Natural Units

A result written in inverse GeV can represent a length or a time, depending on which physical quantity was calculated. Restoring units requires that identification before inserting constants. This worksheet applies Metric and Units to relativistic conversion problems and cross sections. The broader Natural Units translator also covers choices used outside relativistic quantum mechanics.

Required background. Metric and Units supplies the dimensional and electromagnetic convention package.

Energy units determine conversion factors

In $\hbar=c=1$ units, choose an energy unit $Q$, such as $1\,\mathrm{MeV}$ or $1\,\mathrm{GeV}$. The corresponding physical units are:

Quantity represented	Natural-unit dimension	Physical unit associated with $Q$
Energy	Energy	$Q$
Momentum	Energy	$Q/c$
Mass	Energy	$Q/c^2$
Length	Inverse energy	$\hbar c/Q$
Time	Inverse energy	$\hbar/Q$
Cross section	Inverse energy squared	$(\hbar c/Q)^2$

Multiply the appropriate entry by the numerical result expressed in those units. In particular, the natural-unit mass parameter corresponds to a rest energy $mc^2$ when restoring SI mass. A factor such as $1/m$ is not automatically an inverse kilogram.

The conversion constants, rounded here to ten significant digits, are

$$\begin{aligned} \hbar c&\approx197.3269805\ \mathrm{MeV\,fm},\\ \hbar&\approx6.582119570\times10^{-22}\ \mathrm{MeV\,s}. \end{aligned}$$

Thus a natural-unit value of $1\,\mathrm{GeV}^{-1}$ corresponds to approximately $0.1973269805\,\mathrm{fm}$ when it is a length, and $6.582119570\times10^{-25}\,\mathrm{s}$ when it is a time. The two conversions differ by $c$, as their physical dimensions require.

The revised SI fixes $h$, $c$, and the elementary charge $e_{\rm SI}$ exactly. Consequently $\hbar=h/(2\pi)$, $\hbar c$, and the eV-to-joule conversion are exact derived quantities. Their finite printed decimals are approximations, not experimental uncertainties. Particle masses and the dimensionless fine-structure constant remain measured quantities. See the current NIST constant table and SI definitions.

Cross sections in inverse energy squared

If a calculation gives $\sigma=a\,\mathrm{GeV}^{-2}$, restore the physical area as

$$\sigma =a\left(\frac{\hbar c}{1\,\mathrm{GeV}}\right)^2.$$

The useful conversion is

$$\begin{aligned} 1\,\mathrm{GeV}^{-2} &\longleftrightarrow 0.03893793722\ \mathrm{fm}^2\\ &\longleftrightarrow 0.3893793722\ \mathrm{mb}. \end{aligned}$$

The displayed arrows restore physical units; they are not SI dimensional equalities between energy to the minus two and area. A barn is exactly $10^{-28}\,\mathrm m^2=100\,\mathrm{fm}^2$, so $1\,\mathrm{mb}=0.1\,\mathrm{fm}^2$. The barn definition is tabulated in NIST’s guide to units outside the SI.

For example, $2.5\,\mathrm{GeV}^{-2}$ corresponds to approximately $0.97345\,\mathrm{mb}$. If the calculation instead reports $2.5\,\mathrm{MeV}^{-2}$, the area is $10^6$ times larger. The conversion factor must use the same energy unit as the numerical calculation.

An angular differential cross section has the same area conversion; the solid-angle unit does not add a factor of $\hbar$ or $c$. The Mott notebook provides a concrete example with energy, momentum, and area units stated together.

Phases, wave numbers, and Compton scales

Use physical momentum $p=\hbar k$ and energy $E=\hbar\omega$ when restoring a phase:

$$\begin{gathered} e^{-iEt+i\mathbf p\cdot\mathbf x}\\ \longrightarrow\quad \exp\!\left[ -\frac{i}{\hbar} (E_{\rm SI}t-\mathbf p_{\rm SI}\cdot\mathbf x) \right]. \end{gathered}$$

The variables on the left are expressed in a consistent natural-unit system; those on the right are physical SI quantities. Inserting an SI momentum into a formula whose symbol means wave number loses a factor of $\hbar$.

Similarly, a natural-unit length $1/m$ for a particle with SI mass $m_{\rm SI}$ restores to the reduced Compton wavelength

$$\bar\lambda_C=\frac{\hbar}{m_{\rm SI}c}.$$

The ordinary Compton wavelength $h/(m_{\rm SI}c)$ is $2\pi$ times larger. Specify which one an estimate uses. Setting $\hbar=c=1$ is a unit choice; it is neither a classical limit $\hbar\to0$ nor a nonrelativistic limit of the dynamics.

Electromagnetic rationalization is independent

Natural units alone do not decide where factors of $4\pi$ belong. Starting from SI, the dimensionless fine-structure constant is

$$\alpha=\frac{e_{\rm SI}^2} {4\pi\epsilon_0\hbar c}.$$

Two dimensionless positive charge magnitudes can be defined by

$$\begin{aligned} e_{\rm HL}&=\frac{e_{\rm SI}}{\sqrt{\epsilon_0\hbar c}},\\ e_{\rm G}&=\frac{e_{\rm SI}}{\sqrt{4\pi\epsilon_0\hbar c}}. \end{aligned}$$

The rationalized Heaviside–Lorentz convention gives $\alpha=e_{\rm HL}^2/(4\pi)$; the Gaussian convention gives $\alpha=e_{\rm G}^2$. The charge magnitudes differ by $\sqrt{4\pi}$, while the physical coupling $\alpha$ is unchanged. The exact SI charge in coulombs is not the numerical dimensionless charge in either formula.

For arbitrary particles retain signed $q$: an electron has $q=-e$ in the fully declared normalization. Translate the fields, potentials, and couplings together, using Minimal Coupling for the accepted SI package and Mott Scattering for the rationalized natural-unit Coulomb convention. Changing only one $4\pi$ in a copied formula is not a consistent unit conversion.

In the revised SI, exact $e_{\rm SI}$ does not make $\alpha$ exact; $\epsilon_0$ and $\mu_0$ inherit uncertainty from its measured value. This distinction is summarized by NIST’s current discussion of fundamental constants.

Conversion checks

Length versus time. Convert the same inverse-energy coefficient using $\hbar c$ for a length and $\hbar$ for a time. Their ratio should be $c$.

Area versus amplitude. Restore the dimensions of the final cross section, not an isolated amplitude whose external state normalization has not been fixed. Use Relativistic Normalization when comparing amplitude conventions.

Field versus wavefunction. A canonically normalized quantum field and a normalized one-particle wavefunction can carry different dimensions. Do not infer their normalizations merely from the units assigned to their coordinates.

References

• Mohr, Peter J., David B. Newell, Barry N. Taylor, and Eite Tiesinga. “CODATA Recommended Values of the Fundamental Physical Constants: 2022.” Journal of Physical and Chemical Reference Data 54, 033105 (2025). doi:10.1063/5.0279860; NIST-hosted article.
• National Institute of Standards and Technology. The International System of Units (SI), Special Publication 330, section 2. Defining constants and base units.
• National Institute of Standards and Technology. Guide for the Use of the International System of Units (SI), Special Publication 811, chapter 5. Units outside the SI and the barn definition.