Ettore Majorana's 1937 Symmetric Theory

In 1937, an Italian theoretical physicist named Ettore Majorana published a short paper titled Teoria simmetrica dell’elettrone e del positrone — a “symmetric theory of the electron and the positron” — in the journal Il Nuovo Cimento. The paper was dense, eight pages of German-influenced mathematical prose translated into Italian, addressed to a community that had just absorbed Dirac’s 1928 theory of the electron and the 1932 discovery of the positron.

The central idea was striking in its simplicity. Dirac’s theory described electrons and positrons as two halves of the same object, linked by charge conjugation: the positron was the antiparticle of the electron. For charged particles, particle and antiparticle are physically distinguishable — an electron carries negative charge, a positron positive. But what if the particle were neutral? Could a neutral fermion be identical to its own antiparticle?

Majorana’s paper showed that the answer was yes, provided one was willing to write the particle’s quantum-field equations in a specifically symmetric form. The mathematical construction was exquisite. The physical consequences — which Majorana himself did not live to explore — have shaped particle physics to this day, particularly through their connection to the neutrino.

The theoretical problem

Dirac’s 1928 equation described a relativistic spin-1/2 particle using four complex components — two for the particle, two for its antiparticle. For the electron, which carries electric charge, this structure fits perfectly: an electron is unambiguously distinct from a positron, because their charges differ.

But Dirac’s theory was constructed for charged fermions. For neutral fermions — the neutron, postulated in 1932, and the neutrino, postulated in 1930 — the question arose: is the particle-antiparticle distinction physically real, or is it a redundancy of the Dirac formalism?

Majorana’s 1937 paper answered: it depends. Using specific representations of the Dirac matrices (now called the “Majorana representation”), one can write a self-conjugate version of the Dirac equation. A fermion satisfying this version has half the degrees of freedom of a standard Dirac fermion — the particle is its own antiparticle. Such a construction is only possible for electrically neutral fermions, because charge conjugation flips the sign of electric charge; a particle identical to its antiparticle can have no charge to flip.

Mathematically: ${\psi }_{\text{Majorana}}={\psi }^c$ where ${\psi }^c$ denotes charge-conjugated field. This equality is impossible for charged fields (it would require a particle with both $+e$ and $-e$ charge), but entirely allowed for neutral ones.

Why it mattered — eventually

Majorana’s construction was, in 1937, a mathematical possibility without a compelling experimental application. The neutrino existed only as Pauli’s postulate; it had not been directly detected. The neutron had just been discovered, but was quickly identified as a composite of three quarks, not a fundamental fermion suitable for Majorana’s construction.

The theoretical significance emerged gradually. Several decades later, physicists realised that Majorana’s self-conjugate construction carried a profound physical implication: the lepton-number symmetry that protects $\bar{\nu }\ne \nu$ in the Dirac formalism is broken in the Majorana formalism. If neutrinos are Majorana particles, lepton number is not conserved.

This has observational consequences:

Neutrinoless double-beta decay ($0\nu \beta \beta$) is the flagship. In a nucleus where single-beta decay is energetically forbidden but two-simultaneous-beta decay is allowed, the decay produces two electrons plus two antineutrinos — which, if the neutrinos are Majorana, can annihilate internally before leaving the nucleus. The result is a final state of two electrons and no neutrinos, a signature that would be forbidden if neutrinos were Dirac.

Small masses via the seesaw. The seesaw mechanism for generating small neutrino masses depends on Majorana mass terms for right-handed neutrinos at a very high scale. These mass terms exist only if neutrinos are Majorana particles. The observed smallness of neutrino masses is therefore often interpreted as evidence — though not proof — of Majorana character.

Leptogenesis. The matter-antimatter asymmetry of the universe can be generated by CP-violating decays of heavy Majorana neutrinos in the early universe, with electroweak sphalerons converting the lepton asymmetry to a baryon asymmetry. This scenario requires that neutrinos be Majorana.

The experimental status

Whether the neutrino is actually a Majorana particle remains, in 2026, the single most important open question in neutrino physics that is directly testable in the laboratory. The programme to answer it has two prongs:

Direct search via $0\nu \beta \beta$. Experiments including KamLAND-Zen (Japan), LEGEND (Italy, Germany), CUORE/CUPID (Italy), nEXO (Canada), and several others search for the single-peak signature at the endpoint of two-electron spectra in candidate isotopes (${}^{76}$Ge, ${}^{136}$Xe, ${}^{130}$Te, ${}^{100}$Mo). No signal has been detected as of 2026. Current limits set the effective Majorana mass $⟨m_{\beta \beta }⟩$ below approximately 36–150 meV, depending on nuclear matrix element choices. Next-generation experiments will reach 10–20 meV, covering the inverted-ordering parameter space.

Indirect evidence via cosmology and oscillation. The absence of Majorana effects in oscillation probabilities (where they cancel) means oscillation experiments cannot distinguish Dirac from Majorana. Cosmological limits on the sum of neutrino masses combined with oscillation-measured mass splittings constrain the Majorana parameter space. A mass-ordering determination from JUNO or DUNE, combined with $0\nu \beta \beta$ null results at sub-10-meV sensitivity, would substantially narrow the question.

Majorana the mathematician

Ettore Majorana joined Enrico Fermi’s Rome group in 1928 at age 22. His contributions to theoretical physics between 1928 and 1937 were sparse in volume but profound in depth. Beyond the 1937 symmetric theory, he developed:

• A general formalism for particles of arbitrary spin (Majorana equation, 1932)
• Early theoretical work on the neutron-proton interaction
• The Majorana representation of the Dirac matrices
• Theoretical analyses that influenced Heisenberg’s and Wigner’s later work

Fermi reportedly said of him: “There are several categories of scientists in the world; those of second or third rank do their best but never get very far. Then there is the first rank, those who make important discoveries, fundamental to scientific progress. But then there are the geniuses, like Galilei and Newton. Majorana was one of these.”

The disappearance

In March 1938, Majorana withdrew a substantial sum of money from his account, wrote several letters of farewell or ambiguous intent, and boarded a ferry from Palermo to Naples. He was never seen again, according to the conventional account.

The circumstances are disputed. Some of the letters suggested suicide; others suggested retreat to a monastic community. Witnesses reported a person matching his description on later ferries or in Venezuelan monasteries; investigators at the time found inconsistencies in the evidence. The Italian judiciary reopened the case in 2008 after a photograph from Argentina was published that purportedly showed Majorana in his forties. Handwriting analysis, voice matching (from surviving notes), and investigation produced no definitive identification.

Whether Majorana died in 1938 or lived a long life elsewhere, the scientific community lost its access to him at the moment when his 1937 paper was beginning to be cited widely. Almost all the subsequent development of Majorana physics — seesaw mechanism, $0\nu \beta \beta$ theory, leptogenesis, the specific mathematics of Majorana fields in quantum field theory — was done by other hands.

Legacy

Eighty-eight years after the paper’s publication, Majorana’s construction is one of the most important topics in experimental particle physics. A positive $0\nu \beta \beta$ detection would be the single most important particle-physics measurement of the decade — a direct experimental verification of Majorana’s 1937 theoretical possibility.

Majorana’s symmetric theory also has applications beyond neutrinos. Majorana fermions appear as emergent excitations in certain topological superconductors; they are being actively pursued in condensed-matter physics as building blocks for topologically protected quantum computation. Even in systems where actual Majorana particles do not appear fundamentally, the mathematical framework remains essential.

Whether the neutrino turns out to be Dirac or Majorana, Majorana’s 1937 paper has become one of the most quietly influential papers in 20th-century physics. It asks a question — are there neutral fermions identical to their antiparticles? — that remains, for the most important fermion in the universe, genuinely open.

The answer is expected within the next decade. Majorana himself, wherever he was after 1938, would have been the first to want to know.