QFunity – Emergence of Causality, Light, and Information

Emergence of Causality, Light, and Information in QFunity

A unified framework for the origin of time, causality, the speed of light, and the nature of information

From Pre-Temporal State to Causal Spacetime

The Pre-Temporal State (EPT): Absence of Time and Causality

The pre-temporal state (EPT) is an acausal, timeless state governed by pure rotation, described by the pre-temporal Hamiltonian:

\[ \mathcal{H}_{\text{pre}} = \int_{\text{EPT}} \left[ \hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon \right] \Psi_{\text{universe}} \, d^3x = \frac{\hbar}{\epsilon} \cdot \mathcal{R}_{\text{total}} \]

Interpretation:

  • \( \left[ \hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon \right] = 0 \): The zero commutator indicates no causal order, with the torsion operator \( \hat{\mathbb{B}}_\epsilon \) and fractal potential operator \( \hat{\mathbb{V}}_\epsilon \) acting without sequence. See Rotation Details.
  • \( \mathcal{R}_{\text{total}} \): The total rotational energy, defined as: \[ \mathcal{R}_{\text{total}} \equiv \int_{\text{EPT}} \left( \| \hat{\mathbb{T}} \|^2 + \| \hat{\mathbb{\Omega}} \|^2 \right) d\mu \] where \( \hat{\mathbb{T}} \sim \nabla \hat{\mathbb{B}}_\epsilon \) is the torsion tensor (\( [\hat{\mathbb{T}}] = L^{-1} \)), \( \hat{\mathbb{\Omega}} \sim \nabla \hat{\mathbb{V}}_\epsilon \) is the vorticity tensor, and \( d\mu \) is the fractal measure with dimension: \[ D_f = 2 + \frac{\log(\epsilon/\epsilon_0)}{\log N} \] Here, \( N \) is the fractal branching factor, and \( \epsilon_0 \sim \ell_P \) is the Planck length reference scale. The fractal volume is: \[ V = \int_{\text{EPT}} d\mu \sim R^{D_f} \] The dimension of \( \mathcal{R}_{\text{total}} \) is \( [L^{D_f – 4}] \), adjusted by fundamental constants to yield energy (\( M L^2 T^{-2} \)).
  • \( \mathcal{H}_{\text{pre}} \): Analogous to the Wheeler-DeWitt equation, it constrains the EPT state without temporal dynamics, with \( \frac{\hbar}{\epsilon} \cdot \mathcal{R}_{\text{total}} \) as the energy eigenvalue.

Symmetry Breaking: Birth of Time and Causality

The transition from the EPT to spacetime occurs via a symmetry breaking, where the commutator becomes non-zero, as described by the QFunity master equation:

\[ \lim_{\epsilon \to 0^+} \left[ \hat{\mathbb{B}}_\epsilon \hat{\mathbb{V}}_\epsilon – \hat{\mathbb{V}}_\epsilon \hat{\mathbb{B}}_\epsilon^2 \right] \Psi = \Lambda \cdot \frac{\Psi}{\|\Psi\|^2 + \epsilon^2} \]

Mechanism:

  • Critical Threshold: Symmetry breaking occurs when the rotational energy density exceeds the Planck density: \[ \mathcal{E}_{\text{EPT}} = \frac{\mathcal{R}_{\text{total}}}{V} > \mathcal{E}_{\text{crit}} \sim \frac{c^7}{\hbar G^2} \]
  • Non-Commutation: The non-zero commutator \( \left[ \hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon \right] \neq 0 \) establishes a fundamental causal order. See Rotation Details.
  • Emergence of Time: Time \( t \) emerges as a parameter in the curvature equation: \[ \eta(t) = \frac{\mathcal{E}_{\text{EPT}}(t)}{\hbar} \cdot \int_{-\infty}^{t} \omega(\tau) e^{-i \frac{\mathcal{E}_{\text{Micro}}(\tau)}{\hbar} (t – \tau)} \, d\tau \] Here, \( \tau \) is a pseudo-temporal parameter representing fractal complexity levels, with the integral filtering coherent rotational states from the EPT.

Emergence of the Speed of Light \( c \)

The speed of light \( c \) emerges as the propagation speed of perturbations in the emergent spacetime, derived by linearizing the master equation around the critical point:

\[ \left( \hat{\mathbb{O}}_1 \partial_t^2 – \hat{\mathbb{O}}_2 \nabla^2 \right) \delta\Psi = 0 \] \[ \hat{\mathbb{O}}_1 \equiv \Lambda \cdot \frac{1}{\hbar \omega_{\text{eff}}}, \quad \hat{\mathbb{O}}_2 \equiv \frac{\ell_P^2}{\hbar} \] \[ c^2 = \frac{\text{v.p.}(\hat{\mathbb{O}}_2)}{\text{v.p.}(\hat{\mathbb{O}}_1)} = \frac{\ell_P^2 / \hbar}{\Lambda / (\hbar \omega_{\text{eff}})} = \frac{\ell_P^2 \omega_{\text{eff}}}{\Lambda} \] \[ c = \frac{\ell_P \sqrt{\omega_{\text{eff}}}}{\sqrt{\Lambda}} \] \[ \omega_{\text{eff}} \sim \sqrt{\frac{\mathcal{E}_{\text{crit}}}{\hbar}} \sim \frac{c}{\ell_P} \]

Derivation:

  • Linearization: Perturbing \( \Psi = \Psi_0 + \delta\Psi \) yields a wave equation, with \( \hat{\mathbb{O}}_1 \) (dimension \( T^2 / (M L^2) \)) linked to the energy scale of the symmetry breaking and \( \hat{\mathbb{O}}_2 \) (dimension \( T / M \)) to the emergent spatial structure.
  • Speed \( c \): If \( \Lambda \sim 1 \), then \( c \approx \ell_P \sqrt{\omega_{\text{eff}}} \), recovering the known value when \( \omega_{\text{eff}} \sim c / \ell_P \).
  • Metric: The emergent metric defines the causal structure: \[ g_{\mu\nu}(\epsilon) = g_{\mu\nu}^{GR} + \frac{\ell_P^2}{\epsilon^2} g_{\mu\nu}^{\text{LQG}} + \alpha’ \cdot g_{\mu\nu}^{\text{strings}} \] The light cone slope, set by \( c \), governs causal information propagation. See Observer Details.

Scale-Dependent Causality

The observer’s scale \( \epsilon \) modulates the perception of causality and \( c \):

Cosmic Horizon:

\[ R_{\text{horizon}} = \frac{c}{H_0} \cdot \sqrt{\frac{\epsilon}{\epsilon_{\text{cosmic}}}} \] \[ \epsilon_{\text{cosmic}} \sim \frac{c}{H_0} \]

Superluminal Correlations:

  • Human Observer (\( \epsilon \sim 1 \, \text{m} \)): Horizon \( R_{\text{horizon}} \approx \frac{c}{H_0} \approx 14 \, \text{Gly} \).
  • Galactic Observer (\( \epsilon \sim 10^{20} \, \text{m} \)): Smaller horizon leads to apparent superluminal correlations for events within the human-scale horizon.

Quantum Collapse:

\[ P(\psi \to \phi) = \frac{|\langle \phi | \psi \rangle|^2}{|\langle \psi | \psi \rangle|^2 + \epsilon_O^2} \]

Planck-Scale Entanglement:

  • Macroscopic Scale (\( \epsilon_O \sim 10^{-3} \, \text{m} \)): Clear collapse, enforcing strict causality.
  • Planck Scale (\( \epsilon_O \sim \ell_P \)): No collapse, preserving entanglement and reflecting the acausal EPT. See Zero Details.

Fate of Light Information at Obstacles

When a photon encounters an obstacle (e.g., a planet), the fate of its information depends on the interaction:

Interaction Types:

  • Absorption: The photon’s energy is converted into heat or chemical energy, localizing and thermalizing its information (direction, frequency, polarization). Most of the original information is lost as a coherent signal.
  • Scattering: The photon’s direction changes (e.g., Rayleigh scattering), preserving but altering information. Frequency and polarization may be modified, leading to corrupted information.
  • Reflection: Information is best preserved, with direction reversed but frequency, phase, and polarization largely maintained (e.g., by a mirror).

Interactions with massive matter involve energy and momentum exchange, degrading the purity of the photon’s information. Light is an excellent information carrier in vacuum but poor through massive matter.

Neutrinos: Transparent Carriers of Information

Neutrinos, interacting only via the weak force and gravity, have an extremely low interaction cross-section:

Properties:

  • Low-Energy Neutrinos: Can traverse the Earth or a light-year of lead with near-zero interaction probability.
  • Information Preservation: Neutrino information (flavor, energy, momentum) remains nearly unaltered, making them ideal messengers from dense environments (e.g., stellar cores, supernovae, early universe).

Massless Particles and EPT: Guardians of Causal Information

Only massless particles (photons, gluons, potentially gravitons) travel at \( c \), defining the light cones:

Key Points:

  • Causal Information: Particles moving at \( c \) are the primary carriers of causally connected information, as they define the light cone structure.
  • EPT Connection: Massless particles are the first excitations of the EPT post-symmetry breaking, with \( c = \ell_P \sqrt{\omega_{\text{eff}}} / \sqrt{\Lambda} \) inherited from the EPT dynamics. They are manifestations of the emergent causal structure.
  • Neutrinos: With near-zero mass, neutrinos approximate this behavior, carrying near-pristine information from the early universe.

Innate vs. Acquired Information

QFunity distinguishes two types of information:

Innate Information (EPT):

\[ \lim_{\epsilon \to 0^+} \left[ \hat{\mathbb{B}}_\epsilon \hat{\mathbb{V}}_\epsilon – \hat{\mathbb{V}}_\epsilon \hat{\mathbb{B}}_\epsilon^2 \right] \Psi_0 = \Lambda \cdot \frac{\Psi_0}{\|\Psi_0\|^2} \]

Innate Information:

  • Origin: Encoded in the initial state \( \Psi_0 \) at symmetry breaking, determining fundamental constants (e.g., \( c \), \( \alpha \)) and primordial fluctuations (e.g., CMB anisotropies).
  • Carriers: Massless particles (photons, gravitons) and the spacetime metric itself.
  • Example: CMB photon temperature variations \( \delta T/T \) reflect the primordial state \( \Psi_0 \).

Acquired Information:

\[ i\hbar \frac{\partial \Psi(t)}{\partial t} = \left( \hat{\mathbb{V}}_\epsilon + \hat{\mathbb{B}}_\epsilon^2 \right) \Psi(t) \]

Acquired Information:

  • Origin: Generated by interactions post-symmetry breaking, encoded in the evolving state \( \Psi(t) \).
  • Carriers: Massive matter (electrons, atoms), gravitational waves from dynamic events, and « polluted » light (e.g., stellar spectral lines).
  • Example: Stellar absorption lines reflect the chemical composition and dynamics of stars.
Characteristic Innate Information (EPT) Acquired Information (History)
Origin Initial state \( \Psi_0 \) at symmetry breaking Evolution of \( \Psi(t) \) via interactions
Equation Master Equation Generalized Schrödinger Equation
Carriers Massless particles, spacetime Massive matter, gravitational waves, polluted light
Nature Fundamental constants, primordial spectrum Stellar composition, cosmic structures, DNA
Alteration Preserved if uninteracted Subject to degradation, thermalization
Example CMB anisotropies Stellar spectral lines

Testable Predictions

QFunity’s framework yields observable phenomena:

  • Superluminal Correlations: Large-scale alignments in the CMB or galaxy distributions may appear acausal at galactic scales (\( \epsilon \sim 10^{20} \, \text{m} \)), testable via CMB anomaly studies or large-scale structure surveys.
  • Planck-Scale Entanglement: Non-locality at \( \epsilon \sim \ell_P \), reflecting the acausal EPT, testable in high-energy physics (e.g., LHC anomalies with missing transverse energy) or Casimir effect experiments. See Micro-EPT Details.
  • Neutrino Information: Neutrinos from supernovae or the early universe carry near-pristine information, testable via neutrino observatories like IceCube.

Implications for Universes with Different \( c \)

Different EPT parameters yield varying \( c’ \):

\[ c’ = \frac{\ell_P \sqrt{\omega_{\text{eff}}’}}{\sqrt{\Lambda}} \]

Interpretation:

  • Higher \( \omega_{\text{eff}}’ \): More energetic symmetry breaking yields \( c’ > c \).
  • Lower \( \omega_{\text{eff}}’ \): Less energetic breaking yields \( c’ < c \).
  • Consequences: Altered fundamental constants (e.g., fine-structure constant) lead to different physical laws, testable in multi-messenger astronomy.