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Relativistic Hamiltonian as an emergent structure from information geometry

Sikarin Yoo-Kong

TL;DR

The paper addresses how relativistic kinematics can emerge from non-relativistic, multiplicative Hamiltonians by statistically fluctuating an auxiliary parameter β and applying the maximum-entropy principle. It reframes the β-family of Hamiltonians as a statistical manifold endowed with the Fisher–Rao metric, and shows that scale-invariant constraints (⟨1/β^2⟩ and ⟨ln β⟩) uniquely determine the β-distribution ρ(β) ∝ β^-4 e^{-1/β^2}, leading, upon ensemble averaging, to the relativistic form ⟨H⟩ = sqrt(p^2 c^2 + m^2 c^4) with c^2 = 2λ^2. The information-geometric structure explains the invariance and the logarithmic geometry that underpins the square-root dispersion, without assuming Lorentz symmetry a priori. This work links multiplicative dynamics, entropy-based inference, and statistical-manifold geometry to show how relativistic dispersion can be an emergent, inference-driven outcome with potential extensions to more complex kinematics and spacetime contexts.

Abstract

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.

Relativistic Hamiltonian as an emergent structure from information geometry

TL;DR

The paper addresses how relativistic kinematics can emerge from non-relativistic, multiplicative Hamiltonians by statistically fluctuating an auxiliary parameter β and applying the maximum-entropy principle. It reframes the β-family of Hamiltonians as a statistical manifold endowed with the Fisher–Rao metric, and shows that scale-invariant constraints (⟨1/β^2⟩ and ⟨ln β⟩) uniquely determine the β-distribution ρ(β) ∝ β^-4 e^{-1/β^2}, leading, upon ensemble averaging, to the relativistic form ⟨H⟩ = sqrt(p^2 c^2 + m^2 c^4) with c^2 = 2λ^2. The information-geometric structure explains the invariance and the logarithmic geometry that underpins the square-root dispersion, without assuming Lorentz symmetry a priori. This work links multiplicative dynamics, entropy-based inference, and statistical-manifold geometry to show how relativistic dispersion can be an emergent, inference-driven outcome with potential extensions to more complex kinematics and spacetime contexts.

Abstract

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.
Paper Structure (12 sections, 35 equations, 2 figures)

This paper contains 12 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Emergent relativistic Hamiltonian
  3. The principle of maximum entropy
  4. The first constraint:
  5. The second constraint:
  6. The third constraint:
  7. Statistical manifold and Fisher-Rao metric
  8. Fisher-Rao metric
  9. Regulated construction
  10. Scale invariance and logarithmic geometry
  11. Constraints revisited: geometric origin of the relativistic Hamiltonian
  12. Conclusion

Figures (2)

  • Figure 1: The distribution shape of $\rho(\beta)$.
  • Figure 2: Relation between $\beta$ and geodesic coordinate $u=\ln \beta$ on the Fisher-Rao statistical manifold. Equal increment in $\beta$ correspond to larger statistical distances near $\beta \rightarrow 0$, reflecting the scale-invariant geometry induced by information metric.