Chapter 9: Fundamental Physics

Section 14: Elementary Particles

Spin [of particles]

Even when they appear to be of zero size, particles exhibit intrinsic angular momentum known as spin. The total spin is always a fixed multiple of the basic unit \[HBar]: 1/2 for quarks and leptons, 1 for photons and other ordinary gauge bosons, 2 for gravitons, and in theory 0 for Higgs particles. (Observed mesons have spins up to perhaps 5 and nuclei up to more than 50.) Particles of higher spin in effect require more information to specify their orientation (or polarization or its analog). And in the context of network models it could be that spin is somehow related to something as simple as the number of places at which the core of a particle is attached to the rest of the network. Spin values can be thought of as specifying which irreducible representation of the group of symmetries of spacetime is needed to describe a particle after momentum has been factored out. For ordinary massive particles in d-dimensional space the group is Spin(d), while for massless particles it is E(d-1) (the Euclidean group). (For tachyons, it would be fundamentally non-compact, forcing continuous spin values.) For small transformations, Spin(d) is just the ordinary rotation group SO(d), but globally it is its universal cover, or SU(2) in 3D. And this can be thought of as what allows half-integer spins, which must be described by spinors rather than vectors or tensors. Such objects have the property that they are not left invariant by 360° rotations, but only by 720° ones—a feature potentially fairly easy to reproduce with networks, perhaps even without definite integer dimensions. In the standard formalism of quantum field theory it can be shown that (above 2D) half-integer spins must always be associated with fermions (which for example satisfy the exclusion principle), and integer spins with bosons. (This spin-statistics connection also seems to hold for various kinds of objects defined by extended field configurations.)

From Stephen Wolfram: A New Kind of Science [citation]