Relativistic Hamiltonian as an emergent structure from information geometry

This paper demonstrates that the relativistic energy-momentum relation emerges as an effective ensemble-averaged structure from a multiplicative Hamiltonian under maximum entropy inference, where scale-invariant constraints derived from Fisher-Rao geometry naturally yield the relativistic dispersion relation without initially imposing Lorentz symmetry.

The Gist

Imagine you are trying to understand why a speeding car behaves differently than a slow one. In our everyday world, if you double the speed, the energy goes up by four times (it's a simple square relationship). But in the world of Einstein's relativity, things get weird: as you get closer to the speed of light, it takes infinite energy to go any faster. This is usually described by a very specific, "square-root" formula.

This paper asks a bold question: What if that weird "square-root" formula isn't a fundamental rule of the universe, but actually just a statistical accident?

Here is the story the paper tells, broken down into simple steps:

1. The "Magic" Multiplier

The author starts with a strange, made-up version of energy (called a Hamiltonian). Instead of the usual formula, this one multiplies the energy by a mysterious, fluctuating number called $\beta$ (beta).

• The Analogy: Imagine you are baking a cake. The recipe (the physics) is standard, but you have a magical, invisible ingredient ($\beta$) that changes the taste every time you bake. Sometimes it's a little bit of vanilla, sometimes a lot. You don't know exactly how much is in the mix; it just fluctuates.

2. The Guessing Game (Maximum Entropy)

Since we don't know the exact amount of this magical ingredient $\beta$ at any given moment, we have to guess its distribution. How do we make the fairest guess possible without making up facts? We use a rule called Maximum Entropy.

• The Analogy: Think of a detective trying to solve a crime with very little evidence. The "Maximum Entropy" rule says: "Don't assume anything extra. Just spread your suspicion as evenly as possible, but respect the few hard facts you do have."
• In this paper, the "hard facts" are two specific rules about how $\beta$ behaves:
  1. It has a certain average "scale" (how big the fluctuations are).
  2. It behaves the same way whether you zoom in or zoom out (it's "scale-invariant").

3. The Magic Happens

When the author takes all the possible versions of this "magic cake" (all the different values of $\beta$) and averages them out using this fair-guessing rule, something miraculous happens.

• The messy, complicated, exponential math of the individual "magic cakes" cancels out.
• What remains is the exact square-root formula that describes Einstein's relativistic energy.
• The Result: The weird, relativistic behavior didn't exist in the original ingredients. It emerged naturally just by averaging out the fluctuations of the hidden ingredient.

4. The Hidden Map (Information Geometry)

The paper goes a step further to explain why those specific rules for guessing (the constraints) were chosen. It uses a branch of math called Information Geometry.

• The Analogy: Imagine the different values of $\beta$ are points on a landscape. Usually, we think of this landscape as a flat map where one inch equals one mile. But the author shows that for this specific problem, the map is actually a funnel or a trumpet shape.
• In this "funnel landscape," the distance between points isn't measured in miles, but in how "different" they look statistically.
• The rules the author used to guess the distribution ($\langle \ln \beta \rangle$ and $\langle 1/\beta^2 \rangle$) turn out to be the natural "coordinates" of this funnel landscape. They aren't random choices; they are the only way to measure distance on this specific map correctly.

The Big Takeaway

The paper claims that Relativity might not be a fundamental law we have to impose on the universe. Instead, it might be a natural consequence of:

1. Having a system with a multiplicative structure (like the magic ingredient).
2. Having incomplete information about a hidden variable (the fluctuating $\beta$).
3. Using the most logical, unbiased way to fill in the missing information (Maximum Entropy).

In short: The author suggests that if you look at the universe through the lens of "what we know and what we don't know," the strange rules of Einstein's relativity pop out automatically, like a pattern emerging from a cloud of fog, without needing to be programmed in from the start.

Note: The paper strictly limits this to the math of a single particle's energy and momentum. It does not claim to explain gravity, black holes, or how to build a time machine.

Technical Summary

Technical Summary: Relativistic Hamiltonian as an Emergent Structure from Information Geometry

Problem Statement
The paper addresses the foundational question of whether relativistic kinematical features, specifically the relativistic energy–momentum relation (dispersion relation), can emerge from non-relativistic structures without presupposing Lorentz symmetry. While standard formulations of physics rely on additive Lagrangians and Hamiltonians to produce familiar equations of motion, non-standard multiplicative formulations offer alternative perspectives. The central problem is to determine if a relativistic square-root Hamiltonian can be derived as an effective, ensemble-averaged structure from a family of non-relativistic multiplicative Hamiltonians, governed by statistical inference and information geometry rather than fundamental relativistic axioms.

Methodology
The authors employ a three-pronged methodology combining multiplicative Hamiltonian formalism, maximum entropy inference, and information geometry:

1. Multiplicative Hamiltonian Framework: The study begins with a $\beta$-dependent multiplicative Hamiltonian, $H_\beta(p) = m\lambda^2\beta^2 e^{p_N^2 / (2m\lambda^2\beta^2)}$, where $p_N$ is spatial momentum and $\beta$ is an auxiliary parameter representing an underlying physical or effective scale. Unlike standard Hamiltonians, this form exhibits exponential dependence on momentum squared.
2. Ensemble Averaging via Maximum Entropy: The parameter $\beta$ is treated as a fluctuating variable with an unknown distribution $\rho(\beta)$. To determine $\rho(\beta)$ without bias, the authors apply Jaynes' maximum entropy principle. The entropy $S[\rho]$ is maximized subject to three constraints:
   • Normalization: $\int \rho(\beta) d\beta = 1$.
   • Fixed average scale: $\langle 1/\beta^2 \rangle = C_1$.
   • Scale-invariant location: $\langle \ln \beta \rangle = C_2$.
3. Information Geometry Analysis: To justify the specific constraints used in the entropy maximization, the authors construct a statistical manifold where the parameter $\beta$ serves as a coordinate. They define a one-parameter family of probability weights derived from the Hamiltonian and compute the Fisher–Rao metric on this manifold. This geometric analysis is used to demonstrate that the chosen constraints correspond to intrinsic geometric properties (geodesic coordinates and metric density) of the statistical manifold.

Key Results

• Emergence of the Relativistic Hamiltonian: By solving the maximum entropy problem, the authors derive a unique probability distribution $\rho(\beta) \propto \beta^{-4} e^{-1/\beta^2}$. When this distribution is used to ensemble-average the multiplicative Hamiltonian $H_\beta$, the resulting effective Hamiltonian $\langle H \rangle$ takes the form:
  $$\langle H \rangle = \frac{2m\lambda^2}{\sqrt{1 - \frac{p_N^2}{2m\lambda^2}}}$$
  By identifying $2\lambda^2 = c^2$ (where $c$ is the speed of light), this expression simplifies to the standard relativistic Hamiltonian for a massive free particle, $\langle H \rangle = \sqrt{p^2c^2 + m^2c^4}$ (or $\gamma mc^2$).
• Derivation of the Relativistic Lagrangian: Applying a Legendre transformation to the averaged Hamiltonian yields the relativistic Lagrangian $\langle L \rangle = -mc^2\sqrt{1 - \dot{x}^2/c^2}$.
• Geometric Justification of Constraints: The Fisher–Rao metric analysis reveals that the statistical manifold possesses a line element $ds^2 = d\beta^2/\beta^2$. In the logarithmic coordinate $u = \ln \beta$, this geometry is flat (Euclidean). The authors demonstrate that the constraints used in the entropy maximization are not arbitrary:
  • $\langle \ln \beta \rangle$ fixes the mean position along the geodesic (the natural affine coordinate).
  • $\langle 1/\beta^2 \rangle$ controls the average metric scale (local density of statistical distinguishability).
  • The specific form of $\rho(\beta)$ is the unique distribution that respects the scale invariance and geometric structure of this manifold.

Significance and Claims
The paper claims that the relativistic dispersion relation is not necessarily a fundamental axiom but can arise as a natural consequence of statistical averaging over a family of non-relativistic multiplicative Hamiltonians. The key insight is that the "square-root" structure of relativistic energy is a reflection of the norm induced by the statistical distinguishability in the logarithmic geometry of the parameter space.

The authors emphasize that this derivation requires no initial imposition of Lorentz symmetry. Instead, relativistic kinematics emerges from the interplay between:

1. The multiplicative structure of the Hamiltonian.
2. The principle of maximum entropy under scale-invariant constraints.
3. The intrinsic geometry (Fisher–Rao metric) of the statistical manifold associated with the Hamiltonian family.

Limitations and Scope
The authors explicitly state that their analysis is limited to single-particle kinematics and the emergence of the dispersion relation. The work does not address full aspects of special relativity, such as Lorentz transformations, multi-particle interactions, or the emergence of spacetime structure, which are deferred to future work. The paper positions itself as a demonstration of how information geometry can provide a "compass" for exploring the foundations of classical and quantum contexts, suggesting that minimal geometric complexity may suffice to generate relativistic features.