The fact that elementary particles have half-integer spins is a fundamental characteristic observed in quantum mechanics and is known as the spin-statistics theorem. This theorem establishes a connection between the intrinsic properties of particles, such as spin, and their statistical behavior.
In quantum mechanics, particles are described by wave functions, which are mathematical functions that contain information about their properties. The behavior of these wave functions under particle exchange plays a crucial role in determining the statistics of particles.
Particles can be divided into two broad categories based on their statistical behavior: fermions and bosons.
Fermions: Fermions are particles that follow the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. Examples of fermions include electrons, quarks, and neutrinos. Fermions have half-integer spins (e.g., 1/2, 3/2) and are subject to the Fermi-Dirac statistics.
Bosons: Bosons, on the other hand, do not obey the Pauli exclusion principle and can occupy the same quantum state. Examples of bosons include photons, W and Z bosons, and the Higgs boson. Bosons have integer spins (e.g., 0, 1, 2) and follow Bose-Einstein statistics.
The spin-statistics theorem, proved by Wolfgang Pauli and independently by Julian Schwinger, states that particles with half-integer spins (fermions) must follow Fermi-Dirac statistics, while particles with integer spins (bosons) must follow Bose-Einstein statistics. This connection between spin and statistics has been experimentally verified and plays a crucial role in our understanding of quantum mechanics and the behavior of elementary particles.
The specific values of 1/2, 3/2, 5/2 for the spins of elementary fermions (such as electrons and quarks) arise from the mathematical formalism of quantum field theory, which describes these particles as excitations of underlying quantum fields. The half-integer spin values are a consequence of the underlying symmetries and dynamics of these fields, as well as the principles of quantum mechanics.