Length scale

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In physics, length scale is a particular length or distance determined with the precision of one order (or a few orders) of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple. The decoupling of different length scales makes it possible to have a self-consistent theory that only describes the relevant length scales for a given problem. Scientific reductionism says that the physical laws on the shortest length scales can be used to derive the effective description at larger length scales. The idea that one can derive descriptions of physics at different length scales from one another can be quantified with the renormalization group.

In quantum mechanics the length scale of a given phenomenon is related to its de Broglie wavelength $\ell = \hbar/p$ where $\hbar$ is the reduced Planck's constant and $p$ is the momentum that is being probed. In relativistic mechanics time and length scales are related by the speed of light. In relativistic quantum mechanics or relativistic quantum field theory, length scales are related to momentum, time and energy scales through Planck's constant and the speed of light. Often in high energy physics natural units are used where length, time, energy and momentum scales are described in the same units (usually with units of energy such as GeV).

Length scales are usually the operative scale (or at least one of the scales) in dimensional analysis. For instance, in scattering theory, the most common quantity to calculate is a cross section which has units of length squared and is measured in barns. The cross section of a given process is usually the square of the length scale.

[edit] Examples

The atomic length scale is $\ell_a \sim 10^{-10}$ meters and is given by the size of hydrogen atom (i.e., the Bohr radius (approximately 53 pm)) which is set by the electron's Compton wavelength times the fine-structure constant: $\ell_{a} \sim 1/\alpha m_e$ .

The length scale for the strong interactions (or the one derived from QCD through dimensional transmutation) is around $\ell_{s}\sim 10^{-15}$ meters (or in natural units 100 MeV), and the "radii" of strongly interacting particles (such as the proton) are roughly comparable. This length scale is determined by the range of the Yukawa potential. The lifetimes of strongly interacting particles, such as the rho meson, are given by this length scale divided by the speed of light: $10 - 23$ seconds. The masses of strongly interacting particles are several times the associated energy scale (500 MeV to 3000 MeV).

The electroweak length scale is shorter, roughly $\ell_{w} \sim 10^{-18}$ meters and is set by the rest mass of the weak vector bosons which is roughly 100 GeV. This length scale is would be the distance where a Yukawa force mediated by the weak vector bosons. The magnitude of weak length scale was initially inferred by the Fermi constant measured by neutron and muon decay.

The Planck length (Planck scale) is much shorter yet - about $\ell_{P}\sim 10^{-35}$ meters ( $1019$ GeV in natural units), and is derived from Newton's gravitational constant which has units of length squared.