The spin of elementary particles, such as electrons and quarks, is a fundamental property in quantum mechanics. It is not related to the physical rotation or angular velocity that we typically associate with macroscopic objects. Rather, spin is an intrinsic property of particles that is described by quantum numbers and is related to their angular momentum.
In quantum mechanics, particles are described by wave functions, which are mathematical representations that contain information about their properties, including spin. The spin of a particle is quantized, meaning it can only take specific values dictated by the laws of quantum mechanics. The quantization of spin is a fundamental aspect of the theory and is experimentally verified.
Fermions are a class of particles that have half-integer spin values (e.g., 1/2, 3/2, 5/2, and so on). Examples of fermions include electrons, quarks, and neutrinos. The fact that all fermions have a spin of 1/2 is a consequence of the underlying symmetries and properties of the quantum mechanical framework.
In quantum field theory, which is used to describe elementary particles, the spin of particles is related to their behavior under rotations in space. Mathematically, it is determined by the representation theory of the relevant symmetry group. The fundamental fermionic fields in nature are described by representations of the Lorentz group, which is the symmetry group underlying special relativity.
The representation theory of the Lorentz group reveals that the fundamental fermionic fields can be associated with spin-1/2. This means that all fermions, such as quarks and electrons, are described by fields with half-integer spin. The specific value of 1/2 arises due to the particular mathematical properties of the fields and their transformation properties under rotations and boosts.
It is important to note that the spin of a particle does not refer to a physical rotation in space, but rather represents an intrinsic property that influences the particle's behavior and interactions. The value of 1/2 for fermions is a consequence of the underlying symmetries and mathematical structure of quantum field theory.