# Lorentz and Unification of Forces

Coulomb's law is not Lorentz-covariant, nor is the law governing the force on a moving charge in magnetic field. Lorentz combined these two forces and observed that the combined force becomes covariant in the Lorentzian world. Lorentz in this way unified electric and magnetic interactions.

- The Lorentz force formula was written in terms of electric and
magnetic fields which are combined into one tensor representation.
This electromagnetic interaction can also be written in terms of
a four-vector potential in the Lagrangian formulation of classical
mechanics and also in quantum field theory.
In the 1930s, physicists were interested in understanding weak interactions, and Fermi's Golden rule was used for calculating transition rates. Fermi's rule was like a perturbation formula, but it was strictly a phenomenological formula without a consistent field theory.

- Thanks to the works of Glashow, Salam, and Weinberg in the late 1969s,
it became possible to construct a renormalizable field theory based on
the Yang-Mill theory, spontaneous symmetry breaking, and Higg's
mechanism.
This renormalizable field theory requires placing electromagnetic and weak interactions into the same symmetry group. Indeed, this constitutes a unification of these two interactions. These days, we use the word "electro-weak integration."

During this process, there is one interesting step where two-component massless particles become massive after each eating up a one-component boson. This (2 + 1 = 3) mechanism comes from the spontaneous symmetry break down and is widely discussed in textbooks, but it is still of interest to see whether this can be done purely from considerations of the fundamental space-time symmetry.

## Another Gift from Hendrik Lorentz

- In 1939, Eugene Wigner introduced his little groups to describe internal
space-time symmetries of elementary particles. The little group is
the maximal subgroups of the Lorentz group whose transformations leave
the four-momentum of a given particle invariant. The little group for
a massive particle at rest is the three-dimensional rotation group, which
allows us to define particle spins in a Lorentz-covariant way.
The little group for a massless particle cannot be brought to the rest frame. For a given momentum, rotations around its momentum leave it invariant.

- In addition, Wigner found two other transformations which leave the
four-momentum invariant. He noted also that these transformations with
the rotational degree of freedom form a group isomorphic to the
two-dimensional Euclidean group, and they correspond to the two
translational degrees of freedom in the E(2) group.
It is not difficult to associate the rotational degree of freedom with the helicity of the massless particle. However, the translational degrees of freedom had a very stormy history. It was indeed a evolutionary process for physicists to realize that they correspond to the gauge degree of freedom. However, the massless particle has only one, not two gauge degrees.

- I was fortunate enough to discuss this problem with Professor Wigner during
his late years and published two papers on this subject. The result is
illustrated in the figure shown on this webpage.
The massive spin-1 particle can have three components and has the O(3) symmetry, which is described by a sphere and illustrated as a circle in the figure. However, to a moving observer, this circle becomes split into a pancake-like and football-like ellipsoids.

As the frame speed becomes large and becomes close to the speed of light, the pancake becomes a flat plane, and the football becomes a cylinder. The longitudinal rotation remains as the rotation. The transverse rotations become translations on the flat plane. These two rotations collapse into one (up and down) translational degree of freedom on the cylindrical surface. This is precisely the gauge degree of freedom.

- For the cylindrical symmetry of massless particles, you may read
the paper by Kim and Wigner published in
J. Math. Physics.
If you do not have an electronic access to the pdf files of this
journal, you can read the
preprint of this article.
This cylindrical mechanism allows 3 becoming to (2 + 1), and also (2 + 1) becoming 3, according to the geometry of the Lorentz group. Indeed, this aspect is another gift from Hendrik Antoon Lorentz.