Planck Length

Prerequisites: Dimensional Analysis, Fundamental Constants. Full path (3 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

There are many indications that in quantum gravity there might exist a minimal observable distance on the order of the Planck length. The emergence of a minimal length is usually considered a dynamical phenomenon, related to the fact that at the Planck scale there are violent fluctuations of the metric and even topology changes, as in Wheeler space-time foam/ from Quantum groups, gravity, and the generalized uncertainty principle by Michele Maggiore

if one wants to probe an event in the length scale of Planck length with a photon, by the uncertainty principle, the particle has to have roughly Planck energy. Now according to general relativity, a photon on such energy scales causes a gravitational collapse and therefore it does not yield any information of the event. The gravitational collapse is caused by the fact that the Schwarzschild radius of a particle with Planck energy is approximately equal to the Planck length. Consequently, due to the uncertainty principle and the Schwarzschild radius, the very measurement of an event in this length scale creates a black hole and no information about this event will emerge. The region with Planck-length of radius, therefore becomes in a sense noncontinuous, i.e. experimentally not accessible.Source

Intuitive

Nature gives us three fundamental constants that set the scale of three different areas of physics: the speed of light (relativity), Planck's constant (quantum mechanics) and Newton's constant (gravity). From these three numbers one can build only one combination with the dimension of a length (up to a pure number), and this is the Planck length. It is about a hundred millionth of a billionth of a billionth of a billionth of a metre, roughly twenty orders of magnitude smaller than a proton.

The Planck length marks where we expect quantum effects and gravity to become important at the same time. A simple way to see this: to look at a very small region, we need a probe of very short wavelength, and by quantum mechanics short wavelength means high energy. By general relativity enough energy in a small enough region forms a black hole. If we try to probe a region of about the Planck length, the energy needed is so large that the probe would collapse into a black hole about as big as the region we wanted to see. Below this scale our current theories stop giving sensible answers.

For this reason many physicists expect that a theory of quantum gravity is needed to describe physics at the Planck length, and some suspect that distances shorter than it have no operational meaning. These are expectations, since no experiment comes close to probing such distances.

Concrete

The Planck length follows from dimensional analysis. We look for $l = G^\alpha\hbar^\beta c^\gamma$ with $[G] = \text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$, $[\hbar] = \text{kg}\,\text{m}^2\,\text{s}^{-1}$, $[c] = \text{m}\,\text{s}^{-1}$. Matching the powers of kg, m and s gives

$$-\alpha + \beta = 0, \qquad 3\alpha + 2\beta + \gamma = 1, \qquad -2\alpha - \beta - \gamma = 0 ,$$

with the unique solution $\alpha = \beta = \frac12$, $\gamma = -\frac32$:

$$l_P = \sqrt{\frac{G\hbar}{c^3}} \approx 1.6\times10^{-35}\ \text{m} .$$

In the same way one gets the Planck time $t_P = l_P/c \approx 5.4\times10^{-44}$ s, the Planck mass $m_P = \sqrt{\hbar c/G}\approx 2.2\times10^{-8}$ kg and the Planck energy $E_P = m_Pc^2 \approx 1.2\times10^{19}$ GeV. Some authors use the reduced Planck mass $\sqrt{\hbar c/8\pi G}\approx 2.4\times10^{18}\ \text{GeV}/c^2$ instead, so factors of $\sqrt{8\pi}$ differ between references.

Heuristic argument. A particle of mass $m$ cannot be localized better than its Compton wavelength $\lambda_C = \hbar/mc$. Its mass would form a black hole of Schwarzschild radius $r_s = 2Gm/c^2$. The two are equal when

$$\frac{\hbar}{mc} = \frac{2Gm}{c^2} \quad\Rightarrow\quad m = \sqrt{\frac{\hbar c}{2G}} = \frac{m_P}{\sqrt2}, \qquad r_s = \sqrt2\, l_P .$$

So up to factors of order one the Planck length is the scale where a localized quantum particle becomes its own black hole.

Strength of gravity. In quantum electrodynamics the coupling is the dimensionless fine-structure constant $\alpha \approx 1/137$. Newton's constant has dimensions, and the dimensionless measure of gravitational strength between two particles of mass $m$ is

$$\frac{G m^2}{\hbar c} = \left(\frac{m}{m_P}\right)^2 .$$

For two protons this is about $6\times10^{-39}$. For a scattering process at energy $E$ the analogous ratio is $(E/E_P)^2$, so gravity becomes as strong as the other forces only at energies near $E_P$, equivalently at distances near $l_P$.

• The Planck Length by John Baez

Abstract

With $\hbar = c = 1$, Newton's constant has mass dimension $-2$ and $l_P^2 = G$. In $D$ spacetime dimensions $G_D$ has mass dimension $2-D$ and the Planck length is $l_P = G_D^{1/(D-2)}$.

The dimensionful coupling is the reason why the Planck length is the cutoff of perturbative quantum gravity. Expanding the Einstein-Hilbert action

$$S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R$$

around flat space with $g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}$, $\kappa^2 = 32\pi G$, gives a canonically normalized graviton $h_{\mu\nu}$ with interactions suppressed by powers of $\kappa$. Each order in the loop expansion comes with an extra factor $G E^2 = (E\, l_P)^2$, so the theory is power-counting non-renormalizable. Pure gravity is finite on-shell at one loop ('t Hooft and Veltman) but has a divergence at two loops (Goroff and Sagnotti) that requires a new counterterm cubic in the Riemann tensor. As an effective field theory general relativity gives well defined quantum corrections at energies $E \ll 1/l_P$, while its predictive power is lost for $E \sim 1/l_P$.

The Planck length also sets the scale of black hole thermodynamics. The Bekenstein-Hawking entropy is

$$S = \frac{k_B\, A}{4\, l_P^2} ,$$

one quarter of the horizon area in Planck units, which is often taken as evidence that a fundamental theory of gravity has a finite number of degrees of freedom per Planck area.

Arguments for a minimal length combine the uncertainty principle with gravity. Including the gravitational disturbance of the measured region by the probe energy leads to relations of the form

$$\Delta x \gtrsim \frac{\hbar}{\Delta p} + \beta\, l_P^2\, \frac{\Delta p}{\hbar} ,$$

with $\beta$ of order one, which is minimized at $\Delta x \sim l_P$. These arguments are heuristic. A sharp minimal length is in tension with Lorentz invariance if it is read as a fixed length in every frame, and the approaches to quantum gravity treat it differently: in loop quantum gravity area and volume operators have discrete spectra in units set by $l_P$, while in string theory the relevant short-distance scale is the string length, related to $l_P$ through the string coupling.

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1. Fundamental Constants
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