A Coleman: "Spin is a concept that applies only to particles with mass"

[Kostik]

TL;DR Summary

Sidney Coleman in his lectures stated that "spin is a concept that applies only to particles with mass", because spin is the angular momentum in the rest frame. (Obviously a massless particle has no "rest frame".)

Coleman ("Quantum Field Theory Lectures of Sidney Coleman", p. 400) states:

"Spin is a concept that applies only to particles with mass, because only for a particle of non-zero mass can we Lorentz transform to its rest frame and there compute its angular momentum, which is its spin. For a massless particle, there is no rest frame, so we can't talk about the spin. We can however talk about its helicity, the component of angular momentum along the direction of motion."

Is it not possible to define / discuss spin without referring to the rest frame? In some cases, an expression for total angular momentum $\textbf{L} = \textbf{J} + \textbf{S}$ has one piece which is coordinate invariant, which can therefore be identified as the inherent angular momentum, or spin.

Why then do we routinely refer to the spin of the photon as $\pm 1$ (or $\pm \hbar$)?

 

[PeterDonis]

Coleman here is using "spin" in a different sense than the term is used when referring to a photon. Note that Coleman also refers to "helicity", which does apply to photons, and which is informally referred to as "spin" even though that is not technically correct.

 

[Demystifier]

If spin is different from helicity, is there a third word which unifies them, so that we can say that spin and helicity are different kinds of this third thing? Maybe intrinsic angular momentum? (Those are three words, but who counts. )

 

[Kostik]

TL;DR Summary: Sidney Coleman in his lectures stated that "spin is a concept that applies only to particles with mass", because spin is the angular momentum in the rest frame. (Obviously a massless particle has no "rest frame".)

Coleman ("Quantum Field Theory Lectures of Sidney Coleman", p. 400) states:

"Spin is a concept that applies only to particles with mass, because only for a particle of non-zero mass can we Lorentz transform to its rest frame and there compute its angular momentum, which is its spin. For a massless particle, there is no rest frame, so we can't talk about the spin. We can however talk about its helicity, the component of angular momentum along the direction of motion."

Is it not possible to define / discuss spin without referring to the rest frame? In some cases, an expression for total angular momentum $\textbf{L} = \textbf{J} + \textbf{S}$ has one piece which is coordinate invariant, which can therefore be identified as the inherent angular momentum, or spin.

Why then do we routinely refer to the spin of the photon as $\pm 1$ (or $\pm \hbar$)?

Coleman here is using "spin" in a different sense than the term is used when referring to a photon. Note that Coleman also refers to "helicity", which does apply to photons, and which is informally referred to as "spin" even though that is not technically correct.

Can you elaborate on how spin can be defined in a way that precludes its application to a photon? Are you using "spin" as the a.m. in the rest frame? In that case, since helicity is usually defined as the projection of spin in the direction of motion, how can the helicity of a photon be defined in the absence of a definition of spin?

 

[div_grad]

Can you elaborate on how spin can be defined in a way that precludes its application to a photon?

You do not deliberately look for the expression for its "spin", but rather you calculate just the angular momentum of the electromagnetic (EM) field. The angular momentum of the free EM field is
$$
\mathbf{J}_{em} \propto \int\mathrm{d}^3x \,\mathbf{r} \times \left(\mathbf{E}\times\mathbf{B}\right)
$$
and if you replace the field $\mathbf{B}$ using vector potential $\mathbf{A}$, then Fourier-expand $\mathbf{A}$ into plane waves, you will obtain $\mathbf{J}_{em}$ in the form of a sum of two terms, one of them being
$$
\propto \int \frac{\mathrm{d}^3k}{(2\pi)^3}\, \mathbf{k}\, \left[|a_+(\mathbf{k})|^2 - |a_-(\mathbf{k})|^2\right]
$$
where $a_\pm$ are the Fourier expansion coeffcients corresponding to right- ("+") and left-circular ("-") polarizations. This part of the EM angular mometum is position-independent, and if you really want to call it "spin" (by analogy with position-independent part of the angular momentum of, e.g., the massive electron) then you see from the above expression why you can then refer to "photon's spin" being $\pm 1$. But it is better, in the case of light, to speak of helicities becase:

(...) since helicity is usually defined as the projection of spin in the direction of motion (...)

As stated in the Coleman lecture you quoted, helicity $h$ is the projection of the (total) angular momentum $\mathbf{J}$ of the particle onto the direction of its motion (i.e., onto the unit vector $\mathbf{n} = \mathbf{p} / |\mathbf{p}|$, where $\mathbf{p}$ is the particle's momentum). Now, for a massive particle, e.g., the electron, you can indeed write $\mathbf{J}=\mathbf{L}+\mathbf{S}$ with the appropriate orbital angular momentum related to the electron's position $\mathbf{r}$ and its momentum through $\mathbf{L} = \mathbf{r} \times \mathbf{p}$. So you see that the helicity
$$
h = \mathbf{J}\cdot\mathbf{n} \propto \left(\mathbf{r}\times\mathbf{p}\right)\cdot\mathbf{p} + \mathbf{S}\cdot\mathbf{p} = \mathbf{S}\cdot\mathbf{p}
$$
reduces to the "projection of spin in the direction of motion", but this is a consequence of $\mathbf{p}\times\mathbf{p}=0$ and not of the definition of $h$ as the (solely) "spin-projection".

Similarly to this, in the case of the EM angular momentum above (note that you also can compute the momentum of the EM field, onto which $\mathbf{J}_{em}$ will then be projected) only the position-independent part of $\mathbf{J}_{em}$ will survive the computation of helicity, and you will therefore conclude that for a photon there are two possible helicities, $\pm1$ (in appropriate units).

 

[PeterDonis]

helicity is usually defined as the projection of spin in the direction of motion

For particles with mass, yes. But note that for particles with mass, helicity, defined this way, is not an invariant.

For massless particles, like photons, no; helicity is defined to be the same as chirality, which is an invariant and doesn't require a rest frame for its definition.

 

[div_grad]

For massless particles, like photons, no; helicity is defined to be the same as chirality (...)

@PeterDonis I'll take the opportunity to clarify something: Namely, I thought that there's a difference in that helicity is a physical quantity (this angular momentum component along the direction of motion) while the chirality is a mathematical concept used to denote the appropriate representation of the Lorentz group according to which a given field transforms. Is this viewpoint valid, or am I missing something more subtle behind the statement that "helicity is defined to be the same as chirality" (for massless particles)?

I mean, for example, a Weyl spinor belonging to the representation $(\frac{1}{2}, 0)$ can be said to be "left-chiral" (because the "$\frac{1}{2}$" in the label of this representation is on the left - I know it's just a mnemonic) and at the same time the particles created/annihilated by this spinor field can have either positive or negative helicities, depending on their momentum - so that there's no (?) correlation between the two concepts, in this case.

 

[PeterDonis]

helicity is a physical quantity

Not really. As I said, it's not an invariant (at least not for massive particles), and physical quantities, things you can actually measure, have to be invariants.

Note that when you actually measure a particle's spin, you pick a particular axis about which to measure it (by, for example, picking the orientation of a Stern-Gerlach magnet), and the measuring device is in some particular state of motion, so the invariant you're measuring is not just a property of the particle itself; it's a property of the interaction between the particle and the measuring device.

the chirality is a mathematical concept used to denote the appropriate representation of the Lorentz group according to which a given field transforms.

But that mathematical concept represents something physical about the particle; it's not just an abstraction. What physical thing it represents is not very intuitive, but that doesn't mean it's not a physical thing.

 

[div_grad]

(...) physical quantities, things you can actually measure, have to be invariants.

Right, helicity is obviously not Lorentz-invariant for massive particles, but in a given inertial frame it is a constant of motion (it commutes with the Hamiltonian) so I'd assume that it is still perfectly measurable quantity, no? I mean, the particle's energy or the light-absorption frequency for an atomic system are also not Lorentz-invariant, but in a given inertial frame (e.g, the "lab frame") they are still measurable quantities of practical interest - things like the Doppler shifts in atomic spectra, etc.

And if in such measurements it is the appropriate "system + measuring device" interaction that has to be invariant (rather than the "bare" system property, like helicity, on its own), then why not apply this principle here:

(...) it's not an invariant (at least not for massive particles), and physical quantities, things you can actually measure, have to be invariants.

(...) the measuring device is in some particular state of motion, so the invariant you're measuring is not just a property of the particle itself; it's a property of the interaction between the particle and the measuring device.

and say that, while helicity is not invariant (for massive particles) it is still a physical, measurable quantity, because the interaction between the measuring device probing this quantity and the physical system to which it pertains is of invariant character, and that's what matters?

But that mathematical concept represents something physical about the particle; it's not just an abstraction. What physical thing it represents is not very intuitive, but that doesn't mean it's not a physical thing.

Of course. I used the term "mathematical concept" for chirality in a similar vein that one would call the phase space a "mathematical concept" - it does not preclude that these concepts have a physical interpretation (albeit, perhaps not as direct as in the case of, e.q., the orbital angular momentum).

 

[PeterDonis]

in a given inertial frame (e.g, the "lab frame") they are still measurable quantities of practical interest - things like the Doppler shifts in atomic spectra, etc.

The physical thing here is not a quantity "in a given inertial frame", but the relative motion of the source and the measuring device. For example, in order to measure shifts in atomic spectra, you want the atom to be at rest relative to the lab. (And the fact that you can never actually do this exactly is why you get things like spectral line broadening.) Similar remarks apply to trying to measure spin for massive particles.

For massless particles, the point is that the relative motion of the particle and the measuring device doesn't matter. The helicity (chirality) is the same whether your measuring device is in a lab at CERN or in a rocket flying past Earth at close to the speed of light relative to Earth.

 

[Nugatory]

The physical thing here is not a quantity "in a given inertial frame", but the relative motion of the source and the measuring device.

It’s possible to overstate the equivalence between invariants and “physical” things. Certainly we must be aware of the distinction between frame-dependent and invariant quantities and understand when the former will mislead and the latter must be used.

But a frame-dependent quantity that can be predicted by theory, can be measured, and provides a useful way of thinking about a physical system deserves a bit more ontological respect than, for example, ether or fairy dust.

 

[PeterDonis]

a frame-dependent quantity that can be predicted by theory, can be measured,

As I've said, what is actually measured is an invariant. Many things that are informally referred to as "frame-dependent quantities" are actually invariants when you look under the hood.

For example, the energy of an object is often referred to as a frame-dependent quantity. But when you look at what is actually measured when "energy" is measured, it's an invariant: the inner product of the object's 4-momentum with the measuring device's 4-velocity. When we call this thing "frame-dependent", we're implicitly assuming that we have to use the measuring device's rest frame to calculate and predict this quantity, so if we change frames, we have to change measuring devices to one that's at rest in the new frame--but we don't. The measuring device's 4-velocity is a geometric object that is independent of any frame, and nothing requires us to do all calculations in the measuring device's rest frame, or to change measuring devices just because we decided to change frames in our math. Indeed, you don't even need to choose a frame to write down the inner product I just described; it's a tensor equation, valid in any frame.

Once we've taken considerations like the above into account, I stand by my statement that any physical quantity that can be measured is an invariant. It just takes more care than is taken in many informal discussions to be clear about exactly how the invariant in question is specified.