Introduction

renphysics

To be completed

Black hole entropy

Black hole thermodynamics shows that entropy scales with horizon area rather than volume, linking gravity, relativity, quantum mechanics, and statistical physics. This area law implies an upper bound on information set by the boundary of a region, motivating the holographic principle that a gravitating system’s degrees of freedom can be encoded on a lower‑dimensional boundary.

Symmetries

Symmetries play a crucial role in modern theoretical physics.

IR and UV Properties of Fundamental Interactions

Conventions. Natural units ($\hbar=c=1$). “Coupling mass dimension” means the energy (mass) dimension of the interaction strength. Spacetime dimension is $d=4$ unless noted; the last row is $d=3$.

Interaction	Coupling mass dimension	IR / low energy / large scales	UV / high energy / short scales
Gravity (GR)	$E^{-1}$ for $\kappa=\sqrt{32\pi G_N}$; $[G_N]=E^{-2}$	Einstein field equations; EFT expansion in $E/M_{\mathrm{Pl}}$	Perturbatively nonrenormalizable; dimensionless $g(\mu)\sim \mu^2 G_N$ grows; needs UV completion
Electromagnetism (QED)	$E^0$ for $e$; $\alpha=e^2/4\pi$	Maxwell equations; long‑range $1/r^2$; IR divergences cancel in inclusive rates	Renormalizable; $\beta(\alpha)>0$ (screening); Landau pole at exponentially high $\mu$
Weak interaction	$E^{-2}$ for Fermi $G_F$ (IR EFT); $E^0$ for SM $g,g'$ (UV completion)	Fermi 4‑fermion EFT; nonrenormalizable; valid for $\mu\ll m_W$	Renormalizable $SU(2)_L\times U(1)_Y$ gauge theory; spontaneous breaking; massive $W^\pm,Z$
Strong interaction (QCD)	$E^0$ for $g_s$; $\alpha_s=g_s^2/4\pi$	Strong coupling; confinement; chiral symmetry breaking; hadrons as IR dof	Asymptotic freedom; $\alpha_s(\mu)\to 0$ as $\mu\to\infty$; perturbative at high $\mu$
$\lambda\phi^4$ in $d=3$	$E^{+1}$ for $\lambda$	Interacting IR fixed point (Wilson–Fisher CFT); e.g. $N{=}1$ $\to$ 3D Ising	Super‑renormalizable; Gaussian UV fixed point; $\tilde\lambda(\mu)=\lambda/\mu\to 0$

Brief explanations

• IR vs. UV. “IR/low energy/large scales” refers to physics at momenta $\mu$ much smaller than the relevant heavy scales; “UV/high energy/small scales” refers to $\mu$ large compared to those scales.

• Terminology.

  • Renormalizable: finitely many parameters absorb all divergences to all loop orders (e.g., QED, QCD, electroweak).
  • Nonrenormalizable (as EFT): infinite tower of higher‑dimension operators suppressed by a cutoff; predictive in a low‑energy expansion (e.g., gravity, Fermi theory).
  • Super‑renormalizable: only a finite set of divergent structures; UV controlled by the Gaussian fixed point (e.g., $\phi^4$ in $d=3$).

• Gravity as an EFT. The loop expansion is organized by operators suppressed by $M_{\mathrm{Pl}}$. The dimensionless strength $g(\mu)\sim \mu^2 G_N$ increases with $\mu$, so the EFT breaks near $M_{\mathrm{Pl}}$ and requires a UV completion (e.g., asymptotic safety or string theory, model‑dependent).

• QED running and the Landau pole. One‑loop running:

  $$\frac{d\alpha}{d\ln\mu}=\frac{2N_f}{3\pi}\alpha^2+\cdots>0$$

  Screening implies $\alpha(\mu)$ grows logarithmically and formally hits a Landau pole at an exponentially high scale, so QED is best viewed as an EFT embedded in the full Standard Model.

• QCD asymptotic freedom. One‑loop running:

  $$\frac{d\alpha_s}{d\ln\mu}=-\frac{11-\tfrac{2}{3}N_f}{2\pi}\alpha_s^2+\cdots<0\quad (N_f\lesssim 16)$$

  Antiscreening by gluons drives $\alpha_s(\mu)\to 0$ at large $\mu$; in the IR, confinement and a mass gap dominate the dynamics.

• Weak interaction: IR EFT vs. UV completion.

  • Low energies: Fermi theory with coupling $G_F$ (dimension $E^{-2}$), a nonrenormalizable contact interaction valid for $\mu\ll m_W$.
  • Matching relation: $G_F/\sqrt{2}=g^2/(8m_W^2)$.
  • High energies: renormalizable electroweak theory $SU(2)_L\times U(1)_Y$ with dimensionless $g,g'$, spontaneously broken to electromagnetism; $W^\pm,Z$ acquire masses.

• $\phi^4$ in $d=3$ and the Wilson–Fisher fixed point. Writing a dimensionless coupling $\tilde\lambda(\mu)=\lambda/\mu$, the $\beta$‑function near $d=4-\varepsilon$ has the schematic form

  $$\beta(\tilde\lambda)=-\varepsilon\,\tilde\lambda+c\,\tilde\lambda^2+\cdots,\quad c>0,$$

  giving an interacting IR fixed point at $\tilde\lambda_\ast\sim \varepsilon/c$ (set $\varepsilon=1$ for $d=3$). The theory is super‑renormalizable with finitely many primitive divergences.

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ER=EPR

• Einstein–Rosen bridge (Wormhole)
• Einstein–Podolsky–Rosen paradox (EPR paradox)

The entangled spins: I feel you near me even when we are apart.

Further reading

• Read about how-to guides in the Diátaxis framework