Glossary: The Lexicon of Nature

Vector Cosmology III: The Natural Generator delves into the most abstract foundations of mathematical physics—complex geometry, Lie algebras, and operator theory. To help readers cross the cognitive threshold of these concepts, this glossary redefines the core vocabulary in the book from the perspective of “natural generation.”

1. Core Constants & Operators

• $e$ (Natural Constant)

  The generative engine of the universe. It is the base of the unique function whose derivative equals itself ($d/dt(e^t)=e^t$). Physically, it represents the self-driven mechanism of “existence as propulsion”, unfolding infinitesimal generators (Lie algebras) into macroscopic continuous evolution (Lie groups). It is the mathematical bridge connecting discrete QCA with continuous spacetime.

• $i$ (Imaginary Unit)

  The orthogonal rotation operator of the universe. On the complex plane, multiplying by $i$ means rotating 90 degrees. In quantum mechanics, it ensures that the evolution operator $e^{-iHt}$ is unitary (modulus-preserving), thus taming pure exponential growth into eternal phase rotation. It is the geometric guardian of conservation laws.

• Generator

  An element in a Lie algebra, corresponding to conserved quantities in physics (such as Hamiltonian $H$, momentum $P$). It is the “seed of evolution” contained in the infinitesimal neighborhood of the tangent space. The entire history of the universe is mathematically just the exponential map of the generator $H$.

• $H$ (Hamiltonian/Energy)

  The generator of time evolution. In the modular flow hypothesis, it is no longer a fundamental quantity, but a physical manifestation of the state’s entanglement structure (modular Hamiltonian $K$). Energy is essentially the logarithm of the probability distribution.

2. Time & Thermodynamics

• Wick Rotation

  A mathematical transformation that converts real time (Minkowski spacetime) into imaginary time (Euclidean space) by letting $t\to -i\tau$. It reveals that quantum mechanical wave evolution ($e^{-iHt}$) and thermodynamic statistical distribution ($e^{-\beta H}$) are homologous in complex geometry.

• KMS Condition (Kubo-Martin-Schwinger Condition)

  The strict definition of thermal equilibrium states in quantum field theory. It points out that thermal states have periodicity $\beta$ in the imaginary time direction. This proves that temperature is the circumference of imaginary time; heat is not chaos, but geometric circulation.

• Modular Flow

  A one-parameter unitary evolution group ${\Delta }^{it}$ automatically generated by the entanglement structure of the quantum state itself through Tomita-Takesaki theory. It shows that time is an intrinsic property of the state, not an external background. Every entangled system has its own “private time.”

• Thermal Time Hypothesis

  The view that the physical time passage perceived macroscopically is essentially the modular flow of the statistical state in which the system is located. Time originates from the observer’s ignorance of microscopic degrees of freedom (entropy).

3. Information & Perception

• $\ln$ (Natural Logarithm)

  The decoder of the universe. It is the inverse operation of the exponential function $e$. Observers (life) compress the exponentially exploding information of the universe into linear macroscopic experiences (such as entropy, decibels, magnitudes) through logarithmic perception (Weber-Fechner Law).

• Entropy ($S$)

  The logarithm of the number of microscopic states $W$ ($S=k\ln W$). Geometrically, it is the dimensional reduction compression rate when projecting from high-dimensional Hilbert space to low-dimensional macroscopic thermodynamic space.

• Tangent Space

  In differential geometry, a locally flat space established at a point on a manifold. Human linear logic and causal cognition essentially result from forcibly projecting the curved universe manifold onto local tangent spaces.

4. Ultimate Concepts

• Euler’s Formula ($e^{i\pi }+1=0$)

  The mathematical map of the universe’s trinity. $\pi$ (structure), $i$ (rotation), and $e$ (generation) are unified here, revealing the self-consistent logic of the physical world from void (0) to existence (1).

• Causa Sui (Self-Cause)

  The philosophical essence of $e$. It means that the state itself is the driving force of its change. In Vector Cosmology, the universe needs no first mover because it is a self-referential exponential function $e^{\text{Dao}}$.

• The Dao

  In the ultimate formula $|\Psi ⟩=e^{\text{Dao}}$, the Dao is the operator located in the Exponent position. It is unmanifested potential, rules, and generators, which manifest as all things in the universe we see through the mechanism of $e$.