# Planck Scale

# PLANCK SCALE

The Planck scale is named in honor of the famous German physicist Max Planck. He was the first to realize that three constants of nature can be combined to give fundamental units of mass

The Planck mass, length, and time are equivalent ways to describe the Planck scale. They are constructed from the speed of light *c* , the gravitational constant *G* , and the quantum of angular momentum ħ. In particle physics units, *M*_{P} works out to be 1.2 × 10^{19} GeV/*c*^{2}, about 10^{19} times larger than the mass of a proton. The question of why the Planck mass is so large—or why the proton mass is so small—is at the heart of modern particle physics.

Why is the Planck scale important? It contains *c* , *G* , and ħ, so it connects relativity, gravity, and quantum mechanics. In fact, the Planck scale marks the place where quantum gravity replaces Einstein's relativity. At energies higher than the Planck mass, at distances shorter than the Planck length, or at times shorter than the Planck time, classical notions cease to hold. Planck wrote his expressions in 1899—before relativity and quantum mechanics were discovered—even before he presented his famous formula for blackbody radiation! Planck's remarkable intuition has stood the test of time.

Physics at the Planck scale is very different from the physics of the everyday world. At the Planck scale, gravity is a strong force—so strong that it changes the behavior of subatomic particles. Space-time itself is torn apart by quantum fluctuations. In 1957 American physicist John Wheeler proposed that Planck-scale space-time is a quantum foam, bubbling with virtual processes. Wheeler was the first to recognize the Planck scale's role in quantum gravity.

In quantum electrodynamics, the photon couples to the electron through a gauge coupling denoted by *e.* The fine-structure constant, α, is measured to be At short distances, quantum effects renormalize α. In a scattering experiment with particles of energy *E* , the renormalized coupling increases logarithmically with energy.

In gravity, the analog of the photon is called the graviton. It couples to mass—or energy—through a gravitational coupling *g* . The strength of this coupling changes *linearly* with energy, *g* = *E* √*G* Therefore in gravity, the analog of the fine-structure constant is not constant, but In today's particle physics experiments, in which elementary particles are scattered with energies *E* ≈ 300 GeV, the force of gravity is about a factor of 10^{-32} weaker than electromagnetism. Gravity is weak because the Planck mass is large.

At very short distances, the story is different. In a scattering experiment with Planck-scale energies, the gravitational coupling is of order 1, and quantum effects are critically important. Einstein gravity is nonrenormalizable, which means that it is not a consistent theory of quantum theory. It is a low-energy, long-distance approximation to a deeper, more fundamental theory of quantum gravity. At the Planck scale, Einstein's relativity becomes part of whatever takes its place.

At present, string theory is the best candidate for a fundamental theory of gravity. In string theory, point particles are tiny strings, Planck length in size. Viewed from afar, the strings appear to be points. But with Planck-scale resolution, their stringlike character becomes evident. It is believed that string theory gives a consistent theory of quantum gravity—but only in ten (or eleven) dimensions.

In string theory, all the known particles—quarks, lepton, even gravitons—appear as quantized vibrations of strings. In this sense, string theory unifies all the forces and particles of nature. At present, there is no direct experimental evidence in favor of strings. Nevertheless, there is compelling indirect evidence for new physics near the Planck scale. As in electro-dynamics, the couplings of the Standard Model gauge particles change logarithmically with energy. In certain extensions of the Standard Model, they become equal at about 10^{16} GeV, close to the Planck scale, 10^{19} GeV. This suggests that the Planck scale—and string theory—might play an essential role in the ultimate unification of physics.

Why, then, is the Planck mass so much larger than the mass of the proton? There are two ways to approach the question. The first is to suppose that the Plank mass is fundamental. If so, it is reasonable to assume that the forces of nature are unified near *M _{P}* . In particular, near

*M*, the "strong" coupling of quantum chromodynamics (QCD) is no different than the other gauge couplings. At lower energies, however, the QCD coupling grows stronger (unlike electrodynamics, where it grows weaker). The coupling changes slowly—only logarithmically—so it is not until approximately 1 GeV that the QCD coupling is strong enough to bind quarks and gluons into protons. The mass of the proton is so much smaller than

_{P}*M*because of the logarithmic evolution of the QCD coupling.

_{P}A second point of view is to assume that the proton mass is close to the fundamental scale. In a theory with extra dimensions, for example, it is possible for the *true* Planck scale to be 10^{3} GeV and the *apparent* Planck scale to be 10^{19} GeV. To see how this works, suppose that quarks and leptons are restricted to a three-dimensional membrane embedded in sixdimensional space-time, where the extra dimensions are not infinite, but circles of radius *R* . Also suppose that gravity is not confined to the membrane, but can extend into the two extra dimensions.

At very short distances, shorter than *R* , the gravitational force law is that of six dimensions. It is not inverse square, but rather where *G*_{*} is Newton's constant in six dimensions.

At large distances, much larger than *R* , the gravitational lines of force cannot extend into the extra dimensions. Therefore at large distances, the force law is inverse square: The effective four-dimensional gravitational constant is

As in four dimensions, a fundamental "Planck mass" can be constructed for the six-dimensional theory. It is where *G*_{*} is the six-dimensional Newton constant. Combining equations, one can write the apparent Planck mass *M*_{P} in terms of the fundamental Planck mass and the radius *R* : In this theory, quantum gravity becomes important at *M*_{*}, which is assumed to be *M*_{*} ≈ 3 × 10^{3} GeV. The apparent Planck mass is *M*_{P} ≈ 10^{19} GeV. The two can be reconciled provided *R* ≈ 0.1 mm—in other words, provided there are new spatial dimensions of macroscopic size! From this point of view, gravity is weak because the extra dimensions are large.

Exquisitely beautiful (and careful) experiments are being carried out to test the inverse-square nature of gravity at submillimeter scales. A deviation from the inverse square law could be a hint of new macroscopic dimensions. The discovery of new dimensions would spark a revolution in physicists understanding of the universe—and humankind's place within it.

*See also:*Grand Unification; String Theory; Unified Theories

## Bibliography

Arkany-Hamed, N.; Dimopoulos, S.; and Dvali, G. "Large Extra Dimensions: A New Arena for Particle Physics." *Physics Today* (February 2002).

Greene, B. *The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory* (W.W. Norton, New York, 1999).

Planck, M. *"über Irreversible Strahlungsvorgänge." Sitzungsberichte der könglich Preussischen Akademie der Wissenshaften zu Berlin***5** , 440 (1899).

Wheeler, J. "On the Nature of Quantum Geometrodynamics." *Annals of Physics***2** , 604 (1957).

Wilczek, F. "Scaling Mount Planck I: A View from the Bottom." *Physics Today* (November 1999).

*Jonathan Bagger*

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