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title: "Complex-Time (Kime) Interpretation of Quantum Mechanics"
author: "SOCR (Ivo Dinov)"
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subtitle: "(Random Draw) Kime-Representations vs. (Branching) Many-worlds Interpretation of Quantum Mechanics"
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# Summary

*Note*: This [TCIU Section in Chapter 6](https://www.socr.umich.edu/TCIU/HTMLs/Chapter6_TCIU_KimeInterpretation_of_QuantumMechanics.html) extends previous work in [Chapter 3 (Radon-Nikodym Derivatives, Kimemeasures, and Kime Operator)](https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_Radon_NikodymDerivatives.html) and [Chapter 6 (Kime Representation)](https://www.socr.umich.edu/TCIU/HTMLs/Chapter6_TCIU_KimeReps_SpacekimeAnalytics_PlanetaryPerihelionPrecession.html). 

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)](https://en.wikipedia.org/wiki/Many-worlds_interpretation) 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.

# 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.

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

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

 1. State preparation $|\psi_{\text{init}}\rangle \rightarrow |\psi(\kappa)\rangle = U_\kappa|\psi_{\text{init}}\rangle ,$
 2. Measurement operator $M$ action $P_\kappa(m) = \int_{-\pi}^{\pi} \langle\psi(\kappa)|M^\dagger M|\psi(\kappa)\rangle\Phi(\theta)d\theta ,$ and
 3. State collapse to a real value, $|\psi(\kappa)\rangle \rightarrow |m\rangle .$

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:

 1. *Born rule* emergence, $\lim_{N\to\infty} \frac{1}{N}\sum_{i=1}^N P_\kappa(m_i) = |\langle m|\psi\rangle|^2,$

 2. *Measurement basis independence*, $\int_{-\pi}^{\pi} \langle\psi(\kappa)| U^{\dagger} M U |\psi(\kappa)\rangle \Phi(\theta) d\theta = \int_{-\pi}^{\pi} \langle \psi(\kappa) | M | \psi(\kappa) \rangle \Phi(\theta) d\theta ,$

 3. *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.,

 1. How does the choice of $\Phi(\theta)$ affect measurement outcomes?
 2. What's the relationship between kime-phase and decoherence time?
 3. How do we handle sequential measurements?
 4. What's the role of environmental interaction?

## Kime-MWI Duality Framework

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:

 1. Phase distribution width $\Delta\theta$ should correlate with decoherence rates,
 2. Sequential measurement statistics should show kime-phase memory effects, 
 3. Environmental coupling strength should modify the effective phase distribution.












# References

 - Everett, Hugh; Wheeler, J. A.; DeWitt, B. S.; Cooper, L. N.; Van Vechten, D.; Graham, N. (1973). DeWitt, Bryce; Graham, R. Neill (eds.). The Many-Worlds Interpretation of Quantum Mechanics. Princeton Series in Physics. Princeton, New Jersey: Princeton University Press. p. v. ISBN 0-691-08131-X.
 - Tegmark, Max (1998). "The Interpretation of Quantum Mechanics: Many Worlds or Many Words?". Fortschritte der Physik. 46 (6–8): 855–862. arXiv:quant-ph/9709032. Bibcode:1998ForPh..46..855T. doi:10.1002/(SICI)1521-3978(199811)46:6/8<855::AID-PROP855>3.0.CO;2-Q.
 - P. Strasberg et al., “First principles numerical demonstration of emergent decoherent histories,” Phys. Rev. X 14, 041027 (2024).
 - A. Bassi et al., “Collapse models: A theoretical, experimental and philosophical review,” Entropy 25, 645 (2023).
 - L. Vaidman, “Many-worlds interpretation of quantum mechanics,” in The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta (Metaphysics Research Lab, Stanford University, 2021), edition Fall; https://plato.stanford.edu/archives/fall2021/entries/qm-manyworlds/.
 - W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003).
 - P. Strasberg and J. Schindler, “Shearing off the tree: Emerging branch structure and Born’s rule in an equilibrated multiverse,” arXiv:2310.06755.


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