**Further Afield**

There are many observations in the quantum world that do not fit into the Standard Model, which may be generously characterized as incomplete. Other physics theories can account for what SM cannot. But they too have flies in their ointment of exposition.

**Lattice World**

In 1928, Paul Dirac described his relativistic approach to characterizing a fermion field. He had in mind electrons, which have both mass and charge.

Within his mathematical solutions, Dirac found the positron, which is the electron’s antiparticle. The positron has the same mass as the electron, but the opposite charge: positive rather than negative. Positrons were experimentally confirmed the year after the Dirac equation appeared, becoming the first antiparticle found.

In 1929, German mathematician and physicist Hermann Weyl showed that Dirac’s equation could be simplified for massless fermions.

The next year, Wolfgang Pauli proposed neutrinos to explain the continuous energy spectrum coming out of radioactive decay. To respect the law of energy conservation, neutrinos had to be chargeless.

Neutrinos were first detected in 1956. Early experimental data suggested that neutrinos lacked mass. From that, it was assumed that Dirac’s neutrinos were merely massless Weyl fermions. Later investigation surmised that the masses of neutrinos are slight, but that remains uncertain.

In 1937, Italian physicist Ettore Majorana took neutrinos to an even more ethereal state, by proposing a class of quanta that was both massless and chargeless. Majorana particles were first glimpsed in 2012.

Whereas Dirac fermions have an antiparticle counterpart, such as electrons and positrons, Majorana fermions are their own antiparticle.

3 distinct classes of fermions have been identified: Dirac (with mass and charge), Weyl (massless, charged), and Majorana (massless, chargeless). What all fermions have in common is the same spin, which is the direction of internal angular momentum relative to the direction of linear momentum. Spin is the property that distinguishes fermions from bosons.

The asymmetrical spin of fermions explains why they cannot occupy the same space at the same time; but bosons can, because their spin is symmetrical. This fermionic limitation is termed the Pauli exclusion principle, which Wolfgang Pauli discovered in 1925.

Don’t think for a Planck moment that fermions always mind their manners. It all depends upon the environment they are in. Fermions might go bosonic when stressed.

The mathematics of existence can be quite slippery. Nature’s fondness for diversity often rides roughshod over textbook behaviors. Such is the case when fermions find themselves in the tight confines of a crystalline space.

Crystals are highly ordered solids which may form any one of 230 distinct lattice structures. Figuring out the extent of lattice space groups was a tour de force of 19th-century crystallography.

In 1930, Werner Heisenberg wondered what would happen if space itself was quantized instead of continuous. Heisenberg was inspired to speculate about a Planck Gitterwelt (lattice world) out of a desire to rid quantum mechanics of the vexatious infinities that kept arising in equations.

(Quantum mechanics’ mathematics was never able to shake off infinity. So, the beautiful symmetries and inscrutable infinities that appear everywhere are purposely broken by spontaneous symmetry breaking, which is a statistical abuse to force equations to behave so that physicists feel comfortable with them.)

What Heisenberg got in Gitterwelt was inexplicably peculiar: electrons might lose their mass, or morph into protons. This strangeness drove him to abandon “this completely crazy idea.”

But Gitterwelt happens. An electron moving through a honeycomb lattice of graphene carbon atoms loses its mass, transforming itself from a Dirac fermion to a Weyl one. If the lattice is superconductive, the electron may drop its charge and change into a Majorana.

Lattices offer even stranger transformations. A Weyl fermion trapped in a lattice world might alter its spin to that of a boson, while still being fermionic in obeying the Pauli exclusion principle. Other quantum oddities of lattice worlds are still being explored.