Harmonic Entanglement TheoryInfrared suppression
\[ {E_\phi=m_P\,\Theta_\phi\,\Pi_\phi^{p}e^{-1/\xi^\ast}}. \]
| Factor | Meaning |
|---|---|
| Eφ | the energy of the infrared mode φ |
| mP | the Planck mass, 1.2209 × 1019 GeV |
| Θφ | the trace of the identity over the charge the object carries, over the channels it propagates through |
| Πφ | the spectral weight Gφ/TrM−1 of the mode against the whole trace of its operator |
| p | the number of field factors, p=1 for a scalar and p=2 for a vector or a fermion mass term |
| e−1/ξ | the descent suppression ℓP/ℓc, the Planck length over the infrared correlation length of the prism vacuum |
Table 1. The factors of the relation above.
Dividing the energy of the infrared mode by its weight leaves the infrared scale, the same for every mode,
\[ m_{\rm IR}=\frac{E_\phi}{\Theta_\phi\Pi_\phi^{p}}=m_P e^{-1/\xi^\ast}=1.2209\times10^{19}\times2.31952\times10^{-16}\ {\rm GeV}=2831.9\ {\rm GeV}. \]
Here \(m_P\) is the saturation energy of the bond at the Bekenstein–Hawking bound, \(V_{\max}=\frac12\kappa_P\mathcal S_\chi^{\,2}=m_P\), and \(\xi^\ast=\mathcal S_\chi-S^\ast\) is the residual capacity the vacuum leaves below the bound. Randall and Sundrum obtain the same Planck-to-weak hierarchy from the warp factor \(e^{-kr_c\pi}\) of a fifth dimension (Randall–Sundrum), with \mbox{\(kr_c\simeq12\)} fixed by a bulk scalar (Goldberger–Wise). Here the exponent is \(1/\xi^\ast=36\), set by the bound and the vacuum. In lengths, with \(\ell_P=1/m_P\) and \(\ell_c=1/m_{\rm IR}\),
\[ \ln\frac{\ell_P}{\ell_c}=-\frac1{\xi^\ast},\qquad \ell_c=\ell_P e^{1/\xi^\ast}=4.31123\times10^{15}\,\ell_P=6.968\times10^{-20}\ {\rm m}, \]
where \(\ell_c=1/m_{\rm IR}\) is the infrared correlation length of the prism vacuum.