Lorentz–Heaviside units

Lorentz–Heaviside units constitute a system of natural units for electromagnetic theory, particularly prominent in theoretical physics and quantum field theory. The system simplifies the fundamental equations of electromagnetism by setting several dimensionful constants to unity, thereby streamlining calculations in high-energy physics. It is named for the pioneering physicists Hendrik Antoon Lorentz and Oliver Heaviside, who made foundational contributions to the modern formulation of classical electrodynamics.

Definition and motivation

The system is defined by setting the permittivity of free space and the permeability of free space each to one, which consequently forces the speed of light in vacuum to also be unity. This choice eliminates the explicit appearance of factors like \(4\pi\) from the electrostatic force law, concentrating them instead in the definitions of the charge and current densities. The primary motivation, championed by Heaviside and later adopted in relativistic quantum field theories, was to produce a more symmetric and aesthetically pleasing form of Maxwell's equations, especially when combined with the theory of special relativity developed by Einstein. This reformulation proved highly advantageous for the covariant description of fields within the framework of Minkowski spacetime.

Comparison with other systems

Lorentz–Heaviside units differ significantly from the SI system used in practical engineering and laboratory measurements, where \(\epsilon_0\) and \(\mu_0\) have defined, non-unity values. Compared to the Gaussian (or cgs) system, another common unit set in theoretical work, Lorentz–Heaviside units treat electric and magnetic fields as having the same dimensionality, a feature shared with SI units but not with the Gaussian system. The Planck unit system, used in studies of quantum gravity and cosmology, also sets \(c = 1\) but further incorporates Newton's constant and Planck's constant, making it more comprehensive for fundamental physics.

Maxwell's equations in Lorentz–Heaviside units

In this unit system, Maxwell's equations in vacuum take their most compact and symmetric form. The two inhomogeneous equations are \(\nabla \cdot \mathbf{E} = \rho\) and \(\nabla \times \mathbf{B} - \partial \mathbf{E} / \partial t = \mathbf{J}\), while the homogeneous equations are \(\nabla \cdot \mathbf{B} = 0\) and \(\nabla \times \mathbf{E} + \partial \mathbf{B} / \partial t = 0\). This formulation, devoid of the factors \(4\pi\) and \(c\) that appear in SI units, clearly reveals the relativistic covariance of the theory when expressed using the field strength tensor \(F^{\mu\nu}\). The force law is written as \(\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})\), identical in form to its SI counterpart but with the fields and charge defined differently.

Relation to Gaussian units

The Lorentz–Heaviside system is a variant of the broader Gaussian units family, specifically differing by a factor of \(\sqrt{4\pi}\) in the definitions of charge and field quantities. A charge in Lorentz–Heaviside units is smaller by this factor than the same physical charge expressed in pure Gaussian units: \(q_{\text{LH}} = q_{\text{G}} / \sqrt{4\pi}\). Consequently, the electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\) are larger by \(\sqrt{4\pi}\) in the Lorentz–Heaviside system. This rescaling moves the \(4\pi\) factors from the field equations to the force law for point charges, a convention preferred in quantum field theory and the formulation of the Standard Model of particle physics.

Advantages and usage

The primary advantage of Lorentz–Heaviside units is the simplification and clarity they bring to theoretical calculations in advanced electrodynamics and quantum field theory. By removing superfluous constants, they make the relativistic structure of the theory manifest and reduce algebraic clutter in Feynman diagram calculations, especially in quantum electrodynamics (QED). The system is the standard convention in many foundational textbooks on quantum field theory, such as those by Peskin and Schroeder, and Weinberg. Its usage remains prevalent in high-energy physics research, string theory, and condensed matter field theory, where the focus is on underlying symmetries and gauge principles rather than numerical comparison with laboratory experiments.

Category:Units of measurement Category:Electromagnetism Category:Theoretical physics