The Last Equation

WE FOUND IT.

There is nothing underneath.

Z = \int \mathcal{D}g\,\mathcal{D}\psi\; e^{\,iS[g,\psi]/\hbar}

(37)

Everything else follows.

biology

chemistry

atoms

particles

fields

geometry

?

17 symbols

General relativity

spacetime, smooth and curved

complete at every scale it can reach

stops at r = 0

Quantum field theory

fields, on a background that does not move

complete wherever gravity is weak

stops where it is not

Fig. 1. The two descriptions, as of the twentieth century. Each is complete on its own side. Neither survives the other's regime.

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\,T_{\mu\nu}

(3)

Einstein, 1915. Gravity.

\begin{aligned}\mathcal{L} = {}& -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} + i\bar{\psi}\gamma^{\mu}D_{\mu}\psi \\ &+ \bar{\psi}_i\, y_{ij}\, \psi_j\, \phi + \text{h.c.} + |D_\mu\phi|^{2} - V(\phi)\end{aligned}

(11)

sign conventions as in [21]; 19 parameters, all measured, none explained. yet.

The Standard Model, 1970s. Everything else, except gravity.

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\,T_{\mu\nu}\,,\qquad \mathcal{L}_{\mathrm{SM}}

S[g,\psi]

Fig. 2b. Compression. Eqs. (3) and (11) enter as the low-energy limit of the action in Eq. (37). What comes out the other side is shorter.

General relativity

spacetime, smooth and curved

complete at every scale it can reach

stops at r = 0

(37)

Quantum field theory

fields, on a background that does not move

complete wherever gravity is weak

stops where it is not

Fig. 4. The same diagram. The gap is not bridged. It is absent.

Low-energy limit
Newtonian gravity	recovered
general relativity, Eq. (3)	recovered
Standard Model, Eq. (11)	recovered
Lorentz invariance	preserved
black-hole entropy	recovered
Hawking temperature	recovered
free parameters	0
unexplained residual	0

Derivation status: closed.

Nothing special happened.

We expected something beautiful. It wasn't.

We expected symmetry. Not particularly.

We expected the constants to reveal themselves as inevitable. They did.

We expected the universe to become less strange. It became stranger in fewer symbols.

Don't believe us.

Pick something.

An electron.

(i\gamma^{\mu}\partial_{\mu} - m)\psi = 0

Dirac, 1928. Spin ½ and antimatter fall out. The magnetic moment comes out as exactly g = 2.

a_e = \frac{g-2}{2} = \frac{\alpha}{2\pi} + \cdots

Schwinger, 1948. The first correction. Measured and calculated to twelve digits since. They agree.

from Eq. (37), fermion sector, App. B: m_e and e are fixed by the spectrum. no input.

Observed	Derived
charge −e	−e
spin ½ ħ	½ ħ
a_e 0.001 159 652 181	0.001 159 652 181
mass 0.510 998 95 MeV/c² (measured)	0.510 998 95 MeV/c² (no input)

residual: 0

A falling apple.

F = \frac{GMm}{r^{2}}\qquad g = \frac{GM_{\oplus}}{R_{\oplus}^{2}} = 9.81\ \mathrm{m/s^{2}}

Newton, 1687. Good to one part in a billion in an orchard.

\begin{gathered}\frac{d^{2}x^{\mu}}{d\tau^{2}} + \Gamma^{\mu}_{\alpha\beta}\,\frac{dx^{\alpha}}{d\tau}\,\frac{dx^{\beta}}{d\tau} = 0 \\[.4em] g_{00} \approx -\left(1 + 2\Phi/c^{2}\right)\end{gathered}

Einstein, 1915. The apple is not pulled. It follows the straightest line available. In the weak field, the straightest line is Newton's.

Eq. (37) → Eq. (3) as ħ → 0 (App. A). then v ≪ c. nothing to adjust.

Observed	Derived
acceleration 9.81 m/s²	9.81 m/s²
inertial mass = gravitational mass	exact
clocks run slow near the ground, 1.1 × 10⁻¹⁶ per metre	1.1 × 10⁻¹⁶ per metre
the apple's quantum state	decoheres in 10⁻²⁰ s. as observed.

residual: 0

A black hole.

r_s = \frac{2GM}{c^{2}}\qquad S_{\mathrm{BH}} = \frac{k_B c^{3} A}{4G\hbar}\qquad T_H = \frac{\hbar c^{3}}{8\pi G M k_B}

Schwarzschild 1916, Bekenstein 1973, Hawking 1975. A horizon, an entropy that scales with area rather than volume, and a temperature. Three results that did not fit in one theory.

Penrose, 1965. Light cones tip over at the horizon. In Eq. (3) the top edge is where the description terminates.

in Eq. (37) the top edge is not there. the geometry is smooth. checked: App. C, three times.

Observed	Derived
entropy ∝ area	entropy ∝ area
temperature T_H	recovered
information	conserved
singularity	absent

residual: 0

The early universe.

H^{2} = \frac{8\pi G}{3}\,\rho - \frac{kc^{2}}{a^{2}} + \frac{\Lambda c^{2}}{3}

Friedmann, 1922. The universe expands. Run it backwards and it gets hot.

T_{\mathrm{CMB}} = 2.7255\ \mathrm{K}\qquad Y_p \approx 0.245\qquad n_s \approx 0.965

Penzias and Wilson 1965; primordial helium; the tilt of the first fluctuations. Three numbers the sky gives you for free.

initial state: the unique regular solution of Eq. (37) (App. D). no tuning. no inflaton put in by hand.

Observed	Derived
helium fraction 0.245	0.245
background temperature 2.7255 K	2.7255 K
spectral index 0.965	0.965
flatness Ω = 1.00	1

residual: 0

Empty space.

\rho_{\mathrm{vac}}^{(\mathrm{QFT})} \sim 10^{113}\ \mathrm{J/m^{3}}\qquad \rho_{\mathrm{vac}}^{(\mathrm{obs})} \approx 5\times 10^{-10}\ \mathrm{J/m^{3}}

Weinberg, 1989. The worst prediction in the history of physics: off by about 10¹²². Empty space should weigh enough to end the universe. It doesn't.

\frac{F}{A} = -\frac{\pi^{2}\hbar c}{240\,d^{4}}\qquad \Lambda \approx 1.1\times 10^{-52}\ \mathrm{m^{-2}}

Casimir, 1948. The vacuum pushes on two plates. It is not nothing. The measured Λ is the small number the vacuum actually weighs.

the 10¹²² cancels identically. not fine-tuned: forced. App. C, Eq. (C.9).

Observed	Derived
Casimir force	recovered
Λ 1.1 × 10⁻⁵² m⁻²	1.1 × 10⁻⁵² m⁻²
vacuum stability	stable
the 10¹²²	0

residual: 0

You.

mass	compatible
energy	compatible
electromagnetic state	compatible
chemical state	compatible
neural activity	compatible
observer	included

The theory does not require an exception for the person reading it.

Sorry.

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[41] This work.

Acknowledgments

We thank everyone who spent a lifetime getting us almost here.

We also wanted thelastequation.com. Someone got there first. They can keep it.

We got the equation.

Correspondence: none required.