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1 Summary

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

  • State Evolution: \(|\psi(\kappa)\rangle = e^{-itA}\mathcal{l}(e^{i\theta})|\psi_0\rangle\), and
  • Expectation Values: \(\langle O(\kappa)\rangle = \int_{-\pi}^{\pi} \langle\psi_0|\mathcal{l}^*(e^{i\theta})e^{itA}Oe^{-itA}\mathcal{l}(e^{i\theta})|\psi_0\rangle \Phi(\theta)d\theta\).

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:

  1. Conjugation: \(\mathcal{l}^*(e^{-i\theta}) = \mathcal{l}(e^{i\theta})\)
  2. Composition: \(\mathcal{l}(e^{i\theta_1})\mathcal{l}(e^{i\theta_2}) = \mathcal{l}(e^{i(\theta_1+\theta_2)})\), and
  3. Action on test functions: \(\langle\mathcal{l}, \phi\rangle = \int_{\mathbb{R}} \mathcal{l}^*(\theta)\phi(\theta) d\theta\).

The physical interpretation is that \(\mathcal{l}(e^{i\theta})\) encodes how quantum states respond to phase uncertainty in measurements through

  1. Phase uncertainty: \(\Delta\theta = \sqrt{\int_{-\pi}^{\pi} \theta^2|\mathcal{l}(e^{i\theta})|^2 \Phi(\theta)d\theta}\), and
  2. Measurement effects: \(P(m) = \int_{-\pi}^{\pi} |\langle m|\mathcal{l}(e^{i\theta})|\psi\rangle|^2 \Phi(\theta)d\theta\)

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:

  1. Provide exact correspondence between kime-phase distributions and MWI probability amplitudes.
  2. Show how the decoherence mechanism works similarly in both frameworks.
  3. Derive the Born rule from both perspectives and show their equivalence.
  4. Establish clear connections between the mathematical structures while respecting the physical interpretations.

2 Kime-phase distributions and MWI probability amplitudes

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:

  1. It assumes discrete measurement outcomes can be mapped to continuous phase distributions.
  2. The relationship between phase space and measurement space needs more rigorous formulation.
  3. The role of decoherence isn’t explicitely addressed.
  4. The Born rule emergence needs explanation in both frameworks.

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.

2.1 Reproducibility of Quantum Statistics via Kime-Phase Distributions

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 \[\Delta\phi_\kappa = \arg(\langle \phi|U_\kappa|\psi_1\rangle) - \arg(\langle \phi|U_\kappa|\psi_2\rangle) .\]

Exam