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Note: This TCIU Section in Chapter 6 extends previous work in Chapter 3 (Radon-Nikodym Derivatives, Kimemeasures, and Kime Operator) and Chapter 6 (Kime Representation).
This TCIU section delves deeper into the theoretical relationship between kime-phase representaiton of repeated measurements in spacekime theory and the many-worlds interpretation (MWI) of quantum mechanics.
The key mathematical connection between spacekime representation and MWI reflect kime-phase sampling maps to parallel worlds. In spacekime theory, each kime-phase \(\theta\) represents a potential measurement outcome. The kime-phase distribution \(\Phi_{[-\pi,\pi)}\) gives probabilities for different phases. Similarly, in MWI each world represents a potential measurement outcome with associated probability amplitude. In terms of measurement and observeabilty, in the core of MWI is dynamic event-branching at each each spatiotemporal location. Whereas in spacekime, random sampling from a stable multiverse is reflecting each event draw at the point of observability. There is no “branching” in spacekime, it’s more like stochastic traversal of the multiverse.
A mapping could be formalized as \[|\psi(\kappa)\rangle = U_{\kappa}(|\psi_0\rangle) = e^{-itA}\mathcal{l}(e^{i\theta})(|\psi_0\rangle) ,\] where \(\theta\sim \Phi_{[-\pi,\pi)}\) represents the specific measurement possibilities (e.g., energy levels) in the kime framework, and the function \(\mathcal{l}(\cdot)\) is a complex-valued distribution function that represents the kime-phase contribution to quantum evolution. This function \(\mathcal{l}(e^{i\theta})\) maps the unit circle to complex numbers \(\mathcal{l}: S^1 \to \mathbb{C}\) and acts on test functions \(\varphi\) by \[\langle \mathcal{l}, \varphi \rangle = \int_{\mathbb{R}} \mathcal{l}^*(\theta)\varphi(\theta) d\theta \in \mathbb{C}.\] A normalization conditions ensures that \(\int_{-\pi}^{\pi} |\mathcal{l}(e^{i\theta})|^2 \Phi(\theta)d\theta = 1\). It plays a role in the kime unitary operator, \(U_\kappa = e^{-itA}\mathcal{l}(e^{i\theta})\) and also appears
For a uniform phase distribution, \(\mathcal{l}_U(e^{i\theta}) = \frac{1}{\sqrt{2\pi}}\), for a Normal prior, \(\mathcal{l}_N(e^{i\theta}) = \frac{1}{\sqrt{Z_N}}e^{-\theta^2/4\sigma^2}\), and for a Laplace phase, \(\mathcal{l}_L(e^{i\theta}) = \frac{1}{\sqrt{Z_L}}e^{-|\theta|/2b}\).
This function leads to the kime-phase action, \(\langle\mathfrak{P}, \phi\rangle = \Psi(x,y,z,t)\langle\mathcal{l}, \phi\rangle\) and has the following properties:
The physical interpretation is that \(\mathcal{l}(e^{i\theta})\) encodes how quantum states respond to phase uncertainty in measurements through
In MWI terms, this would correspond to \(|\psi\rangle = \sum_i c_i|\psi_i\rangle\), Where each \(|\psi_i\rangle\) represents a different world whose likelihood to be randomly observed corresponds to the amplitude \(c_i\).
Both frameworks use unitary operators, \(U_{\kappa}\) in kime theory and standard quantum evolution \(U(t=|\kappa|)\) in MWI. The key difference is that kime theory explicitly models the observed dispersion in real observables as kime-phase uncertainty through the \(\Phi(\theta)\) distribution. In kime theory, the expected outcome is expressed as \(\langle O(\kappa)\rangle_{mean} = \frac{1}{N}\sum_{i=1}^N \langle O(\kappa_i)\rangle\) In MWI, the (observable) operator expected value is \(\langle O\rangle = \sum_i |c_i|^2\langle\psi_i|O|\psi_i\rangle\).
The key challenges are to:
In complex-time representation, \[|\psi(\kappa)\rangle = U_{\kappa}(|\psi_0\rangle) = e^{-itA}\mathcal{l}(e^{i\theta})(|\psi_0\rangle),\] where \(\theta\sim \Phi_{[-\pi,\pi)}\) is the kime-phase distribution.
In MWI, \(|\psi\rangle = \sum_i c_i|\psi_i\rangle\), where \(|c_i|^2\) reflect the probability of observing world \(i\) in teh multiverse. Any kime-MWI correspondence should explicate a mapping between measurement outcomes. For a given observable \(O\), the kime-representaiton \(\langle O(\kappa)\rangle = \langle\psi(0)|U^{\dagger}(\kappa)OU(\kappa)|\psi(0)\rangle\).
Expanding with the kime-phase distribution: \[\langle O(\kappa)\rangle = \int_{-\pi}^{\pi} \langle\psi(0)|e^{itA}\mathcal{l}^*(e^{i\theta})O e^{-itA}\mathcal{l}(e^{i\theta})|\psi(0)\rangle \Phi(\theta)d\theta .\]
In MWI, the expectation value is \(\langle O \rangle = \sum_i |c_i|^2\langle\psi_i|O|\psi_i\rangle\).
An equivalence between these requires \[\int_{-\pi}^{\pi} \mathcal{l}^*(e^{i\theta})\mathcal{l}(e^{i\theta})\Phi(\theta)d\theta = \sum_i |c_i|^2 .\] One correspondence may be \[|c_i|^2 \leftrightarrow \int_{\theta_i}^{\theta_{i+1}} \mathcal{l}^*(e^{i\theta})\mathcal{l}(e^{i\theta})\Phi(\theta)d\theta ,\] where \([\theta_{i},\theta_{i+1}]\) partitions the phase distribution support \([-\pi,\pi)\) into measurement bin outcomes. To ensure proper normalization, the kime distribution function must satisfy \[\int_{-\pi}^{\pi} \mathcal{l}^*(e^{i\theta})\mathcal{l}(e^{i\theta})\Phi(\theta)d\theta = 1 .\]
An exact correspondence also demends \[\mathcal{l}(e^{i\theta}) = \sum_i \sqrt{|c_i|^2}\delta(\theta - \theta_i),\] where \(\theta_i\) are the discrete phase values corresponding to different MWI worlds. Clearly, there are some problems with this derivation:
Alternative strategies need to ensure that the kime-phase distribution reproduces quantum measurement statistics exactly, show how interference effects are handled equivalently, demonstrate that the correspondence preserves quantum entanglement properties, and finally, address the measurement problem in both frameworks consistently.
Starting with a measurement operator \(M\), an observed real value \(m\in\mathbb{R}\), and a state \(|\psi\rangle\), \[\text{Standard QM probability:} \quad P(m) = |\langle m|\psi\rangle|^2.\] In kime-representation, this becomes \(P_\kappa(m) = \int_{-\pi}^{\pi} |\langle m|U_\kappa|\psi\rangle|^2 \Phi(\theta)d\theta,\) where \(U_\kappa = e^{-itA}\mathcal{l}(e^{i\theta})\).
To prove an equivalence, we need \[\int_{-\pi}^{\pi} |\langle m|e^{-itA}\mathcal{l}(e^{i\theta})|\psi\rangle|^2 \Phi(\theta)d\theta = |\langle m|\psi\rangle|^2 .\]
This requires the following normalization constraints \[\int_{-\pi}^{\pi} \mathcal{l}^*(e^{i\theta})\mathcal{l}(e^{i\theta})\Phi(\theta)d\theta = 1\] and \(\int_{-\pi}^{\pi} e^{itA}\mathcal{l}(e^{i\theta})\Phi(\theta)d\theta = \mathbb{1}.\)
Let’s explore the interference between two states \(|\psi_1\rangle\)⟩ and \(|\psi_2\rangle\). In classical QM \[|\langle \phi|(|\psi_1\rangle + |\psi_2\rangle)|^2 = |\langle \phi|\psi_1\rangle|^2 + |\langle \phi|\psi_2\rangle|^2 + 2\text{Re}(\langle \phi|\psi_1\rangle\langle \psi_2|\phi\rangle) .\]
In kime theory \[P_\kappa(\phi) = \int_{-\pi}^{\pi} |\langle \phi|U_\kappa(|\psi_1\rangle + |\psi_2\rangle)|^2 \Phi(\theta)d\theta .\]
Hence, \[P_\kappa(\phi) = \int_{-\pi}^{\pi} [|\langle \phi|U_\kappa|\psi_1\rangle|^2 + |\langle \phi|U_\kappa|\psi_2\rangle|^2 + 2\text{Re}(\langle \phi|U_\kappa|\psi_1\rangle\langle \psi_2|U_\kappa^\dagger|\phi\rangle)]\Phi(\theta)d\theta .\]
The interference term carries phase information through \[\Delta\phi_\kappa = \arg(\langle \phi|U_\kappa|\psi_1\rangle) - \arg(\langle \phi|U_\kappa|\psi_2\rangle) .\]
Examining quantum entanglement preservation, consider an entangled state \(|\Psi\rangle = \alpha |00\rangle + \beta |11\rangle\). The classical QM density matrix \[\rho = |\Psi\rangle\langle\Psi| = |\alpha|^2|00\rangle\langle 00| + \alpha\beta^*|00\rangle\langle 11| + \alpha^*\beta|11\rangle\langle 00| + |\beta|^2|11\rangle\langle 11|.\]
The kime theory must preserve \[\rho_\kappa = \int_{-\pi}^{\pi} U_\kappa|\Psi\rangle\langle\Psi|U_\kappa^\dagger \Phi(\theta)d\theta .\]
The entanglement entropy must also be preserved \(S(\rho) = -\text{Tr}(\rho\log\rho) = S(\rho_\kappa).\) This requires \[\text{Tr}_B[\rho_\kappa] = \int_{-\pi}^{\pi} \text{Tr}_B[U_\kappa|\Psi\rangle\langle\Psi|U_\kappa^\dagger]\Phi(\theta)d\theta .\]
In kime theory, the measurement problem reflects
Therefore, the probability distribution must satisfy \[\int_{-\pi}^{\pi} P_\kappa(m)\Phi(\theta)d\theta = \text{Tr}(M\rho M^\dagger).\]
In addition, there are a set of consistency requirements including:
Born rule emergence, \(\lim_{N\to\infty} \frac{1}{N}\sum_{i=1}^N P_\kappa(m_i) = |\langle m|\psi\rangle|^2,\)
Measurement basis independence, \(\int_{-\pi}^{\pi} \langle\psi(\kappa)| U^{\dagger} M U |\psi(\kappa)\rangle \Phi(\theta) d\theta = \int_{-\pi}^{\pi} \langle \psi(\kappa) | M | \psi(\kappa) \rangle \Phi(\theta) d\theta ,\)
Decoherence mechanism, \(\rho_{\text{reduced}} = \text{Tr}_{\text{env}}[\rho_\kappa] = \sum_i P_i|i\rangle\langle i| .\)
Still, this kime-MWI duality framework still leaves some open questions, e.g.,
First let’s examine the effect of \(\Phi(\theta)\) on Measurement Outcomes. Consider different phase distributions and quantify their impacts. For a general measurement operator \(M\) and state \(|\psi\rangle\), the kime measurement probability is \[P_\kappa(m) = \int_{-\pi}^{\pi} |\langle m|U_\kappa|\psi\rangle|^2 \Phi(\theta)d\theta .\]
For different distributions, the kime measurement probability will be different. FOr instance, for a Uniform distribution \(\Phi_U(\theta) = \frac{1}{2\pi}\), \[P_\kappa^U(m) = \frac{1}{2\pi}\int_{-\pi}^{\pi} |\langle m|e^{-itA}\mathcal{l}(e^{i\theta})|\psi\rangle|^2 d\theta .\] For a (truncated) Normal distribution, \(\Phi_N(\theta) = \frac{1}{Z_N}e^{-\theta^2/2\sigma^2}\), it is \[P_\kappa^N(m) = \frac{1}{Z_N}\int_{-\pi}^{\pi} |\langle m|e^{-itA}\mathcal{l}(e^{i\theta})|\psi\rangle|^2 e^{-\theta^2/2\sigma^2} d\theta .\] And for a (truncated) Laplace phase distribution, \(\Phi_L(\theta) = \frac{1}{Z_L}e^{-|\theta|/b}\), \[P_\kappa^L(m) = \frac{1}{Z_L}\int_{-\pi}^{\pi} |\langle m|e^{-itA}\mathcal{l}(e^{i\theta})|\psi\rangle|^2 e^{-|\theta|/b} d\theta .\]
The choice of distribution affects measurement uncertainty through \[\Delta m_\kappa^2 = \int_{-\pi}^{\pi} (m - \langle m \rangle_\kappa)^2 P_\kappa(m) \Phi(\theta)d\theta .\]
Similarly, the kime-phase and decoherence time relations are affected by the phase prior. The decoherence time \(\tau_D\) relates to the kime-phase width \(\Delta\theta\) via \(\tau_D \sim \frac{\hbar}{\Delta E} \sim \frac{\hbar}{E\Delta\theta}\).
The time evolution of the density matrix under decoherence is \[\rho(t) = \int_{-\pi}^{\pi} U_\kappa\rho(0)U_\kappa^\dagger \Phi(\theta)d\theta e^{-t/\tau_D}.\] Coupling to environment introduces phase damping \[\rho_{nm}(t) = \rho_{nm}(0)e^{-i\omega_{nm}t}e^{-t/\tau_D}F(\Delta\theta),\] where \(F(\Delta\theta)\) is the phase damping function, \[F(\Delta\theta) = \int_{-\pi}^{\pi} e^{i\theta} \Phi(\theta)d\theta .\]
For sequential measurements, \(M_1, M_2, \cdots, M_n\) corresponding to \(\kappa_1, \kappa_2, \cdots, \kappa_n\), \[P_\kappa(m_1, \cdots ,m_n) = \int_{-\pi}^{\pi} \text{Tr}[M_n \cdots M_1\rho(\kappa)M_1^\dagger \cdots M_n^\dagger] \prod_{i=1}^n \Phi(\theta_i)d\theta_i .\] And the corresponding conditional probability is \[P_\kappa(m_2|m_1) = \frac{\int_{-\pi}^{\pi} \text{Tr}[M_2M_1\rho(\kappa)M_1^\dagger M_2^\dagger] \Phi(\theta_1)\Phi(\theta_2)d\theta_1d\theta_2}{P_\kappa(m_1)} .\]
The total system-environment Hamiltonian is \(H_{tot} = H_S + H_E + H_{int}\), and the reduced density matrix after tracing out environment is \[\rho_S(\kappa) = \text{Tr}_E[U_\kappa(\rho_S \otimes \rho_E)U_\kappa^\dagger].\]
Then, the kime-modified master equation becomes \[\frac{\partial\rho_S}{\partial \kappa} = -\frac{i}{\hbar}[H_S,\rho_S] + \mathcal{L}[\rho_S],\] where \(\mathcal{L}\) is the Lindblad superoperator adapted to kime \[\mathcal{L}[\rho_S] = \sum_k \gamma_k(\kappa)\left (L_k\rho_S L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho_S\}\right ),\] where the curly-braces notation \(\{A,B\}\) represents the anticommutator of operators \(A\) and \(B\), \(\{A,B\} = AB + BA\). Hence, in the Lindblad equation, \[\{L_k^\dagger L_k,\rho_S\} = L_k^\dagger L_k\rho_S + \rho_S L_k^\dagger L_k.\] The anticommutator term is crucial because it preserves the trace of \(\rho_S\), ensures positivity of the quantum dynamical map, and maintains the Hermiticity of the density matrix. The decoherence rates \(\gamma_k(\kappa)\) depend on kime as follows \[\gamma_k(\kappa) = \int_{-\pi}^{\pi} \gamma_k e^{i\theta} \Phi(\theta)d\theta .\] This kime-to-MWI duality framework suggests several testable predictions: