Appendix A: The Mathematics of Modular Time

In Chapter 6 “The Modular Flow Hypothesis” of Vector Cosmology III, we proposed a highly subversive physical view: time is not an external parameter, but an intrinsic property generated by the entanglement structure of the quantum state itself. This view is based on the profound Tomita-Takesaki Theory in algebraic quantum field theory.

To prevent this “alchemy of time” from becoming metaphysics, this appendix provides the underlying mathematical proof. We will show how any non-trivial quantum state can automatically “secrete” a one-parameter unitary evolution group through pure algebraic operations—the time we perceive.

A.1 Tomita Operator: The Mirror of Conjugation

Consider a von Neumann algebra $\mathcal{M}$ (representing the set of observables of a local system) acting on Hilbert space $\mathcal{H}$. Suppose there exists a quantum state $|\Omega ⟩$ that is Cyclic and Separating for $\mathcal{M}$.

• Physical meaning: This means $|\Omega ⟩$ is a highly entangled state (such as vacuum state or thermal state) that contains sufficient information to generate the entire algebraic space, and no local operator can annihilate it.

We define an Antilinear Operator $S$, called the Tomita Operator:

$$SA|\Omega ⟩=A^{†}|\Omega ⟩,\forall A\in \mathcal{M}$$

The physical meaning of this operator $S$ is profound: it maps an operator $A$ (creating some physical effect) to its conjugate operator $A^{†}$ (undoing that effect). This is actually an attempt to perform time reversal or logical negation.

A.2 Modular Operator and Modular Hamiltonian

The Tomita operator $S$ is usually not unitary, but it can undergo Polar Decomposition:

$$S=J{\Delta }^{1/2}$$

Here appear two key objects:

1. $J$ (Modular Conjugation Operator): An anti-unitary operator representing the mirror symmetry between the system and its environment (or its complement) (a generalization of CPT symmetry).

2. $\Delta$ (Modular Operator): A positive definite self-adjoint operator defined as $\Delta =S^{†}S$.

This $\Delta$ is the “clockwork” we mentioned in the main text. We can use it to define a Hermitian operator $K$ (Modular Hamiltonian):

$$\Delta =e^{-K}⟹K=-\ln \Delta$$

A.3 Generation of Modular Flow: The Birth of Time

According to Stone’s Theorem, any Hermitian operator can generate a unitary evolution group. For the modular Hamiltonian $K$, the evolution it generates is:

$$U(t)=e^{-iKt}={\Delta }^{it}$$

The core conclusion of the Tomita-Takesaki theorem is that this evolution group ${\Delta }^{it}$ maps the algebra $\mathcal{M}$ back to itself.

$${\sigma }_t(A)={\Delta }^{it}A{\Delta }^{-it}\in \mathcal{M}$$

This is the Modular Automorphism Group, which is what we call the Modular Flow.

Physical conclusion:

• We did not introduce any Hamiltonian $H$.

• We did not introduce any time parameter $t$.

• We merely gave a state $|\Omega ⟩$ and an algebra $\mathcal{M}$.

• The mathematical structure automatically produces a parameter $t$ and an evolution flow ${\sigma }_t$.

This means: Where there is entanglement, there is evolution. Time $t$ is just the parametrization marker of this intrinsic evolution flow.

A.4 KMS Condition and the Emergence of Temperature

To prove that this “modular flow” is physically the “thermal flow,” we need to verify whether it satisfies the boundary conditions of thermodynamics.

For the modular flow ${\sigma }_t$ defined above, it can be strictly proven that it satisfies the KMS Condition (Kubo-Martin-Schwinger Condition), with parameter $\beta =-1$ (normalized temperature).

This means that at the imaginary time $t\to t+i$ defined by the modular flow, the state returns to the origin.

In physical systems, if we relate the modular Hamiltonian $K$ to the real physical Hamiltonian $H$ (e.g., for Gibbs states $K={\beta }_{phys}H$), then the modular flow parameter $t$ establishes a direct conversion relationship with physical time $\tau$:

$$t=\tau /{\beta }_{phys}$$

Or:

$$\text{Physical Time}=\text{Modular Flow Parameter}\times \text{Temperature}$$

This mathematically strictly proves our assertion in Chapter 5 of the main text: Time is complex temperature. The higher the temperature, the faster the modular flow rotates, and the faster the subjective rate of physical time passage. This is why the concept of time undergoes a phase transition at the Planck temperature of the Big Bang—because the geometric radius of the modular flow contracts to the limit.