What have electrons, black holes and Clifford Algebra ever done for us?

  Consider a spherical Schwarzschild horizon of radius $R$. Its energy, temperature, and entropy are $$E=\frac{c^{4}R}{2G},\qquad k_BT_H=\frac{\hbar c}{4\pi R},\qquad S=\frac{k_BA}{4L_P^2},$$ where $A=4\pi R^2$ and $L_P^2=\hbar G/c^3$. 1. Quantized Horizon The horizon's thermal period defines an angular frequency $$\omega_H=\frac{2\pi k_BT_H}{\hbar}=\frac{c}{2R}.$$ Treat the Euclidean horizon cycle as a periodic quantum motion and define its action by $dI=dE/\omega_H$. Since $dE=c^4dR/2G$, we obtain $dI=c^3RdR/G$, and therefore $$I=\frac{c^3R^2}{2G} =\frac{\hbar R^2}{2L_P^2} =\frac{\hbar A}{8\pi L_P^2}.$$ Equivalently, using $dE=T_HdS$, $$I=\frac{\hbar S}{2\pi k_B}.$$ Quantizing this periodic motion gives $$I_n=\left(n+\frac12\right)\hbar, \qquad n=0,1,2,\ldots$$ where the half-unit is the ground-state contribution associated with the regular centre of the Euclidean horizon geometry. Hence $$A_n=4\pi(2n+1)L_P^2, \qquad R_n=\sqrt{2n+1},L_P.$$ Thus the first nonzero hor...

  A positive cosmological constant sets the future cosmic event-horizon radius of an expanding FLRW universe: $$l_\Lambda=\sqrt{\frac{3}{\Lambda}}.$$ Associate two lengths with a positive mass $m$: $$\bar\lambda_C=\frac{\hbar}{mc} \qquad r_g=\frac{2Gm}{c^2}$$ where $\bar\lambda_C$ is the reduced Compton wavelength and $r_g$ is the gravitational radius. Requiring both lengths not to exceed the cosmic horizon, $$\bar\lambda_C\le l_\Lambda \qquad r_g\le l_\Lambda,$$ gives $$\boxed{ m_{\min}= m_s=\frac{\hbar}{cl_\Lambda} =\frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}} }$$ $m_s$ is the minimum mass first pointed out by Wesson , 2003. This was also adopted by  Bohmer and Harko , 2005 to derive a  minimum density for static bodies  using the cosmological Buchdahl inequality. and $$\boxed{ m_{\max}=\frac{c^2l_\Lambda}{2G} =\frac{c^2}{2G}\sqrt{\frac{3}{\Lambda}} \equiv m_{\rm CEH}. }$$ Wesson also conjectured a maximum mass, but was off by a factor of 2, as well ...

 $\Lambda$CDM and Holographic Dark Energy   Holographic dark energy (HDE) starts from a simple thought: the energy inside a region of size $L$ shouldn't exceed that of a black hole of the same size. Saturate that bound, up to a dimensionless number $c$ (not the speed of light), and $$\rho_{\rm DE}=3M_p^2\frac{c^2}{L^2}, \qquad\text{i.e.}\qquad c=HL\sqrt{\Omega_{\rm DE}}.$$ Units $\hbar=k=1$, $M_p^2=1/8\pi G$, prime $=d/d\ln a$, flat universe, $\Omega_X=\rho_X/3M_p^2H^2$. Li's Model In Li's 2004 model, $L$ is the future event horizon , the farthest a light ray sent today will ever get: $$L_e=a\int_t^\infty\frac{dt'}{a(t')}, \qquad \dot L_e=HL_e-1.$$ Li took $c$ constant, giving $w_{\rm DE}=-\frac13\left(1+\frac{2\sqrt{\Omega_{\rm DE}}}{c}\right)$. And this is where the problems begin. With $c=1$ and $\Omega_{\rm DE,0}=0.6889$, $w_0=-0.887$, about $5\sigma$ from Planck 2018 $w_0=-1.03\pm0.03$. Does not match observations! And any constant $c<1$ gives $w<...

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\...

  In the static patch of de Sitter spacetime , the cosmological horizon behaves thermodynamically in ways closely analogous to a physical interface. One can assign it entropy, temperature, and even an effective surface tension . Remarkably, the familiar Young–Laplace pressure relation from surface physics appears naturally at the horizon.      In the static patch of de Sitter spacetime, the cosmological horizon admits a useful surface-thermodynamic description. It has an effective surface tension and pressure satisfying a Young–Laplace-like relation. Use $G=c=\hbar=k_B=1$. The metric is $$ds^2=-f(r),dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2, \qquad f(r)=1-\frac{r^2}{L^2}, \qquad L=\sqrt{\frac3\Lambda}.$$ The horizon is at $r=L$. 1. Horizon Data The area, entropy, and temperature are $$A=4\pi L^2, \qquad S=\frac A4=\pi L^2, \qquad T=\frac{1}{2\pi L}.$$ The Misner–Sharp energy is $M(r)=\frac r2(1-f)=\frac{r^3}{2L^2}$, so at the horizon $M(L)=L/2$. This equals both th...

This not a proof of the Riemann Hypothesis (RH) .  Its purpose is to isolate a precise mathematical problem whose solution would imply RH. Let $\xi(s)$ be the completed Riemann $\xi$-function, and let $Q_W$ denote the Weil quadratic form. Weil's criterion states, for the standard admissible test-function class, $$\mathrm{RH}\iff Q_W(f)\ge0\quad\text{for all }f.$$ The program is to obtain this positivity from horizon modular/reflection positivity.   1. Horizon Modular Flow is Logarithmic Translation For a non-extremal horizon, the local modular flow is Rindler/boost flow. On a horizon half-line $x>0$, $$(U_u\psi)(x)=e^{u/2}\psi(e^u x).$$ The half-density factor makes $U_u$ unitary on $L^2(\mathbb R_+,dx)$. Its generator is $$K_{\rm sc} =-i\left(x\frac{d}{dx}+\frac12\right).$$ Introduce $q=\ln x$ and $\phi(q)=e^{q/2}\psi(e^q)$. Then $$U_u\phi(q)=\phi(q+u), \qquad K_0=-i\frac{d}{dq}.$$ Thus the geometric horizon Hamiltonian is simply translation in logarithmic coo...