Skip to main content

Zeta Zeros as Logarithmic Spiral Waves

Zeta Zeros as Logarithmic Spiral Waves Matthew R. Watkins gives a striking geometric interpretation of the nontrivial zeros of the Riemann zeta function as generating "spiral wave" contributions whose superposition encodes fluctuations in Chebyshev's prime-counting function $\psi(x)$. Watkins observes that $x^\rho$ maps the positive real axis onto a logarithmic spiral, while conjugate zeros combine to produce real logarithmically rescaled waveforms. See: Matthew R. Watkins, " Encoding the Zeta Zeros "    The algebra below is elementary complex exponentiation; Watkins's contribution is the geometric spiral-wave interpretation. Let $\rho=\beta+i\gamma$ and $t=\ln x$. Then $$x^\rho=e^{\rho\ln x} = e^{\beta t}e^{i\gamma t} = e^{\beta t}\left(\cos(\gamma t)+i\sin(\gamma t)\right).$$ So $\beta$ is the amplitude growth rate in logarithmic time, and $\gamma$ is the angular frequency in logarithmic time. The frequency in cycles per unit $t$ is $f=|\gamma|/(2\...
Post a Comment

Zitter at the Edge of Spacetime

 In a previous post we introduced the idea that our current Universe has boundary conditions. 
 
We also showed a diagram similar to Figure 1 below. Except here, we are once again thinking about the future dS state.

Figure 1. For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$

 
Figure 2. The maximum entropy of the Universe (credit: Lineweaver).  

Now, a comoving volume of the Universe, when considered together with its associated cosmic event horizon, forms a thermodynamically closed system obeying the generalised second law, $$dS_{\text{bulk}} + dS_{\text{horizon}} \ge 0.$$ The maximum entropy of a closed system, in this case  (Figure 2) with $L=2 \pi l_{\Lambda}$, the circumference of a circle with radius  $l_{\Lambda}$,  is obtained when total $E_{H}$ is worked, i.e. degraded into the smallest bits possible. All energy is converted into minimal energy dark photons with wavelengths as large as the system $L$. Because the relevant thermodynamics and quantum degrees of freedom are governed by the Holographic Principle, these physical modes do not propagate through the 3D bulk; they are strictly confined to the 2D $S^2$ horizon surface. Consequently, the maximal fundamental wavelength is bounded not by the bulk diameter, but by the topological great circle of the boundary: $\lambda_{max} = 2\pi l_\Lambda$. From the Compton wavelength relation, the minimum quanta of energy $E_{s}$ of this system is then: \begin{equation}
\notag
E_{s}=\frac{hc}{\lambda_{max}}=\frac{\hslash c}{l_{\Lambda}}=m_{s}c^2
\end{equation}

This defines $m_s$ (discussed in a previous post) as the irreducible mass gap of the holographic surface. Within this framework, $m_s = \hbar / (rc)$ acts as a universal spectral mass function linked by UV/IR duality ($r \leftrightarrow 2L_P l_\Lambda / r$). It resolves to $M_P / 2$ at the UV boundary ($r = 2L_P$) and $\hbar / (cl_\Lambda)$ at the IR boundary ($r = l_\Lambda$).  In 2014, Barrow and Gibbons confirmed this lower-bound analysis, and also pointed out that the classical upper bound (which we term $m_0$) for any mass in a Universe with a positive cosmological constant is (here $m_{CEH}$ is the mass of the cosmic event horizon of our Figure 1 system):   \begin{equation}m_{0} = \frac{c^{2}}{G}\sqrt{\frac{3}{\Lambda }} =\frac{c^2 l_{\Lambda}}{G}=2 \ m_{CEH} 
\end{equation} However, as we know from cosmological observations that $m_{CEH}$ is the observed (effective) mass of the CEH, I posit that $m_{0}$ is unphysical (aka unobservable), being an upper bound of dS spacetime, and therefore $m_{0}$ is a "bare" mass (and I'm sure this could be proved in QFT).  

Now, quantum field theory can be formulated in terms of harmonic oscillators. So, if we consider our Figure 1 system as a quantum harmonic oscillator,  with $E_s = m_s c^2$ as the ground state (zero-point) energy and  $\omega_{s}$ being the ground state angular frequency.    
\begin{equation}
\notag
E_{s}=\hslash \frac{\omega_{s}}{2}
\end{equation}  Now, the de Broglie relation is then:
\begin{equation}
\notag
E_{s}=\hslash \omega_{B}
\end{equation} Of course, these are not the same $\omega$ unless we compare internal vs. external clock rates (exactly what happens in zitterbwegung). Because the fundamental modes are confined to the $S^2$ boundary, their evolution is not merely a scalar translation but is generated by spatial rotors (bivectors) in geometric algebra ($C\ell_{3,0}$). 

Wait, I hear some of you saying, $H_0$ is usually presented as (km/s/Mpa) rather than ($s^{-1}$),  and you are right, but that simply due to cosmologists confusing things, using more convenient units.  

Anyhow, if we model the vacuum state at this boundary not as a naive 3D bosonic oscillator, but as a field of spatial rotors, the zero-point ground state energy remains $E_s$. As  $\omega_{s}=2\omega_{B}$ we might consider $\omega_{s}$ as the zitter frequency of vacuum. Things get even more interesting when we realise that  $\omega_{B} = c/l_{\Lambda} =H$ implying that the Hubble Constant can be considered as a `de Broglie'  (or effective) frequency of vacuum; or equivalently, as per Chappell et al, 2016, Time as a Geometric Property of Space. That is, the macroscopic value $H = c / l_\Lambda$ is the irreversible scalar component ($t$) of the universe's evolution, while the vacuum frequency $\omega_s = 2H$ is the bivector component ($jn$). Thus, the universe's expansion is not just scalar; it possesses an intrinsic quaternionic rotation at the boundary, reconciling the bosonic geometry of macroscopic expansion with the fermionic Zitterbewegung of the quantum vacuum. This also all implies the horizon vacuum is a something like a highly degenerate spin liquid, a macroscopic condensate of phase-synchronised spatial rotors bounded at the IR cutoff.   

  

Comments

Post a Comment

Popular posts from this blog

Blurring the horizon - the quantum width of the cosmic event horizon

A  paper by Zurek applied a random walk argument to a black hole horizon. Credit, Zurek, 2021 Zurek  ( Snowmass 2021 White Paper: Observational Signatures of Quantum Gravity )  called this a blurring of the horizon — a fuzzy, or uncertain horizon — and went through derivations supporting the idea that this length scale is the quantum uncertainty in the position of the black hole horizon: a dynamic quantum width of an event horizon. This is a concept which fundamentally applies to the Universe's own Cosmic Event Horizon (CEH). The Bekenstein-Hawking entropy gives the number of quantum degrees of freedom that can fluctuate. Below, we step out our own cosmic de Sitter derivation of the random walk argument. To do this, let $l_{\Lambda}$ represent the generalised de Sitter horizon scale. Due to the holographic UV/IR correspondence, this scale manifests dually: at the fundamental microscopic limit as $l_{UV} = 2L_p$ (the gravitational/casual limit, aka the Schwarzschild radi...
Post a Comment
Read more

The Cosmic Strange Metal

      Strange metals, quantum spin liquids, and SYK-like systems share a striking transport pattern: no quasiparticles, strong collective dynamics, Planckian relaxation, near-minimal viscosity, and maximal chaos. Their characteristic data are $$\frac{\eta}{s}=\frac{\hbar}{4\pi k_B}, \qquad \lambda_L=\frac{2\pi k_BT}{\hbar}, \qquad \tau_P=\frac{\hbar}{k_BT}.$$ The claim is not that the three-dimensional de Sitter bulk is literally a strange metal. The sharper claim is: $$\boxed{\text{The de Sitter stretched horizon belongs to the same transport universality class as a Planckian strange metal.}}$$ The correspondence applies to the horizon membrane, not to bulk spacetime. Membrane Paradigm and the KSS Value In the membrane paradigm, an event horizon behaves for exterior observers as a stretched viscous membrane with transport coefficients fixed by Einstein gravity. This does not require an assumed AdS/CFT dual. For de Sitter, $$\ell_\Lambda=\frac{c}{H}, \qquad T_{dS...
Post a Comment
Read more

Our cosmic event horizon on a string

A Cosmic Stringy Adventure ! As we previously discussed, our spacetime characterised by a positive cosmological constant $\Lambda$. The natural bounds are then a minimal  ultraviolet (UV) length $l_{UV} = 2L_P$ and an infrared (IR) cosmological horizon $l_{\Lambda}$.  This dual-boundary spacetime enforces a fundamental Compton–gravitational duality . Every geometric scale $r$ carries two natural mass definitions: $$m_C(r) = \frac{\hbar}{rc}, \qquad m_G(r) = \frac{c^2}{4G}\ r$$ The product of these masses, $m_C \ m_G = M_P^2/4$, is scale-independent. They intersect exclusively at the UV boundary $r = l_{UV}$, defining a maximal local force in GR: $F_{max} = c^4 / 4G$. At the opposite extreme, the Compton mass evaluated at the IR horizon yields the fundamental  spectral gap  (not a particle) of the universe: $m_s = \hbar / (l_{\Lambda} c)$.  In this post, to explore how energy propagates through this dual-scale geometry, we model the mass gap $m_s$ as a null-ener...
Post a Comment
Read more