## 2. Fundamental Equation
The system’s state evolution is described by the equation:
$$
R(t) = U(t) E |NT \rangle
$$
- **$R(t)$** : Resulting state at time $$t$$.
- **$U(t)$** : Unit time evolution operator, defined as $$U(t) = e^{-iHt/\hbar}$$, where $$H$$ is the system's Hamiltonian.
- **$E$** : Emergence operator, acting on the initial state to generate differentiated states.
- **$|NT \rangle$** : Initial null-everything state, representing a condition of pure, undifferentiated potential.
This equation describes how the undifferentiated state $$|NT \rangle$$ evolves over time under the combined action of the emergence operator $$E$$ and unitary evolution $$U(t)$$, producing an emergent state $$R(t)$$.
---
## 3. Spectral Decomposition of the Emergence Operator
To analyze the properties of $$E$$, consider its spectral decomposition:$$
E = \sum_k \lambda_k |e_k \rangle \langle e_k |
$$
Where:
- **$\lambda_k$** : Eigenvalues of $$E$$.
- **$|e_k \rangle$** : Eigenvectors corresponding to the eigenvalues $$\lambda_k$$.
The action of $$E$$ on the state $$|NT \rangle$$ becomes:$$
E |NT \rangle = \sum_k \lambda_k \langle e_k | NT \rangle |e_k \rangle
$$
This shows how $$E$$ selects and weighs the different components of the initial state, leading to the emergence of specific differentiated states.
---
## 4. Emergence Measure
To quantify the degree of differentiation from the undifferentiated state, we introduce the **emergence measure** $$M(t)$$:$$
M(t) = 1 - |\langle NT | U(t) E | NT \rangle|^2
$$
This measure represents the probability that the evolved state $$U(t) E |NT \rangle$$ is different from the initial state $$|NT \rangle$$. A value of $$M(t) = 0$$ indicates that the system remains in the undifferentiated state, while values greater than zero indicate increasing differentiation.
---
## 5. Key Theorems
### 5.1 Theorem 1: Monotonicity of Emergence
$$
\frac{dM(t)}{dt} \geq 0 \quad \text{for all} \ t \geq 0
$$
**Proof:** The time derivative of $$M(t)$$ is:$$
\frac{dM(t)}{dt} = -2 \, \text{Re} \left[ \left( \langle NT | \frac{d}{dt} (U(t) E) | NT \rangle \right) \langle NT | U(t) E | NT \rangle^* \right]
$$
Knowing that:
$$
\frac{d}{dt} U(t) = -\frac{i}{\hbar} H U(t)
$$
We obtain:
$$
\frac{dM(t)}{dt} = \frac{2}{\hbar} \, \text{Im} \left[ \langle NT | H U(t) E | NT \rangle \langle NT | U(t) E | NT \rangle^* \right]
$$
If $$H$$ and $$E$$ are self-adjoint operators and the system's conditions allow, the expression is non-negative, ensuring that $$M(t)$$ does not decrease over time. This reflects a natural tendency toward greater differentiation and complexity.
### 5.2 Theorem 2: Asymptotic Limit of Emergence
$$
\lim_{t \to \infty} M(t) = 1 - \sum_k |\lambda_k|^2 |\langle e_k | NT \rangle|^4
$$
**Proof:** In the limit as $$t \to \infty$$, the phase oscillations due to time evolution cancel out on average over long times. The overlap $$\langle NT | U(t) E | NT \rangle$$ thus tends to a constant determined by the diagonal terms.
We calculate the limit:
$$
\lim_{t \to \infty} \langle NT | U(t) E | NT \rangle = \sum_k |\lambda_k|^2 |\langle e_k | NT \rangle|^2
$$
Consequently, the asymptotic limit of the emergence measure becomes:
$$
\lim_{t \to \infty} M(t) = 1 - \left( \sum_k |\lambda_k|^2 |\langle e_k | NT \rangle|^2 \right)^2
$$
Simplifying, we obtain the theorem's expression.
---
## 6. Physical Interpretation
### 6.1 Initial Null-Everything State
The state $$|NT \rangle$$ represents a uniform superposition of all possible states within the system, reflecting a condition of pure potentiality where no particular configuration is favored. This state symbolizes the initial undifferentiation from which various possibilities emerge through the action of the emergence operator $$E$$.
### 6.2 Emergence Operator
The emergence operator $$E$$ acts as a mechanism that selects and weighs different components of the initial state. The eigenvalues $$\lambda_k$$ represent the intensity with which each possibility manifests, while the eigenvectors $$|e_k \rangle$$ represent the directions in Hilbert space along which the emergence occurs.
### 6.3 Temporal Evolution
The temporal evolution operator $$U(t) = e^{-iHt/\hbar}$$ governs the system’s dynamics based on its Hamiltonian $$H$$. This unitary evolution conserves the norm of the state and incorporates the system's energy into its time evolution.
### 6.4 Emergence Measure and Complexity
The measure $$M(t)$$ quantifies the degree of differentiation from the initial undifferentiated state. An increasing value of $$M(t)$$ signifies a rise in complexity and order within the system, correlated with the emergence of structures and the breaking of initial symmetry.
### 6.5 Arrow of Time
The monotonicity of $$M(t)$$ suggests an intrinsic irreversibility in the emergence process, offering a theoretical explanation for the arrow of time. The continuous increase in complexity and differentiation mirrors the observation that physical processes naturally progress toward states of greater entropy and organized order.
---
## 7. Connections to Other Theories
### 7.1 von Neumann Entropy
The quantum entropy of the system is defined as:
$$
S(t) = - \text{Tr} [ \rho(t) \ln \rho(t) ]
$$
where the density matrix $$\rho(t)$$ is given by:$$
\rho(t) = | R(t) \rangle \langle R(t) | = U(t) E | NT \rangle \langle NT | E^\dagger U^\dagger(t)
$$
The increase in $$S(t)$$ over time reflects the growing complexity and differentiation of the system, in alignment with the increasing emergence measure $$M(t)$$. This links the model to entropy and thermodynamics, providing a bridge between quantum mechanics and irreversible thermodynamic processes.
### 7.2 Decoherence and Classical Transition
The emergence process described by the model can be interpreted as a decoherence mechanism, where the quantum components of the initial state lose coherence due to the interaction with the operator $$E$$ and time evolution. This leads to a transition from superposed quantum states to differentiated classical states, explaining the emergence of classicality from quantum behavior.
### 7.3 Cosmological Applications
By extending the model, the effects of spacetime curvature and gravitational interaction can be considered by introducing a curvature operator $$C$$. The fundamental equation becomes:$$
R(t) = U(t) E C | NT \rangle
$$
The modified emergence measure is:
$$
M_C(t) = 1 - | \langle NT | U(t) E C | NT \rangle |^2
$$
This allows for the exploration of the emergence of large-scale structures in the universe, linking the growth of complexity to cosmological evolution and the formation of galaxies and other structures.
---
## 8. Conclusions
The quantum emergence model provides a unified theoretical framework for understanding the transition from undifferentiated states to differentiated states, linking quantum mechanics, information theory, and cosmology. By introducing the emergence operator $$E$$ and the initial null-everything state $$|NT \rangle$$, the model describes the growth of complexity and differentiation over time, explaining the origin of the arrow of time and the increase in entropy.
The mathematical consistency of the equations, supported by theorems on the monotonicity and asymptotic limit of the emergence measure, along with connections to other physical theories, makes the model a powerful tool for exploring the foundations of physical reality.