I apologize for the confusion in my previous response. It seems there was an error. Bosons are actually particles with integer spins, such as 0, 1, 2, and so on. The correct statement is that fermions, not bosons, have spin 1/2.
The spin of a particle is a fundamental property that characterizes its intrinsic angular momentum. In quantum mechanics, particles are described by their wave functions, and the spin of a particle is related to the behavior of its wave function under rotations.
The spin of a fermion, like an electron or a quark, is determined by the Dirac equation, which describes the behavior of relativistic fermions. The Dirac equation predicts that fermions have spin 1/2. This means that when you rotate a fermion by 360 degrees, its wave function picks up a phase change of -1. This property is known as "fermionic statistics" and is responsible for the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously.
On the other hand, bosons, like photons or gluons, have integer spins such as 0, 1, 2, and so on. Bosons obey "bosonic statistics," which means their wave functions do not acquire a phase change under rotations. Consequently, multiple identical bosons can occupy the same quantum state, as observed in phenomena like Bose-Einstein condensation.
In summary, the reason for saying the spin of a fermion is 1/2 is based on experimental observations and theoretical frameworks like the Dirac equation, while bosons have integer spins according to quantum mechanics.