1 Introduction

Complex-time (kime) is defined as follows. Complex-time (kime) representation of repeated measurements longitudinal data localizes measureable quantitities according to their space-kime coordinates, $({\bf{x}}=(x,y,x),\kappa = t e^{i\theta}$), where the kime-magnitude $|\kappa|=t$ is the event temporal ordering, time, and the kime-phase corresponds to random draws from a time-dependent kime-phase distribution, $\theta \sim \Phi(t)$, supported on $[-\pi,\pi)$. As the kime-phase reflects a random draw from a phase distribution, it is unobservable. It’s worth clarifying the derivative operator formulation when $\theta\sim\Phi$, rather than a deterministic coordinate. One may consider the need for using Itô calculus, distributional derivatives, or another framework to clarify the role of $\theta$ and $\frac{d}{d\theta}$ and explicate the derivative operator

In QM, $\hat{x}$ and $\hat{p}$ operate on wavefunctions $\psi(x)$ in $L^2(\mathbb{R})$, where $x$ is a continuous spatial variable. The derivative $\frac{d}{dx}$ is well-defined as an operator on differentiable functions. In the kime context, $\kappa = t e^{i\theta}$ introduces $\theta$ as a phase, randomly sampled from $\Phi(\theta; t)$ for each measurement at time $t$. Approach 1 probes $\Phi(\theta; t)$ via test functions, e.g., $f_\alpha(\theta)$, computing expectations like $\langle \Phi, f_\alpha \rangle = \int f_\alpha(\theta) \Phi(\theta; t) d\theta$. In Approach 1A reformulation, we defined operators on a Hilbert space, e.g., $L^2[-\pi, \pi]$, acting on functions $\psi(\theta)$, where $\psi(\theta)$ represent $\sqrt{\Phi(\theta; t)}$ or another state-like object. When $\theta$ is a random draw, $\frac{d}{d\theta}$ seems ill-defined without a continuous trajectory. Yet, $\Phi(\theta; t)$ is a density over $\theta$, suggesting a functional space where $\theta$ is a variable of integration, not a stochastic process evolving in time (as in Itô calculus).

Since $\Phi(\theta; t)$ is a probability density over $\theta \in [-\pi, \pi]$ at fixed $t$, we treat $\theta$ as a coordinate in a function space, not a random variable evolving dynamically. Thus,

Hilbert Space: $L^2[-\pi, \pi]$, where functions $\psi(\theta)$ are square-integrable over $\theta$, and $\Phi(\theta; t) = |\psi(\theta)|^2$ (analogous to a probability density in QM).
State: $\psi(\theta)$ could be $\sqrt{\Phi(\theta; t)}$ (assuming $\Phi$ is real and non-negative), representing the state of the kime-phase distribution at time $t$.
Operators: Define $\hat{F}_\alpha$ and $\hat{G}_\beta$ as operators on $\psi(\theta)$, with expectations computed against $\Phi(\theta; t)$. The differentiation operator $\hat{G}_\beta = -i \frac{d}{d\theta}$ in $L^2[-\pi, \pi]$, where $\theta$ is the independent variable of the function space, not a random draw per se. The random sampling occurs when we generate data (e.g., $\theta_n(t) \sim \Phi$), but the operators act on the density or its square root. Thus, the multiplication operator $\hat{\Theta} \psi(\theta) = \theta \psi(\theta)$ and the differentiation operator $\hat{G}_\beta \psi(\theta) = -i \beta \frac{d}{d\theta} \psi(\theta)$, where $\beta$ scales the momentum (frequency-like) contribution. Since $\theta \in [-\pi, \pi]$ is a circular domain (phase wraps around), we impose periodic boundary conditions, $\psi(-\pi) = \psi(\pi)$, ensuring $\hat{G}_\beta$ is well-defined on differentiable functions in $L^2[-\pi, \pi]$.

Then, the commutator is $[\hat{\Theta}, \hat{G}_\beta] \psi(\theta) = -i \beta \theta \frac{d}{d\theta} \psi(\theta) - (-i \beta \psi(\theta) - i \beta \theta \frac{d}{d\theta} \psi(\theta)) = i \beta \psi(\theta)$, where

\[\hat{\Theta} \hat{G}_\beta \psi(\theta) = \theta (-i \beta \frac{d}{d\theta} \psi(\theta)) = -i \beta \theta \frac{d}{d\theta} \psi(\theta)\]

\[\hat{G}_\beta \hat{\Theta} \psi(\theta) = -i \beta \frac{d}{d\theta} (\theta \psi(\theta)) = -i \beta \left( \psi(\theta) + \theta \frac{d}{d\theta} \psi(\theta) \right).\]

Thus, $[\hat{\Theta}, \hat{G}_\beta] = i \beta I$ confirms non-commutativity, with a commutator proportional to $\beta$. Itô calculus does not apply to stochastic processes where a variable, $\theta(t)$, evolves randomly over time $t$, governed by a stochastic differential equation (SDE). In kime, $\theta$ is sampled independently at each $t$ from $\Phi(\theta; t)$, with no explicit temporal dynamics (e.g., $d\theta = \mu dt + \sigma dW_t$). And the derivative $\frac{d}{d\theta}$ operates over the $\theta$-domain of the density $\Phi(\theta; t)$ at fixed $t$, not over $t$. Thus, Itô calculus is unnecessary here. The randomness is in the sampling process, not in a continuous evolution of $\theta$. A distributional derivative (e.g., weak derivative) would be needed if $\psi(\theta)$ or $\Phi(\theta; t)$ were not differentiable (e.g., included Dirac deltas). For simplicity, assume $\Phi(\theta; t)$ is smooth (e.g., von Mises), so the standard derivative suffices. If $\Phi$ were less regular, we’d define $\hat{G}_\beta$ in the sense of distributions, $\langle \hat{G}_\beta \psi, \phi \rangle = -i \beta \int_{-\pi}^\pi \psi(\theta) \frac{d}{d\theta} \phi(\theta) d\theta$, for test functions $\phi \in C^\infty[-\pi, \pi]$.

2 Wheeler-DeWitt Equation Formulation in Spacekime

To extend the classical Wheeler-DeWitt equation to space-kime (complex-time), we incorporate the kime-phase $\theta$ as an additional quantum variable. Here’s the mathematical derivation:

Extended Hilbert Space: The wave functional depends on the spatial metric $g_{ij}$ and the kime-phase $\theta$, $\Psi[g, \theta] \in L^2(\text{Super space}) \otimes L^2[-\pi, \pi),$ where Super space is the space of 3-metrics $g_{ij}$, and $L^2[-\pi, \pi)$ handles the $\theta$-dependence.

Kime-Phase Operators: Define operators acting on the $\theta$-Hilbert space

Position operator: $\hat{\Theta} \psi(\theta) = \theta \psi(\theta)$,
Momentum operator: $\hat{G}_\beta = -i\beta \frac{d}{d\theta}$, with commutator $[\hat{\Theta}, \hat{G}_\beta] = i\beta I$.

Extended Wheeler-DeWitt Equation: The Hamiltonian constraint includes spatial gravity and kime-phase contributions $\left[\hat{\mathcal{H}}_{\text{space}} + \hat{\mathcal{H}}_{\theta}\right] \Psi[g, \theta] = 0,$ where the spatial Hamiltonian (standard Wheeler-DeWitt) is \[ \hat{\mathcal{H}}_{\text{space}} = -16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right). \], the kime-phase Hamiltonian is \[ \hat{\mathcal{H}}_{\theta} = -\frac{\beta^2}{2\mu} \frac{d^2}{d\theta^2} + V(\theta), \] and $\mu$ a coupling constant and $V(\theta)$ a potential (e.g., enforcing $\Phi(\theta; t)$).

Periodic Boundary Conditions: For $\theta \in [-\pi, \pi)$, impose $\Psi[g, -\pi] = \Psi[g, \pi], \quad \forall g.$

2.1 Assuming Separation of Variables

Assume $\Psi[g, \theta] = \Psi_{\text{space}}[g] \psi(\theta)$. This splits the equation: \[ \hat{\mathcal{H}}_{\text{space}} \Psi_{\text{space}}[g] = E \Psi_{\text{space}}[g], \] \[ \hat{\mathcal{H}}_{\theta} \psi(\theta) = -E \psi(\theta). \]

Fourier Decomposition: Expand $\psi(\theta)$ in eigenfunctions of $\hat{G}_\beta$ \[ \psi(\theta) = \sum_{n=-\infty}^\infty c_n e^{in\theta}, \quad n \in \mathbb{Z}. \] Substituting into $\hat{\mathcal{H}}_{\theta}\psi = -E\psi$ gives: \[ \left(\frac{\beta^2 n^2}{2\mu} + V(n)\right) c_n = -E c_n. \]

Modified Spatial Equation: For each Fourier mode $n$, the spatial Wheeler-DeWitt equation becomes \[ \left[-16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right) + \frac{\beta^2 n^2}{2\mu} + V(n)\right] \Psi_{\text{space}}^{(n)}[g] = 0. \] This introduces a kime-phase energy term $\frac{\beta^2 n^2}{2\mu} + V(n)$, modifying the effective cosmological constant $\Lambda$.

The physical interpretation of the kime-phase $\theta$ contributes discrete energy levels indexed by $n$, analogous to angular momentum. Each mode $n$ corresponds to a universe with a distinct effective energy density, leading to a multiverse-like spectrum of solutions.

The space-kime Wheeler-DeWitt equation extends general relativity by quantizing the kime-phase $\theta$. The resultant theory couples spatial geometry to a compact internal dimension ($\theta$), introducing quantized energy corrections. This formalism bridges complex-time stochasticity with quantum gravity, offering a pathway to unify measurement statistics with cosmological dynamics. \[ \boxed{\left[-16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right) - \frac{\beta^2}{2\mu} \frac{d^2}{d\theta^2} + V(\theta)\right] \Psi[g, \theta] = 0} \]

2.2 Example using von Mises Mixture Kime-Phase Distribution

Let’s work our an example to illustrate the special case of the Wheeler-DeWitt Equation (WDE) in spacekime, when the kime-phase distribution is a two-component mixture of von Mises ($vM$) where we have good estimates of the 4 parameters (the mixture distribution weight, $w$) and time-varying ($t$-dependent) locations ($\mu_1(t)$ and $\mu_2(t)$) and concentrations ($k_1(t)$ and $k_2(t)$), i.e., $\Phi(\theta) = w vM(\theta;\mu_1(t), k_1(t)) + (1-w)\,vM(\theta;\mu_2(t),k_2(t)).$

In this case, the kime-phase distribution at time $t$ is a mixture $\Phi(\theta; t) = w \, vM(\theta; \mu_1(t), k_1(t)) + (1-w) \, vM(\theta; \mu_2(t), k_2(t)),$ where $w \in [0,1]$ is the mixture weight, $\mu_1(t), \mu_2(t)$ are time-dependent mean angles, and $k_1(t), k_2(t)$ are time-dependent oncentration parameters.

The von Mises density is $vM(\theta; \mu, k) = \frac{e^{k \cos(\theta - \mu)}}{2\pi I_0(k)},$ where $I_0(k)$ is the modified Bessel function of order $0$.

Assuming that the kime-phase wavefunction is $\psi(\theta; t) = \sqrt{\Phi(\theta; t)} = \sqrt{w \, vM_1 + (1-w) \, vM_2}$ ensures $|\psi(\theta; t)|^2 = \Phi(\theta; t)$.

Then, the kime-phase Hamiltonian is \[\hat{\mathcal{H}}_\theta = -\frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t),\] where $V(\theta; t)$ is designed to enforce the mixture $\Phi(\theta; t)$. Substituting $\psi(\theta; t)$ into the eigenvalue equation $\hat{\mathcal{H}}_\theta \psi = -E \psi$, we solve for $V(\theta; t)$ \[V(\theta; t) = -E + \frac{\beta^2}{2\mu} \frac{\nabla_\theta^2 \psi(\theta; t)}{\psi(\theta; t)}.\]

Let’s explicitly derive the potential by computing $\nabla_\theta^2 \psi$, \[\nabla_\theta^2 \psi = \frac{w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2}{2\sqrt{\Phi}} - \frac{\left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{4\Phi^{3/2}}.\] Substituting into $V(\theta; t)$ yields \[V(\theta; t) = -E + \frac{\beta^2}{4\mu \Phi} \left[ w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2 \right] - \frac{\beta^2 \left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{8\mu \Phi^2}.\]

For more specificity, assume a simplified von Mises case with static locations (means) and concentrations (dispersions). Consider $\mu_1, \mu_2, k_1, k_2$ are constant (time-independent) and use the von Mises property $\nabla_\theta vM = -k \sin(\theta - \mu) vM$, the potential becomes \[V(\theta) = -E + \frac{\beta^2}{4\mu \Phi} \left[ w k_1 \left(k_1 \sin^2(\theta - \mu_1) - \cos(\theta - \mu_1)\right) vM_1 + (1-w)k_2 \left(k_2 \sin^2(\theta - \mu_2) - \cos(\theta - \mu_2)\right) vM_2 \right].\]

Therefore, the extended WDE with von Mises mixture phase (true model), the full spacekime WDE becomes \[\boxed{ \left[ -16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right) - \frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t) \right] \Psi[g, \theta] = 0 },\] where $V(\theta; t)$ is defined above.

The corresponding physical interpretation of the {} $V(\theta; t)$ combines two “wells” centered at $\mu_1(t)$ and $\mu_2(t)$, modulated by $k_1(t), k_2(t)$. The mixture weight $w$ controls their relative influence. With regard to the spatial geometry coupling, the spatial Hamiltonian $\hat{\mathcal{H}}_{\text{space}}$ is modified by the kime-phase energy $-E$, effectively renormalizing the cosmological constant $\Lambda \rightarrow \Lambda + \Delta\Lambda$, where $\Delta\Lambda \propto E$.

Let’s consider an example with separated modes where $\mu_1 \approx \mu_2 + \pi$ and $k_1, k_2 \gg 1$ and the mixture approximates a bimodal distribution. The potential $V(\theta; t)$ resembles a double-well on a circle, with minima at $\mu_1, \mu_2$. The spatial WDE then describes a universe with two dominant geometric states correlated with the kime-phase peaks.

This example depicts how non-Gaussian kime-phase mixture distributions generate structured potentials $V(\theta; t)$, which modify the effective energy-momentum content of the Wheeler-DeWitt equation. The formalism bridges stochastic measurement statistics (via $\Phi(\theta; t)$) with quantum gravitational dynamics.

2.3 Assuming No Separation of Variables

An alternative, slightly more general, extension of the classical Wheeler-DeWitt equation to complex-time (kime) without separability assumptions.

The classical Wheeler-DeWitt equation in quantum gravity is formulated as $\hat{\mathcal{H}}\Psi[g_{ij}] = 0$, where $\hat{\mathcal{H}}$ is the Hamiltonian constraint operator and $\Psi[g_{ij}]$ is the wave functional of the universe, depending on the 3-metric $g_{ij}$.

In the ADM formalism of general relativity, the Hamiltonian constraint arises from the Einstein-Hilbert action \[S = \int d^4x \sqrt{-g}R = \int dt \int d^3x \sqrt{g}\left(K_{ij}K^{ij} - K^2 + {}^{(3)}R\right),\] where $K_{ij}$ is the extrinsic curvature, $K$ is its trace, and ${}^{(3)}R$ is the scalar curvature of the spatial slice.

The quantized Hamiltonian constraint becomes \[\hat{\mathcal{H}} = G_{ijkl}\frac{\delta^2}{\delta g_{ij}\delta g_{kl}} - \sqrt{g}{}^{(3)}R ,\] where $G_{ijkl} = \frac{1}{2\sqrt{g}}(g_{ik}g_{jl} + g_{il}g_{jk} - g_{ij}g_{kl})$ is the DeWitt supermetric.

In the kime framework, $\kappa = te^{i\theta}$, where the kime-magnitude $|\kappa| = t$ represents the usual time ordering, and the kime-phase $\theta$ is drawn from a time-dependent distribution $\Phi(\theta; t)$ supported on $[-\pi, \pi)$.

To extend to space-kime, we first modify the ADM metric to incorporate complex time \[ds^2 = -N^2d\kappa d\bar{\kappa} + g_{ij}(dx^i + N^i d\kappa)(dx^j + N^j d\kappa),\] where $\bar{\kappa}$ is the complex conjugate of $\kappa$, and $N$ and $N^i$ are the complex-valued lapse and shift functions.

The modified action becomes \[S = \int d\kappa d\bar{\kappa} d^3x \sqrt{g}\left(K_{ij}(\kappa)K^{ij}(\bar{\kappa}) - K(\kappa)K(\bar{\kappa}) + {}^{(3)}R\right),\] where the extrinsic curvature now depends on $\kappa$ \[K_{ij}(\kappa) = \frac{1}{2N}\left(\nabla_i N_j + \nabla_j N_i - \frac{\partial g_{ij}}{\partial \kappa}\right).\]

Since $\theta$ is randomly drawn from $\Phi(\theta; t)$, we must integrate over this distribution to obtain physical observables. The expectation value of any observable $\mathcal{O}$ becomes \[\langle\mathcal{O}\rangle = \int_{-\pi}^{\pi} \mathcal{O}(t,\theta) \Phi(\theta; t) d\theta\]

Without the separability assumption, the extended Wheeler-DeWitt equation in space-kime becomes \[\hat{\mathcal{H}}_{\kappa}\Psi[g_{ij}, \kappa] = 0,\] where $\hat{\mathcal{H}}_{\kappa}$ is the kime-dependent Hamiltonian constraint \[\hat{\mathcal{H}}_{\kappa} = G_{ijkl}\frac{\delta^2}{\delta g_{ij}\delta g_{kl}} - \sqrt{g}{}^{(3)}R + i\hat{\Theta}_{\kappa}, \] where $\hat{\Theta}_{\kappa}$ is an operator representing the kime-phase contribution.

To properly account for the probabilistic nature of $\theta$, we expand the wave functional in a kime-phase basis \[\Psi[g_{ij}, \kappa] = \int_{-\pi}^{\pi} \psi[g_{ij}, t, \theta] \Phi(\theta; t) d\theta .\]

Using the Hilbert space structure $L^2[-\pi, \pi]$ for the kime-phase, we define

Multiplication operator: $\hat{\Theta} \psi(\theta) = \theta \psi(\theta)$
Differentiation operator: $\hat{G}_{\beta} \psi(\theta) = -i\beta \frac{d}{d\theta}\psi(\theta)$

With these operators, we can formulate the space-kime Wheeler-DeWitt equation \[\left[G_{ijkl}\frac{\delta^2}{\delta g_{ij}\delta g_{kl}} - \sqrt{g}{}^{(3)}R + \frac{\partial^2}{\partial t^2} + i\beta\frac{\partial}{\partial \theta}\right]\psi[g_{ij}, t, \theta] = 0 ,\] subject to the periodic boundary condition $\psi[g_{ij}, t, -\pi] = \psi[g_{ij}, t, \pi]$.

The commutation relation $[\hat{\Theta}, \hat{G}_{\beta}] = i\beta I$ introduces a quantum uncertainty principle in the kime domain, analogous to the position-momentum uncertainty in standard quantum mechanics. This creates a richer structure that may help address the “problem of time” in quantum gravity.

To explore solutions in the semi-classical limit, we can use the WKB approximation \[\psi[g_{ij}, t, \theta] = A[g_{ij}, t, \theta]e^{iS[g_{ij}, t, \theta]/\hbar}.\]

This leads to the kime-modified Hamilton-Jacobi equation \[\left(\frac{\delta S}{\delta g_{ij}}\right)G^{ijkl}\left(\frac{\delta S}{\delta g_{kl}}\right) + \sqrt{g}{}^{(3)}R + \left(\frac{\partial S}{\partial t}\right)^2 + \beta\frac{\partial S}{\partial \theta} = 0 .\]

This space-kime formulation introduces a statistical ensemble of universes, each with different kime-phase $\theta$ drawn from the distribution $\Phi(\theta; t)$. Physical observables are computed by averaging over this ensemble. This approach may provide new insights into the emergence of time in quantum gravity, as the kime-phase can be interpreted as a quantum degree of freedom that complements the classical notion of time represented by $t$.

The non-commutative structure in the kime domain suggests a fundamental limitation on the simultaneous determination of time and its conjugate variable, analogous to the energy-time uncertainty in quantum mechanics. This space-kime extension of the Wheeler-DeWitt equation offers a novel framework for addressing longstanding problems in quantum gravity, particularly the “problem of time,” by introducing a richer structure that captures both the deterministic and probabilistic aspects of time evolution.

3 Potential Reformulation of WDE in Spacekime

We need to clarify the role of $\theta$ as a coordinate in $L^2[-\pi, \pi)$ and as a a random draw ($\theta \sim \Phi(\theta; t)$). Considering $\theta$ as a random draw at each $t$, how does $\Psi[g, \theta]$ evolves dynamically, as $\theta$ lacks a temporal trajectory (unlike a stochastic process).

Assuming $\psi(\theta) = \sqrt{\Phi(\theta; t)}$ (where $\Phi = |\psi|^2$) may be contradictory. In quantum mechanics (QM), $|\psi|^2$ is a probability density over the configuration space, but in spacekime, $\Phi(\theta; t)$ is a pre-defined distribution from data. This conflates the quantum state with a statistical distribution, risking inconsistency.

$\Phi(\theta; t)$ is time-dependent, but the Hilbert space $L^2[-\pi, \pi)$ and operators ($\hat{\Theta}, \hat{G}_\beta$) are defined statically. The $t$-dependence of $\Phi$ should influence the operators or the Hilbert space structure dynamically.

Itô calculus isn’t needed as $\theta$ isn’t a stochastic process evolving in $t$, yet, the operator $\hat{G}_\beta$ acts on $\psi(\theta)$, which is tied to $\Phi(\theta; t)$. If $\Phi$ is non-smooth (e.g., a mixture with sharp peaks), the derivative $\frac{d}{d\theta}$ may require a weak formulation.

What may be the physical meaning of the scaling parameter $\beta$ in $\hat{G}_\beta$? Is it a fundamental constant, or does it depend on the system (e.g., related to $\Phi$)? This ambiguity affects the interpretation of the commutator and uncertainty relation. Let’s try to derive the implied commutator uncertainty relation $\Delta \Theta \cdot \Delta G_\beta \geq \frac{\beta}{2}$ to justify the quantum nature of $\theta$.

The separability assumption $\Psi[g, \theta] = \Psi_{\text{space}}[g] \psi(\theta)$ may be overly restrictive. In quantum gravity, the spatial metric $g_{ij}$ and time (or kime) are typically coupled, especially in a framework aiming to address the problem of time. This assumption may oversimplify the dynamics. Also, if the potential $V(\theta)$ enforces the kime-phase distirbution, $\Phi(\theta; t)$, we may need a general form for specificity of the kime-phase Hamiltonian.

How to physically interpret the coupling parameter, e.g., related to kime-phase dynamics or measurement statistics, $\mu$ in $\hat{\mathcal{H}}_{\theta}$. The separation assumption leads to $\hat{\mathcal{H}}_{\text{space}} \Psi_{\text{space}} = E \Psi_{\text{space}}$ and $\hat{\mathcal{H}}_{\theta} \psi = -E \psi$. What is the physical role of $E$ as a shared eigenvalue? Does $E$ represent a physical energy, or is it a mathematical artifact?

In the von Mises mixture example, assuming $\psi = \sqrt{\Phi}$ may conflate the quantum wavefunction with a statistical distribution. In QM, $\psi$ evolves via the Schrödinger equation, but here, $\Phi(\theta; t)$ is externally specified, making $\psi$ static at each $t$. In this example, we assumed constant $\mu_1, \mu_2, k_1, k_2$ for simplification. Can the more general case be worked out ($\mu_i(t), k_i(t)$)? The time-dependence of $V(\theta; t)$ complicates the WDE, as the Hamiltonian becomes time-dependent.

As the potential $V(\theta; t)$ involves terms like $\frac{\nabla_\theta^2 \Phi}{\Phi}$, there may be numerical unstabilities when $\Phi \to 0$, e.g., for high $k_1, k_2$, $\Phi$ has sharp peaks, which ay require regularization. The energy eigenvalue $E$ is treated as a constant, but it may be dependent on $t$ via $\Phi(t)$.

Assessment: The von Mises example is a strong demonstration of the framework’s potential, but the assumption $\psi = \sqrt{\Phi}$ is problematic. The time-dependence of $V(\theta; t)$ needs better handling, and numerical stability should be addressed for practical implementation.

For the non-separability formulation, how to interpret the lapse $N$ and shift $N^i$ as complex-valued due to $d\kappa d\bar{\kappa}$. In standard ADM, $N$ and $N^i$ are real, ensuring a real metric. Complex values may lead to non-physical solutions. The term $i\hat{\Theta}_\kappa$ in $\hat{\mathcal{H}}_\kappa$ is defined as a kime-phase contribution. Does its form, $i\beta \frac{\partial}{\partial \theta}$, require verification of hermiticity?

The expansion $\Psi[g_{ij}, \kappa] = \int \psi[g_{ij}, t, \theta] \Phi(\theta; t) d\theta$ mixes quantum and classical probabilities. $\Phi(\theta; t)$ is a classical distribution, but $\psi[g_{ij}, t, \theta]$ is a quantum wavefunction, creating a conceptual mismatch. The final WDE includes $\frac{\partial^2}{\partial t^2}$, but the classical WDE is typically timeless (Hamiltonian constraint = 0). Having a $t$-derivative may contradict the core problem of time the WDE aims to solve, unless $t$ is reinterpreted as a parameter.

Perhaps clarify the role of $\theta$ and $\Phi(\theta; t)$, e.g.,

Separate Quantum and Classical Roles: Treat $\Phi(\theta; t)$ as a classical distribution for sampling $\theta$, but let $\psi(\theta)$ evolve quantum-mechanically via $\hat{\mathcal{H}}_\theta$. For example, solve a Schrödinger-like equation for $\psi$ $i \frac{\partial \psi(\theta, t)}{\partial t} = \hat{\mathcal{H}}_\theta \psi(\theta, t),$ where $\hat{\mathcal{H}}_\theta = -\frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta, t)$, and $V(\theta, t)$ is chosen to approximate $|\psi|^2 \to \Phi$ over time.
Dynamic Hilbert Space: Allow the Hilbert space to evolve with $t$, reflecting the time-dependence of $\Phi(\theta; t)$. For example, define a family of operators $\hat{\Theta}(t), \hat{G}_\beta(t)$ that adapt to $\Phi(t)$.

Can we formulate the weak derivatives for non-smooth phase $\Phi$ via a weak derivative for $\hat{G}_\beta$ \[ \langle \hat{G}_\beta \psi, \phi \rangle = -i\beta \int_{-\pi}^\pi \psi(\theta) \frac{d\phi}{d\theta} d\theta, \] ensuring robustness for non-smooth $\Phi$ (e.g., mixtures with sharp peaks).

Derive Uncertainty Relation: Explicitly derive the uncertainty relation $\Delta \Theta \cdot \Delta G_\beta \geq \frac{\beta}{2},$$ using the commutator $[\hat{\Theta}, \hat{G}_\beta] = i\beta I$, and validate it numerically for sample $\Phi$.
Generalize $V(\theta; t)$: Derive a general form for $V(\theta; t)$ without assuming $\psi = \sqrt{\Phi}$. For example, minimize a cost function $V(\theta; t) = \arg\min_V \int_{-\pi}^\pi \left| |\psi(\theta)|^2 - \Phi(\theta; t) \right|^2 d\theta,$ where $\psi$ evolves via $\hat{\mathcal{H}}_\theta$.

Is it possible to reformulate separability by coupling $g_{ij}$ and $\theta$? Relax the separability assumption by introducing a coupling term, e.g., $\hat{\mathcal{H}} = \hat{\mathcal{H}}_{\text{space}} + \hat{\mathcal{H}}_{\theta} + \hat{\mathcal{H}}_{\text{couple}},$ where $\hat{\mathcal{H}}_{\text{couple}} = \lambda \frac{\delta}{\delta g_{ij}} \frac{\partial}{\partial \theta}$, with $\lambda$ a coupling constant.

For a physical interpretation of $\beta, \mu$, can we relate $\beta, \mu$ to physical scales, e.g., $\beta \sim \hbar$ (quantum scale), $\mu \sim \frac{1}{G}$ (gravitational scale), and validate through dimensional analysis?

Consider eliminating $\frac{\partial^2}{\partial t^2}$ to preserve the timeless nature of the WDE. Instead, treat $t$ as a parameter in $\Phi(\theta; t)$, and solve \[\left[ G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \sqrt{g} {}^{(3)}R - \frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t) \right] \psi[g_{ij}, \theta; t] = 0,\] where $t$ is fixed, and solutions are computed for each $t$.

Complex Lapse/Shift: To ensure the metric remains real, can we redefine the ADM formalism, e.g., use $d\kappa d\bar{\kappa} = dt^2 - t^2 d\theta^2$, and adjust $N, N^i$ to be real or impose constraints?
Regarding hermiticity of $\hat{\Theta}_\kappa$, can we verify that $i\beta \frac{\partial}{\partial \theta}$ is Hermitian on $L^2[-\pi, \pi)$ with periodic boundary conditions, ensuring the Hamiltonian is physically meaningful?

Consider computing the observable effects of kime-phase, e.g., modifications to the cosmological constant $\Delta\Lambda$ affecting the Hubble parameter $H_0$. Compare with CMB data or cosmological observations. Can we estimate the parameters $\beta, \mu$ from physical constraints, e.g., by matching $\Delta\Lambda$ to observed dark energy density? Is it possible to test the multiverse interpretation by simulating universes for different $n$-modes and comparing their geometric properties (e.g., curvature, expansion rates)?

In the numerical simulations, let’s try to solve the WDE numerically, $\hat{\mathcal{H}}_\theta \psi = -E \psi$ for the von Mises mixture, using finite differences or spectral methods (e.g., Fourier basis $e^{in\theta}$). Perhaps regularize $V(\theta; t)$: Add a small constant to $\Phi$ (e.g., $\Phi + \epsilon$) to avoid division by zero in $V(\theta; t)$, ensuring numerical stability. Computing ensemble averages $\langle \mathcal{O} \rangle$ for sample observables (e.g., curvature $R$) over $\Phi(\theta; t)$, may validate the statistical ensemble interpretation.

Would these revised Hilbert Space and Operators formulations improve the framework?

Hilbert Space: Define the wave functional as $\psi[g_{ij}, \theta; t] \in L^2(\text{Superspace}) \otimes L^2[-\pi, \pi)$, where $t$ is a parameter reflecting the time-dependence of $\Phi(\theta; t)$.
Operators: $\hat{\Theta} \psi(\theta) = \theta \psi(\theta)$, and $\hat{G}_\beta = -i\beta \frac{\partial}{\partial \theta}$, with $[\hat{\Theta}, \hat{G}_\beta] = i\beta I$.
Uncertainty Relation: $\Delta \Theta \cdot \Delta G_\beta \geq \frac{\beta}{2},$ derived from the commutator.

The non-separable WDE Hamiltonian \[\hat{\mathcal{H}} = G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \sqrt{g} {}^{(3)}R - \frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t) + \lambda \frac{\delta}{\delta g_{ij}} \frac{\partial}{\partial \theta},\] where $\lambda$ introduces coupling between $g_{ij}$ and $\theta$.

And the WDE $\hat{\mathcal{H}} \psi[g_{ij}, \theta; t] = 0,$ with periodic boundary condition $\psi[g_{ij}, -\pi; t] = \psi[g_{ij}, \pi; t]$.

Kime-Phase Evolution: Solve a separate Schrödinger equation for $\psi(\theta, t)$ \[i \frac{\partial \psi(\theta, t)}{\partial t} = \left( -\frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta, t) \right) \psi(\theta, t),\] where $V(\theta, t)$ is chosen to minimize $\int | |\psi|^2 - \Phi(\theta; t) |^2 d\theta$.

In the von Mises mixture example consider the potential \[V(\theta; t) = V_0 + \frac{\beta^2}{4(\Phi + \epsilon)} \left[ w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2 \right] - \frac{\beta^2 \left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{8(\Phi + \epsilon)^2},\] with $\epsilon = 10^{-6}$ for numerical stability, the parameters $\beta = \hbar$, $\mu = \frac{1}{G}$, and estimate $V_0$ to match cosmological observations (e.g., $\Delta\Lambda \sim 10^{-52} \, \text{m}^{-2}$).

3.1 Simulation

We’ll demonstrate a numerical R code implementation showing solving the kime-phase WDE and computing the potential $V(\theta; t)$.

library(circular)

# Von Mises mixture
von_mises_mixture <- function(theta, w, mu1, mu2, k1, k2) {
  vm1 <- dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
  vm2 <- dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
  w * vm1 + (1 - w) * vm2
}

# Compute potential V(theta; t)
compute_potential <- function(theta, t, w, mu1_t, mu2_t, k1_t, k2_t, beta = 1, mu = 1, E = 0, eps = 1e-6) {
  # Time-dependent parameters
  mu1 <- mu1_t(t)
  mu2 <- mu2_t(t)
  k1 <- k1_t(t)
  k2 <- k2_t(t)
  
  # Compute Phi
  Phi <- von_mises_mixture(theta, w, mu1, mu2, k1, k2)
  
  # First derivative of vM
  grad_vM1 <- -k1 * sin(theta - mu1) * dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
  grad_vM2 <- -k2 * sin(theta - mu2) * dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
  grad_Phi <- w * grad_vM1 + (1 - w) * grad_vM2
  
  # Second derivative (numerical approximation)
  dtheta <- theta[2] - theta[1]
  grad2_vM1 <- diff(grad_vM1, differences = 1) / dtheta
  grad2_vM2 <- diff(grad_vM2, differences = 1) / dtheta
  grad2_vM1 <- c(grad2_vM1, grad2_vM1[length(grad2_vM1)])  # Extend for length
  grad2_vM2 <- c(grad2_vM2, grad2_vM2[length(grad2_vM2)])
  grad2_Phi <- w * grad2_vM1 + (1 - w) * grad2_vM2
  
  # Potential
  V <- -E + (beta^2 / (4 * mu * (Phi + eps))) * grad2_Phi - 
       (beta^2 / (8 * mu * (Phi + eps)^2)) * grad_Phi^2
  V
}

# Example parameters
w <- 0.6
mu1_t <- function(t) 0.5 * t
mu2_t <- function(t) 0.5 * t + pi
k1_t <- function(t) 5 + 2 * sin(2 * pi * t / 30)
k2_t <- function(t) 5 - 2 * sin(2 * pi * t / 30)

# Compute V at t = 10
theta <- seq(-pi, pi, length.out = 1000)
t <- 10
V <- compute_potential(theta, t, w, mu1_t, mu2_t, k1_t, k2_t)

# Plot
plot(theta, V, type = "l", xlab = "θ", ylab = "V(θ; t=10)", main = "Kime-Phase Potential")

Below we expand this simulation to compute the kime-phase potential $V(\theta; t)$ as a surface over $\theta \in [-\pi, \pi)$ and $t \in [0, 100]$. The graph visualizes the potentail as a 3D surface plot in SVG format using the plotly package.

# Load required libraries
library(circular)
library(plotly)
library(htmlwidgets)  # For saving as SVG

# Von Mises mixture density function
von_mises_mixture <- function(theta, w, mu1, mu2, k1, k2) {
  vm1 <- dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
  vm2 <- dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
  w * vm1 + (1 - w) * vm2
}

# Compute the kime-phase potential V(theta; t) across a grid of theta and t
compute_potential_surface <- function(theta, t, w, mu1_t, mu2_t, k1_t, k2_t, 
                                     beta = 1, mu = 1, E = 0, eps = 1e-6) {
  # Initialize matrix to store V(theta; t)
  V <- matrix(NA, nrow = length(t), ncol = length(theta))
  
  # Loop over time points
  for (i in seq_along(t)) {
    # Time-dependent parameters
    mu1 <- mu1_t(t[i])
    mu2 <- mu2_t(t[i])
    k1 <- k1_t(t[i])
    k2 <- k2_t(t[i])
    
    # Compute Phi
    Phi <- von_mises_mixture(theta, w, mu1, mu2, k1, k2)
    
    # First derivative of vM (numerical approximation)
    dtheta <- theta[2] - theta[1]
    grad_vM1 <- -k1 * sin(theta - mu1) * dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
    grad_vM2 <- -k2 * sin(theta - mu2) * dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
    grad_Phi <- w * grad_vM1 + (1 - w) * grad_vM2
    
    # Second derivative (numerical approximation)
    grad2_vM1 <- diff(grad_vM1, differences = 1) / dtheta
    grad2_vM2 <- diff(grad_vM2, differences = 1) / dtheta
    grad2_vM1 <- c(grad2_vM1, grad2_vM1[length(grad2_vM1)])  # Extend for length
    grad2_vM2 <- c(grad2_vM2, grad2_vM2[length(grad2_vM2)])
    grad2_Phi <- w * grad2_vM1 + (1 - w) * grad2_vM2
    
    # Compute potential V at time t[i]
    V[i, ] <- -E + (beta^2 / (4 * mu * (Phi + eps))) * grad2_Phi - 
              (beta^2 / (8 * mu * (Phi + eps)^2)) * grad_Phi^2
  }
  
  return(V)
}

# Plot the kime-phase potential as a 3D surface using plotly
plot_kime_surface <- function(theta, t, V, title = "Kime-Phase Potential Surface") {
  # Create a plotly surface plot
  fig <- plot_ly(
    x = theta,
    y = t,
    z = V,
    type = "surface",
    colorscale = "Viridis",
    colorbar = list(title = "V(θ; t)")
  ) %>%
    layout(
      title = title,
      scene = list(
        xaxis = list(title = "θ (radians)", range = c(-pi, pi)),
        yaxis = list(title = "t (seconds)", range = c(0, 100)),
        zaxis = list(title = "V(θ; t)"),
        camera = list(eye = list(x = 1.5, y = 1.5, z = 0.8))
      )
    )
  
  return(fig)
}

# Main function to compute and plot the kime-phase potential surface
kime_potential_analysis <- function() {
  # Define parameters
  w <- 0.6
  mu1_t <- function(t) 0.5 * t %% (2 * pi) - pi  # Linearly increasing, wrapped to [-pi, pi)
  mu2_t <- function(t) (0.5 * t + pi) %% (2 * pi) - pi  # Opposite phase
  k1_t <- function(t) 5 + 2 * sin(2 * pi * t / 30)  # Oscillating concentration
  k2_t <- function(t) 5 - 2 * sin(2 * pi * t / 30)
  
  # Define grids for theta and t
  theta <- seq(-pi, pi, length.out = 200)  # Reduced resolution for faster computation
  t <- seq(0, 100, length.out = 100)
  
  # Compute the potential surface
  V <- compute_potential_surface(theta, t, w, mu1_t, mu2_t, k1_t, k2_t)
  
  # Plot the surface
  fig <- plot_kime_surface(theta, t, V)
  
  # Save as SVG
  # htmlwidgets::saveWidget(fig, "kime_potential_surface.html", selfcontained = TRUE)
  # Note: Direct SVG export requires additional tools like `webshot` or manual conversion
  # For now, we save as HTML and suggest manual conversion to SVG using a browser
  
  return(list(V = V, fig = fig))
}

# Run the analysis
results <- kime_potential_analysis()

# Display the plot (in RStudio or a browser)
results$fig

Notes:

The von_mises_mixture() function computes the kime-phase distribution $\Phi(\theta; t) = w vM(\theta; \mu_1(t), k_1(t)) + (1-w) vM(\theta; \mu_2(t), k_2(t))$.
The compute_potential_surface() function calculates $V(\theta; t)$ over a grid of $\theta$ and $t$. A small $\epsilon = 10^{-6}$ is added to $\Phi$ to avoid division by zero. The simulation uses numerical derivatives for $\nabla_\theta \Phi$ and $\nabla_\theta^2 \Phi$, using \[ V(\theta; t) = -E + \frac{\beta^2}{4\mu \Phi} \left[ w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2 \right] - \frac{\beta^2 \left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{8\mu \Phi^2}. \]
The plot_kime_surface() function creates a 3D surface plot using plot_ly. The x-axis is $\theta \in [-\pi, \pi)$, the y-axis is $t \in [0, 100]$, and the z-axis is $V(\theta; t)$. The colorscale = "Viridis" provides a clear color gradient, and the camera angle is adjusted for better visualization.
The time-dependent parameters are defined as $\mu_1(t) = 0.5 t \mod 2\pi - \pi$, $\mu_2(t) = (0.5 t + \pi) \mod 2\pi - \pi$: Linearly increasing means, wrapped to $[-\pi, \pi)$ and $k_1(t) = 5 + 2 \sin(2\pi t / 30)$, $k_2(t) = 5 - 2 \sin(2\pi t / 30)$: Oscillating concentrations with a period of 30 seconds. These choices create a dynamic double-well potential that evolves over time.

Time Complexity, Inferential uncertainty and Spacekime Analytics

Wheeler-DeWitt Equation in Spacekime

SOCR Team

03/29/2025