EQ(vT): Entropic Quantum — Vectorized Time. The Entropic Origin of Temporal Direction in Quantum State Space

This work introduces a geometric framework in which the direction of time is identified with the normalized entropy gradient on quantum state space equipped with the Bures metric. The configuration space Γ = ρ ∈ B (H): ρ ≥ 0, Tr (ρ) = 1 is taken as the fundamental arena of physics — not an auxiliary description overlaid on a pre-existing spacetime, but the space from which spacetime structure may emerge. The von Neumann entropy S (ρ) = −Tr (ρ log ρ) defines a smooth scalar field on (Γ, gB), and the time vector field T (ρ) = ∇S (ρ) /|∇S (ρ) | is its unit gradient. Five geometric consequences follow from this identification: (1) T is irrotational, implying a natural causal ordering; (2) the constant-entropy hypersurfaces Σc foliate Γ, suggesting spatial structure as entropy level sets; (3) the metric deformation g̃_μν = gB_μν − 2T_μT_ν produces Lorentzian signature, with the coefficient −2 uniquely fixed by the requirement that T be unit timelike; (4) S acts as a global time function on (Γ, g̃), establishing stable causality and strongly constraining the formation of closed timelike curves; (5) K_μν = ∇B_μ T_ν is symmetric and generates the extrinsic curvature of the entropy surfaces. These structures provide the geometric foundation from which gravitational dynamics are developed in the companion paper EFT: Entropic Field Theory. Within this framework the Wheeler–DeWitt equation HΨ = 0 can be interpreted as the condition ∇Sᵤniverse = 0: a closed system evolving unitarily preserves all eigenvalues of ρ, and therefore possesses no entropy gradient and no intrinsic time direction. The tension field f (ρ) = |∇S (ρ) | = √ (Var_ρ (log ρ) ) vanishes at two fixed points: the order pole (pure states on ∂Γ) and the chaos pole (the maximally mixed state). Time therefore disappears at both extremes, while physical processes occur in the interior region where f > 0. EQ (vT) establishes the geometric framework; the companion theory EFT provides the dynamical field equations.