The Flow-Space Metric
The Flow-Space Metric The flow-space metric g ij ( F ) is defined as the Hessian of the double-well potential: g ij ( F ) = ∂ F i ∂ F j ∂ 2 Φ ( F ) = λ [ 2 ( F i − F 0 i ) ( F j − F 0 j ) + ( ∥ F − F 0 ∥ 2 − v 2 ) δ ij ] . At the vacuum ( ∥ F − F 0 ∥ 2 = v 2 ): g ij ( F ) = 2 λ ( F i − F 0 i ) ( F j − F 0 j ) . This metric encodes the geometry of the flow space and determines how flows interact. It is rank-1 unless F − F 0 has multiple non-zero components, ensuring that dimensionality emerges dynamically. 2. The Lagrangian The core Lagrangian for TFT can be written as: L = L kinetic + L potential + L interaction , where: 2.1 Kinetic Term The kinetic term describes the propagation of flows: L kinetic = 2 1 g ij ( F ) ∂ μ F i ∂ μ F j . Here, g ij ( F ) is the flow-space metric, and ∂ μ F i represents gradients of the flow field in spacetime. 2.2 Potential Term The potential term governs the self-interaction of flows: L potential ...