The Fubini–Study (FS) metric in quantum projective geometry

In standard quantum theory, the Fubini–Study (FS) metric arises naturally on the complex projective space \( \mathbb{CP}^n \), which represents the space of pure quantum states modulo overall phase. If \( |\psi\rangle \) is a normalized state vector in a Hilbert space and \( |\delta \psi\rangle \) is an infinitesimal variation, the FS line element is given by

\[ ds^2_\text{FS} = g_{i\bar{j}} \, dz^i \, d\bar{z}^j = \frac{\langle \delta \psi | \delta \psi \rangle \langle \psi | \psi \rangle - |\langle \psi | \delta \psi \rangle|^2}{\langle \psi | \psi \rangle^2}. \]

This metric measures the infinitesimal distance between neighboring rays in Hilbert space and thus quantifies quantum distinguishability. It encodes how rapidly two nearby states can be told apart by measurements and is deeply connected to transition amplitudes, path integrals, and the Born rule. In geometric terms, the FS metric endows the projective space of quantum states with a Kähler structure, combining Riemannian, symplectic, and complex structures into a unified geometric framework.

In conventional quantum mechanics, this geometry is static: it does not itself encode entropy production or irreversibility. The FS metric provides a notion of distance and curvature on state space, but it does not distinguish between reversible and irreversible processes. The Theory of Entropicity modifies this picture by allowing the geometry of state space to be dynamically deformed by the entropic field.

Entropy weighting of the Fubini–Study metric in the Spectral Obidi Action

The Spectral Obidi Action is formulated in terms of density operators and relative entropy, capturing the global, operator-level structure of the entropic dynamics. A representative form of the SOA is

\[ \mathcal{A}_\text{Spectral} = \mathrm{Tr} \left[ \rho \, \log \left( \frac{\rho}{\rho_0 e^{S/k_B}} \right) \right], \]

where \( \rho \) is the actual density operator of the system, \( \rho_0 \) is a reference or undeformed state, and the factor \( e^{S/k_B} \) introduces an explicit dependence on the entropic field into the operator structure. Within the language of Tomita–Takesaki modular theory, one can associate a modular operator

\[ \Delta = \rho \otimes \rho_0^{-1} e^{S/k_B}, \]

which governs the modular flow of states and observables. The presence of the entropic factor \( e^{S/k_B} \) implies that the geometry of the underlying state space is no longer purely determined by the Hilbert space structure but is deformed by the entropic field.

This deformation can be expressed geometrically by introducing an entropy-weighted Fubini–Study metric. If \( g^{\text{FS}}_{i\bar{j}} \) denotes the standard FS metric components, the entropy-weighted metric is defined as

\[ g^{(S)}_{i\bar{j}} = e^{S/k_B} \, g^{\text{FS}}_{i\bar{j}}. \]

The exponential factor \( e^{S/k_B} \) acts as a conformal weight on the quantum state-space geometry. Regions of state space associated with higher entropy are effectively “stretched” relative to those of lower entropy, and the curvature of the quantum manifold becomes a function of the entropic field. In this way, irreversible entropy production is encoded directly into the geometry of quantum states, in close analogy with how spacetime curvature in general relativity encodes the presence of energy and momentum.

Role of the entropy-weighted Fubini–Study metric in spectral dynamics

The entropy-weighted FS metric plays a central role in the spectral dynamics generated by the SOA. When one extremizes the spectral action with respect to variations in the state \( \rho \) and the entropic field \( S \), the resulting equations of motion can be interpreted as driving the system along geodesics in the entropy-deformed quantum manifold. These geodesics are no longer those of the standard FS metric but of the entropy-weighted metric \( g^{(S)}_{i\bar{j}} \). The gradient of the entropic field, \( \nabla S \), then acts as an effective entropic force on quantum transitions, biasing the evolution of states toward directions in state space that are consistent with irreversible entropy increase.

This geometric picture provides a natural way to incorporate phenomena such as the Entropic Time Limit (ETL) and the No-Rush Theorem at the operator level. The finite rate at which the entropic field can update its configuration manifests as constraints on the speed with which states can move along entropy-weighted FS geodesics. In accelerated or noninertial settings, the same structure can be related to Unruh-like temperatures, where the entropic deformation of state space encodes the effective thermal behavior perceived by accelerated observers.

The entropy-weighted FS metric also serves as a unification bridge between quantum and classical information geometry. In ToE, the FS metric on quantum state space and the Fisher–Rao metric on classical probability distributions are viewed as projections of a more general entropic manifold equipped with Amari–Čencov \(\alpha\)-connections. The parameter \( \alpha \) interpolates between different dual affine structures, and the entropic weighting by \( e^{S/k_B} \) ties these structures to the dynamics of the entropic field. In this way, the spectral formulation links local and global descriptions, with the entropy-weighted FS metric providing the quantum side of this duality.

Physically, gradients of entanglement entropy defined on the entropy-weighted FS manifold can be interpreted as sources of effective curvature and as generators of particle masses viewed as excitations of the entropic field. The geometry of quantum state space thus becomes directly responsible for emergent properties traditionally attributed to spacetime geometry and mass-energy content. Standard quantum mechanics is recovered in the limit \( S \to 0 \), where the entropy weighting disappears and the FS metric reduces to its usual, entropy-independent form. In this limit, irreversibility and entropic deformation vanish, and one recovers the familiar reversible unitary evolution on a fixed projective Hilbert space.

The incorporation of the Fubini–Study metric into the Spectral Obidi Action therefore achieves more than a formal modification of quantum geometry. It provides a concrete realization of the central claim of the Theory of Entropicity: that entropy is the fundamental field from which both spacetime geometry and quantum structure emerge. By weighting the FS metric with the entropic field, the SOA renders quantum irreversibility geometric and subsumes standard quantum mechanics as a special, entropy-free limit of a more general entropic framework.