✎ Spin [The Physics Travel Guide]

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====== Spin ======

<tabbox Why is it interesting?> 
<blockquote>According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field.
<cite> [[http://aapt.scitation.org/doi/abs/10.1119/1.14580|What is spin?]] by Hans C. Ohanian </cite></blockquote>

**Important Related Concepts:**

* [[advanced_notions:helicity|]]
  * [[advanced_notions:chirality|]]
  * [[advanced_tools:spinors]]

* https://www.scientificamerican.com/article/what-exactly-is-the-spin/
  * https://www.forbes.com/sites/startswithabang/2017/11/21/spin-the-quantum-property-that-nature-shouldnt-possess/#20a3df076349
<tabbox Student> 
Spin is a quantum number like mass or like the electric charge.

We will see in the next section that spin has exactly the same origin as the other quantum numbers and is therefore not as strange as most people believe it to be. Quantum numbers are what we use to label elementary particles and their values are crucial for the behaviour of particles in experiment.

Spin is no different and one of the most important quantum numbers.

In fact, it is so important that is responsible for the definition of the most important categories of elementary particles:

* Particles with integer spin ($0,1,\ldots$) are called **bosons** and are responsible for the fundamental interactions. Examples are the photon, which is responsible for electromagnetic interactions or the gluons which are responsible for the strong interactions. 
  * Particles with half-integer spin ($\frac{1}{2}$) are responsible for matter and are called **fermions**. Examples are electrons and quarks, which are the constituents of atoms.

The reasons for these completely different roles in nature can be understood by taking a close look at the fundamental spacetime symmetry: the Poincare group.

The Poincare group is the set of all transformations that leave the speed of light invariant. The invariance of the speed of light is the basis of Einstein's special relativity and the Poincare group is the symmetry of special relativity.

In physics, we are interested in what a group actually does. In mathematical terms such investigations are carries out by doing "representation theory". Representation theory is a subfield of group theory and deals with how groups act on different mathematical spaces.

Therefore, as physicist, whenever we encounter an important symmetry as one of the first steps we start to do representation theory.

The details of the representation theory of the Poincare group lie beyond the scope of this article, but the final result of this investigation is crucial to understand the notion spin.

This final crucial result is that the representations of the Poincare group can be labelled by two numbers, two quantum numbers:

* Mass
  * Spin

While the mass label can take on arbitrary values, for spin only half-integer and integer values are allowed. At this stage, spin is merely an abstract label for representations and it is not clear how it relates to anything in the real world.

The simplest representations of the Poincare group are the representations that act on the smallest spaces. In the following the mass label is not important for the following discussion and therefore we neglect it.

The three simplest representations of the Poincare group are called:

* the spin $0$ representation,
  * the spin $\frac{1}{2}$ representation and
  * the spin $1$ representation.

Using the tools of representation theory one can derive that Poincare group acts trivially in the spin $0$ representation, which means it does nothing at all. In other words, in the spin $0$ representation all transformations of the Poincare group are simply representation by the number $1$. The objects that the spin $0$ representation acts are called \textbf{scalars}.

<note tip>Scalars are what we use in physics to describe particles and fields with spin $0$. 
</note>

In the spin $\frac{1}{2}$ representation the Poincare group is represented by $2 \times 2$ matrices. The objects these matrices act on are, of course, $2$ component objects and are called \textbf{Weyl spinors}.

<note tip>Weyl spinors are what we use to describe particles and fields with spin $\frac{1}{2}$.
</note>

In the spin $1$ representation the Poincare group is represented by $4 \times 4$ matrices. The objects these matrices act on are $4$ component objects and are called \textbf{vectors}.

<note tip>Vectors are what we use to describe particles and fields with spin $1$.
</note>

As already mentioned above, at this stage spin is just an abstract label for representations of the Poincare group. (To be a bit more precise: what we really do is derive representations of the Lie algebra of the Poincare group. These representations aren't necessarily all representations of the Poincare group, but some only of the double cover of the Poincare group).

However, we will discuss in the next section another occurrence of "spin" that gives a first hint how we can interpret this notion and how it is related to all other quantum numbers. At this point it is not clear how the object that we call "spin" in the next section is related to what we called spin here in this section.

We will put these two puzzle pieces together in the third section and then understand that both things that we named spin are exactly the same thing. In addition, we are then able to discuss how one can actually measure and observe spin.

**The origin of Spin**

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\footnote{In this sense, the usual wave function of quantum mechanics $\psi(x)$ is also a "field".} $\phi(x), \Psi(x), A_\mu(x)$ to describe the dynamics of systems. 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!

To summarize:

<note tip>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 compoents.

In general only the sum of these two parts is conserved.</note>

----

**Recommended Resources:**

* A brilliant discussion of spin, involving real measuring devices, can be found in the Feynman Lectures Vol. 3. 
  * http://math.ucr.edu/home/baez//spin/spin.html
  * For the origin of spin, see http://rickbradford.co.uk/NoethersTheorem.pdf
  * [[http://math.ucr.edu/home/baez/spin/spin.html|Spin]] by Michael Weiss

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Example1#

<--

--> Example2:#

<--

--> What exactly is spin?#

See https://www.scientificamerican.com/article/what-exactly-is-the-spin/

and[[http://people.westminstercollege.edu/faculty/ccline/courses/phys425/AJP_54(6)_p500.pdf| What is spin?]] by HC Ohanian

<--

--> Why does spin align with momentum at high velocity?#

See "Student friendly Quantum Field Theory" by Klauber page 95 and page 99. The illustrations there elucidate this point perfectly.

<blockquote>Note:  In classical relativity, one can show that for a spinning object, as v→c, the rotation spin axis approaches alignment with the momentum vector.  This can be visualized as due to the Lorentz-Fitzgerald shortening of the direction parallel to the momentum vector direction, as v→c.  Imagine a rotating wheel with axis not aligned at low speed to the velocity vector with the dimension in the velocity direction shrinking to zero as speed increases.  The plane of the wheel effectively rotates into the plane perpendicular to velocity.  So, any particle traveling with speed c would be in a pure helicity state.  That is what we have shown quantum mechanically in (12)</blockquote>

<--

--> Spin vs. Helicity vs. Chirality#

<blockquote>Whether spin, helicity or chirality is important depends on the physical question you are
interested in. For free massless [[advanced_tools:spinors]], the spin eigenstates are also helicity eigenstates
and chirality eigenstates. In other words, the Hamiltonian for the massless Dirac equation
commutes with the operators for chirality, γ5, helicity, S⃗·p⃗, and the spin operators, S⃗. The E
QED interaction ψ ̄A/ψ = ψ ̄LA/ψL + ψ ̄RA/ψR is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless

QED interaction ψ ̄A/ψ = ψ ̄LA/ψL + ψ ̄RA/ψR is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless

In the massive case, it is also possible to take the non-relativistic limit. Then it is often better to talk about spin, the vector. Projecting on the direction of motion does not make so much sense when the particle is nearly at rest, or in a gas, say, when its direction of motion is constantly changing. The QED interactions do not preserve spin, however; only a strong magnetic field can flip an electron’s spin. So, as long as magnetic fields are weak, spin is a good quantum number. That is why spin is used in quantum mechanics.

In QED, we hardly ever talk about chirality. The word is basically reserved for chiral theories, which are theories that are not symmetric under L ↔ R, such as the theory of the weak interactions. We talk very often about helicity. In the high-energy limit, helicity is often used interchangeably with chirality. As a slight abuse of terminology, we say ψL and ψR are helicity eigenstates. In the non-relativistic limit, we use helicity for photons and spin (the vector) for spinors. Helicity eigenstates for photons are circularly polarized light.

<cite>Quantum Field Theory by Schwartz</cite></blockquote>
 
<--

-->Why do we not have spin greater than 2?#

See https://physics.stackexchange.com/questions/14932/why-do-we-not-have-spin-greater-than-2/15164#15164
<--
<tabbox History>

For summary see http://www.brannenworks.com/plavchan_feynmancheckerboard.pdf and http://www.fhi-berlin.mpg.de/mp/friedrich/PDFs/ptsg.pdf.

For some additional nice historical perspective, see "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics" by B. Friedrich
</tabbox>

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