Natural Units and Heaviside-Lorentz Units in Theoretical Physics
In theoretical physics, standard measurement systems like SI can make equations unnecessarily cumbersome with repeating constants. To simplify calculations, physicists use natural units and Heaviside-Lorentz units, which elegantly set fundamental constants to one. This approach not only streamlines complex derivations in quantum field theory but also reveals the deep underlying symmetries of nature.
1. The Philosophy of Natural Units
In the International System of Units (SI), we use distinct base dimensions for mass, length, and time (kilograms, meters, and seconds). However, nature provides its own fundamental scales. In high-energy physics, it is standard practice to set the reduced Planck constant (\(\hbar\(), the speed of light (\(c\(), and the Boltzmann constant (\(k_B\() to unity:
\( \hbar = c = k_B = 1 \(By doing this, we fundamentally alter dimensional analysis. Velocity becomes a dimensionless fraction of the speed of light (\(v/c\(), and action becomes a dimensionless multiple of \(\hbar\(. Consequently, all physical quantities can be expressed in terms of a single dimension, typically energy (measured in Electron-volts, \(\text{eV}\().
Dimensional Collisions
Because \(c=1\(, the famous mass-energy equivalence equation \(E = mc^2\( simplifies dramatically. Mass and energy have the exact same dimension:
\( E = m \(Similarly, the relativistic energy-momentum relation \(E^2 = (pc)^2 + (mc^2)^2\( reduces to a simple Pythagorean form:
\( E^2 = p^2 + m^2 \(Because \(\hbar = 1\(, the relation between energy and time (\(E = \hbar / t\() implies that time has the dimension of inverse energy. Therefore, length and time share the same dimension (\([\text{Length}] = [\text{Time}]\(), which is the inverse of mass and energy (\([\text{Energy}]^{-1}\().
2. Heaviside-Lorentz Units
While standard natural units handle mechanics and quantum theory, electromagnetism requires further rationalization. The Heaviside-Lorentz (HL) unit system is the standard choice in Quantum Electrodynamics (QED) and particle physics.
HL units are "rationalized," meaning the factor of \(4\pi\( is strategically placed. In SI units, Maxwell's equations contain \(\varepsilon_0\( and \(\mu_0\(, while Coulomb's law contains a \(4\pi\(. In Gaussian units, the \(4\pi\( is removed from Coulomb's law but infects Maxwell's differential equations. HL units fix this by setting the vacuum permittivity and permeability to unity in natural units:
\( \varepsilon_0 = \mu_0 = 1 \(Maxwell's Equations in HL Units
By defining the units of charge appropriately, the \(4\pi\( is completely eliminated from the fundamental differential equations of electromagnetism. In natural HL units, Maxwell's equations take their most pristine form:
\( \nabla \cdot \mathbf{E} = \rho \( \( \nabla \cdot \mathbf{B} = 0 \( \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \( \( \nabla \times \mathbf{B} = \mathbf{J} + \frac{\partial \mathbf{E}}{\partial t} \(However, because the equations are rationalized, the geometric factor of \(4\pi\( correctly reappears in macroscopic spherical problems, such as Coulomb's Law for the force between two point charges:
\( F = \frac{q_1 q_2}{4\pi r^2} \(3. The Fine-Structure Constant
The fine-structure constant (\(\alpha\() is a dimensionless parameter characterizing the strength of the electromagnetic interaction. Because it is dimensionless, its value remains exactly the same regardless of the unit system chosen. In SI units, it is defined as:
\( \alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \(In natural Heaviside-Lorentz units, where \(\varepsilon_0 = \hbar = c = 1\(, the definition simplifies beautifully to:
\( \alpha = \frac{e^2}{4\pi} \approx \frac{1}{137} \(This reveals that in this unit system, the fundamental charge of an electron \(e\( is a dimensionless number, given by \(e = \sqrt{4\pi\alpha}\(.
4. Restoring SI Units
To convert an equation derived in natural units back to SI units, one must perform dimensional analysis and insert factors of \(\hbar\( and \(c\( until the dimensions balance. For example, if a calculation in natural units yields a decay rate \(\Gamma\( (which has units of Mass/Energy) but we want a time (\(\tau = 1/\Gamma\() in seconds, we multiply by \(\hbar\(:
\( \tau \text{ (seconds)} = \frac{\hbar}{\Gamma \text{ (Joules)}} \(If \(\Gamma\( is in \(\text{eV}\(, we use \(\hbar \approx 6.582 \times 10^{-16} \text{ eV}\cdot\text{s}\(.