Riemannian Parameterization of Physical Quantities

• Riemannian parameterization is a geometric framework that represents physical observables and state configurations as coordinates on a differentiable manifold.
• It enables a coordinate-free dimensional analysis and reconstruction of metrics using geometric invariants like curvature and area-based parameterizations.
• Applications span statistical mechanics, thermodynamics, and quantum information, offering invariant diagnostics of correlations and phase transitions.

Riemannian parameterization of physical quantities is a geometric framework wherein physical, statistical, and thermodynamic structures are encoded as properties of a differentiable manifold endowed with a Riemannian (or pseudo-Riemannian) metric. This paradigm extends classical and quantum physical theories by regarding physical observables, field configurations, or control parameters as coordinates on an abstract manifold, allowing physical quantities and their relations to be reformulated as geometric invariants, distances, curvature, and volume forms. The Riemannian parameterization is central in fields such as statistical mechanics, differential geometry, thermodynamics, quantum information theory, and the mathematical foundations of general relativity.

1. Riemannian Structure on Configuration and State Manifolds

The core construction begins with the assignment of a differentiable manifold $\mathcal{M}$ whose points represent admissible physical or statistical configurations. In the context of equilibrium statistical mechanics, these are macroscopic observables $I^i$ (e.g., energy, particle number, magnetization), and for fixed control parameters $\theta$, one studies a submanifold $\mathcal{M}_\theta \subset \mathcal{M}$ of accessible states (Velazquez, 2010). This manifold is endowed with a smooth, positive-definite metric tensor: $$g_{ij}(I|\theta), \qquad ds^2 = g_{ij}(I|\theta)\,dI^i\,dI^j$$ enabling the definition of distance, geodesics, curvature, and covariant tensor calculus. In general relativity and field theory, a pseudo-Riemannian metric $g_{\mu\nu}$ provides the arena for encoding the causal structure and gravitational interaction (Mana, 2020).

2. Physical Quantities and Intrinsic Dimensional Analysis

The Riemannian parameterization naturally incorporates intrinsic (coordinate-free) dimensional analysis. Each tensorial physical quantity $\mathbb{A}$ is characterized by its absolute or operational dimension, reflecting its physical meaning (such as action, curvature, energy density), which is invariant under coordinate transformations. The components of tensors in local coordinates inherit dimensions from the coordinate system, but the intrinsic dimension remains invariant—ensuring universal, operational consistency (Mana, 2020). For example, the metric tensor can have dimension $L^2$ or $T^2$ depending on the chosen physical units, but its role as the inner product on tangent spaces is coordinate independent.

Fundamental Riemannian tensors such as the Riemann curvature $R^\rho_{~\sigma\mu\nu}$, Ricci tensor $R_{\mu\nu}$, and Einstein tensor $G_{\mu\nu}$ have dimensionless intrinsic character. Raising and lowering indices with the metric and defining proper volume elements (e.g., $$\sqrt{|\det g|}\,dx^0\wedge dx^1\wedge dx^2\wedge dx^3$$) are dimensionally tracked in this intrinsic scheme (Mana, 2020).

3. Metric Reconstruction and Alternative Parameterizations

Riemannian parameterization admits non-metric encodings of physical geometry. In four-dimensional Riemannian manifolds, the ten independent components of the metric tensor $g_{\mu\nu}$ at a point can be faithfully traded for the ten areas of the two-faces of a 4-simplex $\sigma^4$ embedded at that point. Given the areas $A_{ijk}$ of these faces (computed via the Gram determinant from the metric), the full quadratic form $g_{\mu\nu}(P)$ can be uniquely reconstructed using algebraic inversion (e.g., via the Cayley–Menger determinant and Gram matrix techniques) (Mäkelä, 2010). This approach is valuable in loop quantum gravity and thermodynamic interpretations of gravity, where area emerges as a primary quantum or statistical variable, and local curvature may be discretized in terms of area differences.

Quantity	Metric-Based Formulation	Area-Based (Simplex) Parameterization
Local geometry	$g_{\mu\nu}$	$\{ A_{ijk} \}$ (areas of simplex 2-faces)
Reconstruction map	Direct via Gram matrix	Inversion from $\{A_{ijk}\}$ via Cayley–Menger
Physical context	Continuum GR, classical geometry	Quantum gravity, entropic gravity

4. Thermodynamic and Statistical Geometric Structures

In equilibrium statistical mechanics, physical quantities such as entropy, equilibrium distributions, and fluctuation measures receive covariant geometric definitions. The scalar entropy $S_g(I|\theta)$ becomes a coordinate-invariant field on $\mathcal{M}_\theta$, and the equilibrium distribution is: $$dp(I|\theta) = \exp\bigl[S_g(I|\theta)\bigr]\,du(I|\theta)$$ with $du$ the invariant Riemannian volume element (Velazquez, 2010). The metric is identified as the negative covariant Hessian of the scalar entropy,

$g_{ij}(I|\theta) = -D_i D_j S_g(I|\theta)$

and the Riemann curvature derived from $g_{ij}$ measures irreducible statistical dependence between observables: $R = 0$ if and only if the statistical manifold is flat and the joint distribution factorizes into decoupled Gaussians. Nonzero curvature $R \neq 0$ signals irreducible coupling that cannot be transformed away by coordinate changes, highlighting a geometric parallel with gravitational interactions in general relativity (Velazquez, 2010).

Reparametrizations of intensive or extensive thermodynamic variables act as diffeomorphisms on thermodynamic phase space, altering the explicit form of the contact structure and Riemannian metric, but leaving the induced metric (and associated statistical fluctuation structure) on the equilibrium manifold invariant (Pineda-Reyes et al., 2018).

5. Riemannian Parameterization in Quantum Information and Physical Models

The space of quantum states, particularly density matrices for finite-dimensional systems, inherits a family of monotone Riemannian metrics—prominently, the quantum Fisher (Braunstein–Caves) metric. On the Bloch ball of qubit states, this metric takes the form: $$ds^2 = dP \cdot dP + \frac{(P \cdot dP)^2}{1 - |P|^2}$$ where $P$ is the Bloch vector (Cafaro et al., 2011). Quantum channels, such as the depolarizing map ($P \mapsto (1-p)P$), deform the Riemannian structure by contracting the spatial part of the metric. This geometric contraction translates the physical effect of noise into the shortening of distinguishability distances and deformation of geodesic flows, providing a direct geometric handle on quantum decoherence, metrological error bounds, and resource quantification.

In dynamical contexts, Riemannian parameterizations extend to parameter-dependent Lagrangians—e.g., Kawaguchi metrics for spinning particles, where the metric depends on higher-derivative invariants (Frenet curvature, velocity norm) and leads to higher-order covariant variational equations coupled to background curvature (Matsyuk, 2014).

6. Curvature Invariants and Physical Meaning

Curvature invariants constructed from the Riemannian metric—such as the Riemann tensor, Ricci tensor, and scalar curvature—encode operationally significant features of physical systems. In statistical mechanics, the curvature scalar $R(I|\theta)$ quantifies irreducible correlations among macroscopic observables; in general relativity, $R_{\mu\nu\rho\sigma}$ distinguishes purely gravitational effects from inertial ones. These invariants are dimensionless in the intrinsic scheme and play a central role in classifying physical regimes, detecting phase transitions (via singularities in $R$), and constructing partition functions and thermodynamic potentials in a fully geometric form (Velazquez, 2010, Mana, 2020).

7. Applications and Operational Implications

Riemannian parameterization underlies multiple operational approaches:

• Dimensionally consistent physical law formulation: By assigning intrinsic dimension to each abstract tensorial object and verifying dimensional consistency under all coordinate systems, one ensures physical relations are universal and independent of parametrization choices (Mana, 2020).
• Statistical manifold invariants: The structure of the metric and curvature provides invariant diagnostics of correlation, independence, criticality, and phase structure in both classical and quantum statistical models (Velazquez, 2010, Cafaro et al., 2011).
• Quantum gravity and discrete geometry: Area-based parameterization enables the expression of curvature and field equations in terms of discrete geometric quantities suited to quantum gravity, loop quantum gravity, and entropic gravity program (Mäkelä, 2010).
• Thermodynamic coordinate redundancy: Physical thermodynamic properties are invariant under smooth reparametrizations, a consequence of the geometric encoding in the equilibrium manifold’s induced metric (Pineda-Reyes et al., 2018).

These constructs collectively provide a unifying framework for the geometric encoding of physical quantities, ensuring robustness under coordinate changes and supporting precise, operationally meaningful descriptions across classical, quantum, and statistical domains.