Essay: An Attempt of a Theory of Beta Rays

Background
By the early 1930s, the continuous energy spectrum of electrons emitted in beta decay posed a major puzzle for nuclear physics. Conservation of energy and momentum seemed violated unless an unseen particle carried away missing energy. Wolfgang Pauli had proposed a neutral, light particle, the "neutron" later renamed the neutrino by Fermi, but a coherent dynamical description of beta decay was still lacking. Experimental regularities in half-lives and spectra demanded a quantitative framework connecting nuclear transitions to emitted leptons.
Enrico Fermi proposed a simple but profound solution that recast beta decay as a fundamental interaction among four fermions. The theory supplied a calculable transition rate, an explanation for the shape of the electron spectrum, and a single coupling constant that could be compared to experiment, thereby turning beta decay into a field-theoretical problem amenable to quantitative tests.

Central proposal
Fermi treated beta decay as a pointlike "contact" interaction in which a neutron transforms into a proton while emitting an electron and an antineutrino. The process was described by a local product of four fermionic field operators, one for the neutron, one for the proton, one for the electron and one for the neutrino, multiplied by a coupling constant now called the Fermi constant, G_F. This interaction was postulated to be analogous in spirit to the electromagnetic interaction but without mediating quanta: the interaction acted at a point rather than being carried by a long-range field.
The proposal unified diverse beta processes under a single dynamical law and naturally incorporated Pauli's neutrino, giving it a definite role in carrying away energy, momentum and angular momentum. The simplicity of the contact form made perturbative calculation feasible and allowed direct comparison of theoretical decay rates with laboratory measurements.

Mathematical structure
Using the nascent language of quantum field theory, transition matrix elements were constructed from the four-fermion operator and evaluated with first-order perturbation theory. The resulting rate formula combined phase-space factors for the outgoing electron and neutrino with nuclear matrix elements reflecting the nucleon transition. Energy-conserving delta functions led to an explicit expression for the electron energy spectrum, whose continuous shape emerged from integrating over the unobserved neutrino momentum.
Fermi's formalism identified selection rules and angular-momentum constraints for allowed transitions, while leaving room for spin-dependent generalizations. The contact interaction involved different possible couplings between spinor fields, a point later developed into classifications of "Fermi" (vector) and "Gamow–Teller" (axial) transitions when spin-dependent terms were recognized.

Predictions and immediate consequences
The theory produced spectral shapes and rates that matched available beta-decay data much better than previous heuristic accounts. Measurements of end-point energies, spectra and lifetimes could be interpreted to extract the Fermi coupling constant and to test neutrino kinematics. The formalism also clarified how nuclear structure influences decay strengths through nuclear matrix elements, establishing a bridge between nuclear physics and particle dynamics.
Limitations were soon apparent: the contact interaction is nonrenormalizable, and the theory left open the underlying mediator and deeper symmetry structure of weak processes. Parity violation, unknown at the time, lay outside the original formulation and would require later revisions of the coupling structure.

Later developments and legacy
Fermi's theory became the cornerstone of weak-interaction phenomenology. Extensions by Gamow and Teller introduced axial components necessary to account for observed spin-dependent transitions, while later work by Feynman, Gell-Mann, Sudarshan and Marshak reformulated the interaction in vector-minus-axial (V−A) form to incorporate parity violation. The four-fermion interaction remained a useful effective theory at low energies and gave way to the electroweak gauge theory of Glashow, Weinberg and Salam at higher energies, where the contact interaction is understood as the low-energy limit of W-boson exchange.
The legacy endures in the Fermi constant, G_F, a fundamental parameter still used to quantify weak processes, and in the conceptual leap of treating weak decay as a field-theoretical interaction. The 1934 essay transformed the neutrino from a speculative book-keeping particle into an essential actor in a predictive dynamical theory, shaping the course of particle physics for decades.