Intuitive

Noether's famous theorem states that there is a conserved quantity for every symmetry of the Lagrangian.

An interesting subtlety of this theorem is that the corresponding conserved quantity in field theories has two parts.

The reason for this is the following:

In particle theories, we use the position and momenta or velocities to describe what is going on.

In contrast, in field theories we use more abstract notions called fields $\phi(x), \Psi(x), A_\mu(x)$ to describe the dynamics of systems. (In this sense, the usual wave function of quantum mechanics $\psi(x)$ is also a "field".) In mathematical terms, these objects are what we already mentioned in the last section: scalars, spinors and vectors. scalars are 1-component objects, spinors 2-component objects and vectors 4-compontent objects. However, it is important to take note that in a field theory these objects are functions of the position and time, for example $$\Psi = \Psi(x) = \begin{pmatrix} \Psi_1(x) \\ \Psi_2(x) \end{pmatrix},$$ where er use the shorthand notation $x$ for all spatial ($x_1,x_2,x_3$) and the time coordinate ($t$).

Now this means that the action of symmetry generators on such objects do two things. On the one hand, they change the spatial and time coordinates $ x\to G x$, where $G$ denotes a generator. But on the hand, in general, a transformation of an object with several components can also mix these components. The most familiar example, is when we rotate a vector, but the same thing happens for spinors, too! (Scalars have only one component and thus, of course, nothing interesting happens.)

An example: If we look at the vector field $A_\mu= \begin{pmatrix} A_0 \\A_1 \\A_2 \\A_3 \end{pmatrix}$ from a different perspective, i.e. describe it in a rotated coordinate system it can look like $A'_\mu= \begin{pmatrix} A'_0 \\A'_1 \\A'_2 \\A'_3 \end{pmatrix}= \begin{pmatrix} A_0
-A_2
A_1 \\A_3 \end{pmatrix}$. $A'_\mu$ and $A_\mu$ describe the same field in coordinate systems that are rotated by $90^\circ$ around the z-axis relative to each other.

As already mentioned above, this means that the conserved quantity that arises in field theories can have two components. One component arises from the invariance under $ x\to G x$ and the second part from the invariance under the mixing of the components. In general, only the sum of these two parts is conserved!

Concrete

The representations of the Poincare group that act on objects with multiple components, where the components are functions, are known as field representations. They consist of the infinite-dimensional part that acts on functions (the space of functions is infinite-dimensional), and a finite-dimensional part that mixes the components. In the infinite-dimensional representation, the elements of the Poincare group are given by differential operators. For example, the generator of translations is $\partial_x$, because

$$\Phi(x) \rightarrow \Phi(x+\epsilon)= \Phi(x)+ \underbrace{\partial_x \Phi(x)}_{\text{"rate of change" along the x-axis}} \epsilon.$$

(The symbols $\partial^{\nu}$ are a shorthand notation for the partial derivative $\frac{\partial}{\partial_{\nu}}$.)

We can see here that such differential operators have exactly the effect of transforming the arguments of the functions.

Analogously, the generators of rotations and boosts can be written as

\begin{equation} \label{eq:boostrotgenDef} M_{\mu \nu}^{\mathrm{inf }}=i( x^\mu \partial^{\nu} - x^\nu \partial^{\mu}) \end{equation}

In contrast, in the finite-dimensional representation, the elements of the Poincare group are given by matrices, which have the effect that they mix the elements of objects with multiple elements.

The complete transformation is then a combination of a transformation generated by the finite-dimensional representation $M_{\mu \nu}^{\mathrm{fin }}$ and a transformation generated by the infinite-dimensional representation $M_{\mu \nu}^{\mathrm{inf }}$ of the generators: \begin{equation} \Phi_a(x) \rightarrow \left({\mathrm{e }}^{-i \frac{\omega^{\mu \nu}}{2} M_{\mu \nu}^{\mathrm{fin }}}\right)_a^b {\mathrm{e }}^{-i \frac{\omega^{\mu \nu}}{2} M_{\mu \nu}^{\mathrm{inf}}} \Phi_b(x). \end{equation} Because our matrices $M_{\mu \nu}^{\mathrm{fin }}$ are finite-dimensional and constant we can put the two exponents together \begin{equation} \Phi_a(x) \rightarrow \left({\mathrm{e }}^{-i \frac{\omega^{\mu \nu}}{2} M_{\mu \nu}}\right)_a^b \Phi_b(x) \end{equation} with $ M_{\mu \nu} = M_{\mu \nu}^{\mathrm{fin }} +M_{\mu \nu}^{\mathrm{inf }}$.

The transformation of a field $\Psi(x)$, consisting of the two parts described above can be written as

\begin{equation} \label{eq:twoparts} \delta \Phi = \epsilon_{\mu \nu} S^{\mu \nu} \Phi(x) - \frac{\partial \Phi(x)}{\partial x_\mu} \delta x_\mu , \end{equation} with the transformation parameters $\epsilon_{\mu \nu}$, the transformation operator $S^{\mu \nu}$ in the corresponding finite-dimensional representation and a conventional minus sign.

The first part describes how the components get mixed, and the second part how the argument of the functions gets transformed.

$S_{\mu \nu}$ is related to the generators of rotations by $S_i = \frac{1}{2} \epsilon_{ijk} S_{jk}$ and to the generators of boosts by $K_i = S_{0i}$. This definition of the quantity $S_{\mu \nu}$ enables us to work with the generators of rotations and boosts at the same time.

The first part is only important for rotations and boosts because translations do not lead to a mixing of the field components. For boosts, the conserved quantity will not be very enlightening, just as in the particle case, so in fact, this term will become only relevant for rotational symmetry. In addition, the first part plays no role for scalars, because these only have one component.

In the following, we only discuss the conserved quantity that we get from invariance under rotations.

Using Noether's theorem, we can derive that from the invariance under the action of the infinite-dimensional part of the transformation we get the conserved quantity.(Again, we skip here the details and only quote the final result. For the details see, for example, "Physics from Symmetry" by J. Schwichtenberg).

\begin{equation} \label{eq:consrotORBIT} L^i_{\mathrm{orbit}} = \frac{1}{2} \epsilon^{ijk} Q^{jk} = \frac{1}{2} \epsilon^{ijk} \int d^3x ( T^{k0} x^j - T^{j0} x^k), \end{equation}

This is what we usually call orbital angular momentum. For scalar fields this is the complete conserved quantity that follows from invariance under rotations.

However, for representation that act on objects with multiple components, we get an additional contribution to the conserved quantity.

\begin{equation} \delta \Phi = \epsilon_{\mu \nu} S^{\mu \nu} \Phi(x), \end{equation} where $S^{\mu \nu}$ is the appropriate finite-dimensional representation of the transformation in question. (Recall: The finite-dimensional representations are responsible for the mixing of the field components. For example, the two dimensional representation of the rotation generators: $J_i= \frac{1}{2} \sigma_i$, mix the components of Weyl spinors.)

Using Noether's theorem, we get from invariance under such transformations, the conserved quantity:

\begin{equation} L^i_{\mathrm{spin}} = \frac{1}{2} \epsilon^{ijk} \int d^3x \left( \frac{\partial \mathscr{L}}{\partial(\partial_0 \Phi)} S^{jk} \Phi(x) \right) \, , \end{equation} which is called spin.

The complete conserved quantity that follows from invariance under rotations is

\begin{align} L^i &= \frac{1}{2} \epsilon^{ijk} \int d^3x \left( \frac{\partial \mathscr{L}}{\partial(\partial_0 \Phi)} S^{jk} \Phi(x) \right. \notag \\ & \quad \left. + ( T^{k0} x^j - T^{j0} x^k) \right) \end{align} and therefore we write

\begin{equation} L^i = L^{i}_{\mathrm{spin}}+ L^{i}_{\mathrm{orbit}}. \end{equation}

The first part is something new, but needs to be similar to the usual orbital angular momentum we previously considered, because the two terms are added and appear when we consider the same invariance. The standard point of view is that the first part of this conserved quantity is some-kind of internal angular momentum. (In quantum field theory fields create and destroy particles. A spin $\frac{1}{2}$ field creates spin $\frac{1}{2}$ particles, which is an unchangeable property of an elementary particle. Hence the usage of the word "internal". Orbital angular momentum is a quantity that describes how two or more particles revolve around each other.)

One effect of a rotation is that the arguments of the components get transformed $\Phi_1(x) \to \Phi_1(x'),\Phi_2(x) \to \Phi_2(x') ,\ldots $. From the invariance under this effect on the spatial coordinates $x\to x'$, we get the part of the conserved quantitiy that we call orbital angular momentum.

The second effect is that the components of objects get mixed. From the invariance under this internal mixing of the components, we get a second part of the conserved quantity that we interpret as internal angular momentum. The standard name for this internal angular momentum is spin.

Abstract

Why is it interesting?

Using Noether's theorem for field theories, we can understand where spin comes from and why electric charge is conserved.