This is the first in a series of posts explaining fundamental physics concepts in simple terms. I will try to explain as deeply as possible from first principles, but without assuming any math beyond high-school level. However, the footnotes contain details for more advanced readers.
The first topic is spin. Spin is a measure of the internal rotational degrees of freedom of a particle. Consider a particle at rest at the origin. We will assume the particle has nonzero mass for now and discuss the massless case later. What transformations can we make to it that leave it looking externally the same? There are just the three rotations, one around each axis1.
What if we try to make this single number change? Let’s say a 180° rotation around the x, y, or z axis multiplies it by 2. This doesn’t work, because a 180° rotation around x followed by a 180° rotation around y is the same as a 180° rotation around z, which you can check. But the former results in a factor of 4, while the latter gives a factor of 2. So not every possible choice of transformation works: it must be compatible with the behavior of rotations.
The little group
Spin and statistics
Spin and angular momentum
DOFs associated with rotations are known as angular momentum. We have only discussed the internal DOFs (spin), but particles also carry external rotational DOFs called orbital angular momentum. The total angular momentum is the sum of spin and orbital contributions. Since spin representations are finite-dimensional, spin angular momentum is quantized4. A measurement in a particular direction will give a result from for a particle of spin , where is Planck’s constant. You can see there are different values. Because rotations around different axes don’t commute, angular momentum in different directions cannot be measured simultaneously: once angular momentum in one direction is known exactly, the other directions become uncertain.
For spin-1/2, the two values are and , corresponding to states called “spin down” and “spin up” with respect to a particular direction. These states can be visualized as little arrows in the direction of angular momentum:
Spin and magnetism
When electromagnetism is included into the theory, it turns out that spin couples to the magnetic field5. Classically, you can visualize spin angular momentum as arising from a particle literally spinning around:
The potential energy of a current loop in a magnetic field is6:
where is the current, is the area, and is the normal vector in the direction given by the right-hand rule on the current. Using , , and angular momentum , this becomes
Thus, spins tend to “align” with an external magnetic field, since the states with spin in the same direction as the field have lower energy than the states in the opposite direction (for a positive charge ). is known as the gyromagnetic ratio. It turns out in the quantum theory that this ratio is actually twice the value of the classical theory: . So a particle with spin cannot really be thought of as a classical current loop.
Magnetic fields allow us to manipulate and measure spin, as in the Stern-Gerlach experiment.
Massless particles and helicity
Massless particles such as the photon do not have spin, because their little group is different. Special relativity tells us that massless particles must travel at the speed of light. Therefore, we cannot imagine them “at rest”: they are always going in a particular direction. However, we can still make rotations around this direction and leave the particle the same. The little group in this case is , the rotation group in 2 dimensions. This is a very simple group that only has 1-dimensional representations7. In fact, we can find them all here. A helicity-s representation acts on the number by multiplying it by
under a rotation by angle . There is a representation for every integer and half-integer s, where s can be less than 0 now. Note that half-integer representations are fermions again, since a rotation by gives -1.
It would then seem that all massless particles have only one degree of freedom. In fact, another symmetry principle requires us to stick two of these representations together: parity, or symmetry under spatial reflections (). Under parity, a rotation in one direction around the particle’s velocity goes in the opposite direction:
We have reflected both the black rotation arrow and the velocity vector . Particle representations must be invariant under parity. If we include the representation, we also need to include the representation that transforms in the opposite way under rotation. Therefore, massless particles have two DOFs: both the and representations. Each degree of freedom transforms independently under this rotation (see footnote 7). Note that massive particles have no preferred direction when at rest, so representations are automatically invariant under parity. Imagine removing the blue arrows in the above figure; you will see that a rotation reflected is the same rotation.
The photon has helicity 1, and the graviton (the force carrier of gravity) has helicity 2. One fascinating result of quantum field theory is that it is impossible to have a locally interacting theory of massless particles greater than helicity 2.
So much for this whirlwind tour of spin and related topics. For more info, see any standard textbooks on quantum mechanics or quantum field theory. Some I recommend:
- Griffiths, D.J. Introduction to Quantum Mechanics, Ch. 4.
- Schwartz, Matthew. Quantum Field Theory and the Standard Model, Chs. 8, 10, 12.
- Weinberg, Steven. The Quantum Theory of Fields, Ch. 2.4, 2.5.
1 In special relativity, there are also transformations called “boosts”, but these give the particle a constant velocity, so it is no longer at rest.
2 You may have heard that the photon is spin 1. But the photon is massless, so has helicity instead of spin. More on this later.
3 This may seem impossible, since a rotation by 360° is the same as no rotation at all. And you are right! I slightly lied earlier. Technically, we are finding representations of , the double cover of the rotation group . This is because the Lie algebras are identical, , but exponentiating the spinor representation of the algebra produces instead of .
4 Orbital angular momentum is also quantized, but only integer representations exist. This is because the DOFs here are actually fields: functions of space. A rotation by 360° must take the field to the same field, because the position vector itself is a vector representation of rotations.
5 This can be derived by starting with the Lagrangian of quantum electrodynamics and taking the non-relativistic limit, where all energies are smaller than the rest energy of the electron . However, in the spirit of effective field theory, we can also consider writing down all terms consistent with the non-relativistic symmetries: , parity, and gauge invariance. The spin operator is a vector, so it must be dotted with another vector to create an invariant. Actually, it is a pseudovector: is invariant under parity , . It must be dotted with another pseudovector to be invariant under parity. This must be the magnetic field , since the electric field is a vector. It cannot be any other function of the vector potential due to gauge invariance. Thus, the lowest order interaction is
for some constant .
6 A nice derivation is as follows. Start with the integral form of Faraday’s law:
where is the magnetic flux through the loop. Multiply by current and integrate over time:
where , is the current, and is the dissipated power. Thus, it takes an energy
to change the magnetic flux by an amount . It is easiest to draw a picture to get the sign right. This derivation shows that the energy is independent of the shape of the loop.
7 A rotation matrix in 2D is, of course, two-dimensional. However, this is actually a reducible representation made of two irreducible, one-dimensional (complex) representations: helicity +1 and -1. You can see this by noting that and are both eigenvectors under rotation, so do not transform into each other. We are only classifying the irreducible representations here.