿演化过程中实际使用多少规范自由度？对于正则化经典标量场，我们通过相空间动力学直接回答这个问题，确定最小辛维度以重现单个轨迹的自主哈密顿系统。利用辛模型降阶作为结构保持诊断，我们显示自由标量场的最小维度由远小于离散化场变量数量的distinct normal-mode频率控制。在平直空间中，这导致区域大小的面积标度，直到缓慢变化的修正。在最大对称曲空间的测地球中，正曲率引起轻微超面积增长，而负曲率抑制标度，小曲率极限下恢复平直结果。数值实验进一步表明，这种行为在弱相互作用λϕ^4理论中持续存在。除了计数，减少的动力学表现出独特的内部结构：分解为独立振子块，而这些块的线性组合生成更大的 apparent field modes，其泊松括号由投影器而非单位元控制。这揭示了纯经典和动力学机制，通过叠加自由度而不修改规范结构。我们的结果提供了一个受控的场论环境，研究面积标度和叠加现象，帮助识别哪些结构特征——常在全息背景下讨论——可从经典哈密顿动力学中出现。
英文摘要：
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.
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