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，当利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。由此产生的概率分布由尺度不变约束唯一确定，这种约束自然源于关联统计流形的Fisher-Rao几何结构。在此信息几何框架下，相对论色散关系无需预先施加洛伦兹对称性，而是作为统计平均与几何不变性的结果呈现出来。