2. The process is irreversible and leads to an asymptotic maximum of complexity determined by the eigenvalues and eigenvectors of \( E \) and the superposition of the initial state with these eigenvectors.
---
**Formalization of the Axiom:**
1. **Hilbert Space and Initial State:**
- Let \( \mathcal{H} \) be a separable Hilbert space.
- The undifferentiated initial state \( |NT\rangle \in \mathcal{H} \) is defined as:
\[
|NT\rangle = \frac{1}{\sqrt{N}} \sum_{n=1}^{N} |n\rangle
\]
where \( \n\rangle \) is an ortho-normal basis of \( \mathcal{H} \), and \( N \) is the dimension of \mathcal{H} \).
2. **Emergency Operator \( E \):**
- \( E: \mathcal{H} \to \mathcal{H} \) is a self-adjoint operator defined as:
\[
E = \sum_{k} \lambda_k |e_k\rangle \langle e_k|
\]
where \ \( \lambda_k \) are the eigenvalues of \ E \), and \ \( \e_k\rangle \) \) are the corresponding orthonormal eigenvectors.
3. **Operator of Temporal Evolution \( U(t) \):**
- Temporal evolution is governed by the unitary operator \( U(t) \):
\[
U(t) = e^{-i H t / \hbar}
\]
where \( H \) is the Hamiltonian of the system.
4. **Evolved state \( |Psi(t)\rangle \):**
- The state of the system at time \( t \) is given by:
\[
|\Psi(t)\rangle = U(t) E |NT\rangle
\]
5. **Emergency Measure \( M(t) \):**
- The measure \( M(t) \) quantifies the degree of differentiation of the state \( |Psi(t)\rangle \) from the initial state \( |NT\rangle \):
\[
M(t) = 1 - | | NT \langle | \Psi(t) \rangle|^2 = 1 - | | NT \langle | U(t) E | NT \rangle|^2
\]
- \( M(t) \) varies between 0 and 1:
- \( M(t) = 0 \) indicates that the system is still in the undifferentiated state.
- \( M(t) \) close to 1 indicates a high degree of differentiation.
---
**Properties:**
1. **Monotonicity of the Emergency Measure \( M(t) \):**
- The measure \( M(t) \) is a nondecreasing function over time:
\[
\frac{dM(t)}{dt} \geq 0 \quad \forall t \geq 0
\]
- It implies that the complexity of the system increases or remains constant over time, but does not decrease.
2. **Asymptotic limit of \( M(t) \):**
- In the limit \( t \to \infty \), the measure \( M(t) \) reaches a maximum:
\[
\lim_{t \to \infty} M(t) = 1 - \left| \sum_k \lambda_k | \langle e_k | NT \rangle|^2 \right|^2
\]
where \( | |langle e_k | NT \rangle|^2 \right|^2 \) represents the weight of the state \( |e_k\rangle \) in the initial state.
3. **Irreversibility:**
- The process of emergence and differentiation is irreversible; the system cannot spontaneously return to the undifferentiated state \( |NT\rangle \) without external intervention.
4. **Growth of Entropy:**
- The von Neumann entropy \( S(t) \), defined as:
\[
S(t) = -\text{Tr}[\rho(t) \ln \rho(t)]
\]
where \ where \( \rho(t) = |\Psi(t)\rangle \langle \Psi(t)| \), tends to increase, reflecting the growth in complexity.
5. **Decoherence:**
- Quantum coherence decreases during evolution, leading to a classical state with respect to certain observables. This phenomenon is known as decoherence.
---
**Final Synthesis:**
The **Axiom of Quantum Emergence** states that a quantum system initially in an undifferentiated state \( |NT\rangle \) inevitably evolves toward increasingly differentiated states through the combined action of the emergence operator \( E \) and time evolution \( U(t) \). The contingency measure \( M(t) \) increases monotonically over time, reflecting the irreversibility of the process and the growth of system complexity. This axiom provides a basis for understanding the emergence of complex structures.