Why is it interesting?

Using Noether's theorem for field theories, we can understand where spin comes from.

Layman

The Noether theorem in field theories yields for each symmetry a conserved quantity that consists of two parts. One part that corresponds to invariance under the transformation of component functions, and a second part that corresponds to the invariance under the mixing of the components.

In general only the sum of these two parts is conserved.

Student

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 quantitiy.(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.

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