What Gives a Particle Charge and Spin?

[Willem F Esterhuyse]

We got the same problem as with mass i.e. what gives a particle charge, spin, weak isospin?

[studiot]

1 hour ago, Willem F Esterhuyse said:

We got the same problem as with mass i.e. what gives a particle charge, spin, weak isospin?

Aren't these all provided for in 'the standard model'  ?

 

But how is this classical physics ?

[joigus]

In quantum field theory, particles are characterised as irreducible representations of symmetry groups.

Symmetry transformations are different ways to look at particles that leave the quantum state unchanged.

When these transformations can be expressed in terms of a finite number of parameters, these groups of transformations help us classify the particles according to their so-called Noether charges. When the system is symmetric under one of these groups --also called Lie groups--, Noether's theorem guarantees that these "charges" are conserved.

Some transformations have to do with space-time symmetries. Examples are translations and rotations, which have the corresponding conserved quantities that we're familiar with under the name of linear momentum and angular momentum.

In any quantum theory, charged particles are represented by complex wave functions. Everything observable depends on quadratic expressions of the form (field)*(field), where the asterisk represents complex conjugate. Because the physics is indifferent to a global phase change, a conserved quantity exists --by virtue of Noether's theorem-- that we call electric charge.

In the case of angular momentum --due to symmetries under rotations--, it so happens that the spatial coordinates are not enough to represent all the rotational states of particles. Internal variables must be specified describing the orientation of a particle that allow no representation in terms of spatial coordinates. That's what we call spin.

Spin has to do with rotation, although it's not nearly as intuitive as the rotation of a spinning top, eg.

Charge has to do with something even more abstract, which is a phase shift in the wave function. This transformation is sometimes called "internal."

Isospin invariance is not exact, it's only approximately conserved. It is an analogue of electric charge, and also occurs in this "internal space" of elementary particles. It so happens that, if you ignore electromagnetic interaction, a proton and a neutron are very similar when you only consider the strong nuclear force. You can kind of rotate the states smoothly from "being a proton" to "being a neutron." This is in close analogy with electric charge.

There are other analogues of electric charge: baryon number, lepton number, hypercharge... Colour charge is similar, but more complicated, as the previous charges depend on a 1-parameter group called U(1), while colour is defined in terms of a 3-parameter group SU(3).

Mass is very different. It is not a conserved quantum number like the other charges.

So I would say that what gives a particle its charge and spin are its properties under global gauge transformations (global phase shifts), in the case of charge; and rotational properties, in the case of spin. This is a summary of the present theoretical understanding of these things within the context of the standard model of particle physics.

Edited November 30, 2022 by joigus
minor correction

[Markus Hanke]

10 hours ago, joigus said:

That's what we call spin.

Just to add to joigus’ excellent post (+1) - spin also characterises how the mathematical object that represents the particle in question behaves under Lorentz transformations; it reflects what class of mathematical object you need to describe all relevant degrees of freedom of such a particle. Spin-0 particles are described by scalars, spin-½ particles by spinors, spin-1 particles by vectors, and spin-2 particles by rank-2 tensors. Other spins (such as 3/2 or 5/2) would correspond to multi-component spinors. These classes of objects are representations of the Lorentz group, and are valid solutions to the Bargmann-Wigner equations. Thus, spin is intrinsically a relativistic phenomenon - in a spacetime with Euclidean geometry, particles wouldn’t have spin.

The elementary particles we know of have spins 0, ½ or 1; it is also possible to create composite particles with other spins such as 3/2 or 5/2. In flat spacetime, the Weinberg no-go theorem and the Coleman-Mandula theorem imply that no elementary particles with spin-3 or greater can exist.