We show that the relativistic energy-momentum relation can emerge as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints, which are shown to arise naturally from the Fisher-Rao geometry of the associated statistical manifold. Within this information-geometric framework, the relativistic dispersion relation appears without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance.

翻译：我们证明，当使用最大熵推断处理辅助参数的涨落时，相对论性能量-动量关系可以从乘法哈密顿量中作为有效的系综平均结构涌现出来。所得概率分布由尺度不变约束唯一确定，这些约束被证明源于相关统计流形的Fisher-Rao几何自然产生。在此信息几何框架内，相对论性能散关系出现时并未预先强加洛伦兹对称性，而是作为统计平均和几何不变性的自然结果。