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### The spin of gauge bosons: vector particles

Particles have an inherent spin. We explored the case of fermions (“spin-1/2”) in a recent post on helicity and chirality. Now we’ll extend this to the case of vector (“spin-1”) particles which describe gauge bosons—force particles.

By now regular US LHC readers are probably familiar with the idea that there are two kinds of particles in nature: fermions (matter particles) and bosons (force particles). The matter particles are the ‘nouns’ of the Standard Model. The ‘verbs’ are the bosons which mediate forces between these particles. The Standard Model bosons are the photon, gluon, W, Z, and the Higgs. The first four (the gauge bosons of the fundamental forces) are what we call vector particles because of the way they spin.

## An arrow that represents spin

You might remember the usual high school definition of a vector: an object that has a direction and a magnitude. More colloquially, it’s something that you can draw as an arrow. Great. What does this have to do with force particles?

In our recent investigation of spin-1/2 fermions, the punchline was that chiral (massless) fermions either spin clockwise or counter-clockwise relative to their direction of motion. We can convert this into an arrow by identifying the spin axis. Take your right hand and wrap your fingers around the direction of rotation. The direction of your thumb is an arrow that identifies the helicity of the fermion, it is a ‘spin vector.’ In the following cartoon, the gray arrows represent the direction of motion (right) and the big colored arrows give the spin vector.

You can see that a particle has either spin up (red: spin points in the same direction as motion) or spin down (blue: spin points in the opposite direction as motion). It should not surprise you that we can write down a two-component mathematical object that describes a particle. Such an object is called a spinor, but it’s really just a special kind of vector. In can be represented this way:

ψ = ( spin up , spin down )

As you can see, there’s one slot that contains information about the particle when it is spin up and another slot that contains information about the particle when it is spin down. It’s really just a list with two entries.

Don’t panic! We’re not going to do any actual math in this post, but it will be instructive—and relatively painless—to see what the mathematical objects look like. This is the difference between taking a look at the cockpit of a jet versus actually flying it.

All you have to appreciate at this point is that we’ve described fermions (spin-1/2 particles) in terms of an arrow that determine its spin. Further, we can describe this object as a two-component ‘spinor.’

For experts: a spinor is a vector (“fundamental representation”) of the group SL(2,C), which is the universal cover of the Lorentz group. The point here is that we’re looking at projective representations of the Lorentz group (quantum mechanics says that we’re allowed to transform up to a phase). The existence of a projective representation of a group is closely tied to its topology (whether or not it is simply connected); the Lorentz group is not simply connected, it is doubly connected. The objects with projective phase -1 (i.e. that pick up a minus sign after a 360 degree rotation) are precisely the half-integer spinor representations, i.e. the fermions.

## Relativity and spin

Why did we bother writing the spinor as two components? Why not just work with one component at a time: we pick up a fermion and if it’s spin up we use one object and if it’s spin down we use another.

This, however, doesn’t work. To see why, we can imagine what happens if we take the same particle but change the observer. You can imagine driving next to a spin-up particle on the freeway, and then accelerating past it. Relative to you, the particle reverses its direction of motion so that it becomes a spin-down particle.

What does this mean? In order to account for relativity (different observers see different things) we must describe the particle simultaneously in terms of being spin-up and spin-down. To describe this effect mathematically, we would perform a transformation on the spinor which changes the spin up component into the spin down component.

Remark: I’m cheating a little because I’m implicitly referring to a massive fermion while referring to the two-component spinor of a massless fermion. Experts can imagine that I’m referring to a Majorana fermion, non-experts can ignore this because the punchline is the same and there’s not much to be gained by being more rigorous at this stage.

In fact, to a mathematician, this is the whole point of constructing vectors: they’re things which know how to transform properly when you rotate them. In this way they are intimately linked to the symmetries of spacetime: we should know how particles behave when we grab them and rotate them.

## Spin-1 (vector) particles

Now that we’ve reviewed spin-1/2 (fermions), let’s move on to spin-1: these are the vector particles and include the gauge bosons of the Standard Model. Unlike the spin-1/2 particles, whose spin arrows must be parallel to the direction of motion, vector particles can have their spin point in any direction. (This is subject to some constraints that we’ll get to below.) We know how to write arrows in three dimensions: you just write down the coordinates of the arrow tip:

3D arrow = (x-component, y-component, z-component)

When we take into account special relativity, however, we must work instead in four dimensional spacetime, i.e. we need a vector with four components (sometimes called a four-vector, see Brian’s recent post). The reason for this is in addition to rotating our vector, we can also boost the observer—this is precisely what we did in the example above where we drove past a particle on the freeway—so that we need to be able to include the length contraction and time dilation effects that occur in special relativity. Heuristically, these are rotations into the time direction.

So now we’ve defined vector particles to be those whose spin can be described by an arrow pointing in four dimensions. A photon, for example, can thus be represented as:

Aμ = (A0, A1, A2, A3)

Here we’ve used the standard convention of labeling the x, y, and z directions by 1, 2, and 3. The A0 corresponds to the component of the spin in the time direction. What does this all mean? The (spin) vector associated with a spin-1 particle has a more common name: the polarization of the particle.

You’ve probably heard of polarized light: the electric (and hence also the magnetic) field is fixed to oscillate along only one axis; this is the basis for polarized sunglasses. Here’s a heuristic drawing of electromagnetic radiation from a dipole (from Wikipedia, CC-BY-SA license):

The polarization of a photon refers to the same idea. As mentioned in Brian’s post, the electric and magnetic fields are given by derivatives of the vector potential A. This vector potential is exactly the same thing that we have specified above; in a sense, a photon is a quantum of the vector potential.

## Four vectors are too big

Now we get to a very important point: we’ve argued based on spacetime symmetry that we should be using these four-component vectors to describe particles like photons. Unfortunately, it turns out that four components are too many! In other words, there are some photon polarizations that we could write down which are not physical!

Here we’ll describe one reason why this is true; we will again appeal to special relativity. One of the tenets of special relativity is that you cannot travel faster than the speed of light. Further, we know that photons are massless and thus travel at exactly the speed of light. Now consider a photon with is spinning in the same direction as its motion (i.e. the spin vector is perpendicular to the page):

In this case the bottom part of the photon (blue) is moving opposite the direction of motion and so travels slightly slower than the speed of light. On the other hand, the top part of the photon is moving with the photon and thus would be moving faster than the speed of light!

This is a big no-no, and so we cannot have any photons polarized in this way. Our four-component vector contains more information than the physical photon. Or more accurately: being able to write down our theory in a way that manifestly respects spacetime symmetry comes at the cost of introducing extra, non-physical degrees of freedom in how we describe some of our particles.

(If we removed this degree of freedom and worked with three-component vectors, then our mathematical formalism doesn’t have enough room to describe how the particle behaves under rotations and boosts.)

Fortunately, when we put four-component photons through the machinery of quantum field theory, we automatically get rid of these unphysical polarizations. (Quantum field theory is really just quantum mechanics that knows about special relativity.)

## Gauge invariance: four vectors are still too big

Now I’d like to introduce one of the key ideas of particle physics. It turns out that even after removing the unphysical ‘faster than light’ polarization of the photon, we still have too many degrees of freedom. A massless particle only has two polarizations: spin-up or spin-down. Thus our photon still has one extra degree of freedom!

The resolution to this problem is incredibly subtle: some of the polarizations that we could write down using a four-vector are physically identical. I don’t just mean that they give the same numbers when you do the math, I mean that they literally describe the same physical state. In other words, there is a redundancy in this four-vector description of particles! Just as the case of the unphysical polarization above, this redundancy is the cost of writing things in a way which manifestly respects spacetime symmetry. This redundancy is called gauge invariance.

Gauge invariance is a big topic that deserves its own post—I’m still thinking of a good way to present it—but the “gauge” refers to the same thing in term “gauge boson.” This gauge invariance (redundancy in our description of physics) is intimately linked to the fundamental forces of our theory.

Remark, massive particles: Unlike the massless photon, which has two polarizations, the W and Z bosons have three polarizations. Heuristically the third polarization corresponds to the particle spinning in the direction of motion which wasn’t allowed for massles particles that travel at the speed of light. It is still true, however, that there is still a gauge redundancy in the four-component description for the thee-polarization massive gauge bosons.
For experts: at this point, I should probably mention that the mathematical object which really describes gauge bosons aren’t vectors, but rather co-vectors, or (one-)forms. One way to see this is that these are objects that get integrated over in the action. The distinction is mostly pedantic, but a lot of the power of differential geometry and topology is manifested when one treats gauge theory in its ‘natural’ language of fiber bundles. For more prosaic goals, we can write down Maxwell’s equations in an even more compact form: d*F = j. (Even more compact than Brian’s notation! 🙂 )

## Wigner’s classification

Let me take a step back to address the ‘big picture.’ In this post I’ve tried to give a hint of a classification of “irreducible [unitarity] representations of the Poincaré group” by Hungarian mathematical physicist Eugene Wigner in the late 1930s.

At the heart of this program is a definition of what we really mean by ‘particle.’ A particle is something with transforms in a definite way under the symmeties of spacetime, which we call the Poincaré group. Wigner developed a systematic way to write down all of the ‘representations’ of the Poincaré group that describe quantum particles; these representations are what we mean by spin-1, spin-1/2, etc.

In addition to these two examples, there are fields which do nothing under spacetime symmetries: these are the spin-0 scalar fields, such as the Higgs boson. If we treated gravity quantum mechanically, then the graviton would be a spin-2 [antisymmetric] tensor field. If nature is supersymmetric, then the graviton would also have a spin-3/2 gravitino partner. Each of these different spin fields is represented by a mathematical object with different numbers of components that mix into one another when you do a spacetime transformation (e.g. rotations, boosts).

In principle one can construct higher spin fields, e.g. spin-3, but there are good reasons to believe that such particles would not be manifested in nature. These reasons basically say that those particles wouldn’t be able to interact with any of the lower-spin particles (there’s no “conserved current” to which they may couple).

Next time: there are a few other physics (and some non-physics) topics that I’d like to blog about in the near future, but I will eventually get back to this burning question about the meaning of gauge symmetry. From there we can then talk about electroweak symmetry breaking, is the main reason why we need the Higgs boson (or something like it) in nature. (For those who have been wondering why I haven’t been writing about the Higgs—this is why! We need to go over more background to do things properly.)

• Sjoerd

Hi Flip!
Thanks for your enormously interesting series of posts! I really enjoyed reading this one, but I have one question concerning the “forbidden” spin mode of a photon: since a photon is an excitation of a field, it wouldn’t exactly have physical dimensions, right? Therefore, a rotation perpendicular to its direction of motion wouldn’t mean that one “side” of the photon would go faster than c, if I understand it correctly.
I’m sure that you sacrificed some technical details to make the explanation more available to a general public, but could you explain where this argument comes from in a bit more detail?
Cheers, Sjoerd

• Hello Sjoerd, before I attempt to explain further, I should probably emphasize again that this pops out automatically when working with the full machinery of quantum field theory. There are a few different ways (with varying levels of technical detail) of trying to attach words to this.

In fact, perhaps one way of looking at it is purely classically: we know from Maxwell’s electromagnetism that you can never have an electromagnetic wave in vacumm where the electric (or magnetic) field has a component along the direction of motion. This is precisely because of the relativistic argument above. This restricts the physical components of the vector potential, and hence the physical components of the photon.

A slightly more “quantum” way of rephrasing the above argument is that the electric and magnetic fields are somewhat ‘nonlocal’ in the sense that they are spacetime derivatives of the vector potential. Derivatives are nonlocal in the sense that they measure the difference of a field in the limit of infinitesimal sampling distance, and so even though the photon is ultimately pointlike, the electric field is not quite pointlike. (This argument is far from being rigorous and I’m not sure if I trust it myself.) Perhaps a very hand-wavy way of saying this is that a photon of finite energy has some associated compton wavelength which gives a characteristic size?

I’m not sure if this was a satisfying answer. I’ll try to think more carefully of a better one. Thanks for the question!

compton

• David George

Dear Flip,

This is the most informative post that I have seen on this subject (and I have been reading about it for quite a long time). I have tried (unsuccessfully) to figure out the motions of the photon’s electric and magnetic components from the illustration, and came to the (apparently erroneous) conclusion you note: if the “corkscrew” is not perfect, one component must be moving faster than c, and the other slower than c, at any particular instant. But even if the “corkscrew” is perfect, are there not two constantly changing “vectors” that are moving at some angle to the direction of motion? Wouldn’t the velocity along those constantly changing vectors be faster than the velocity along the direction of motion?

In a slightly different vein, I have read that the wave function is “an abstraction modelled on the slight pressure variations in the air that occur when sound passes” (Robert Laughlin, “A Different Universe”, p. 55-56). That description is more illustrative (to me) than the wikipedia pictures. However it is difficult to reconcile that description with the wikipedia picture. In fact the whole scenario associated with the “disturbance in the field” is pretty impossible to picture (for me anyway), unless it is a pressure wave. How would you respond to the description by Laughlin?

• Hi David, thanks for the comment. If I understand your question correctly, you’re saying that even if the electric and magnetic fields are perpendicular to the photon’s direction of motion, from the point of view of any observer the fields should have some component along the direction of motion because their vectors should look something like “transverse part + motion of the photon.” I think if one works this out carefully using special relativity, one *should* find that there’s no component along the direction of motion.

(As a simplified ‘homework’ problem, imagine someone in a train throwing a ball in the forward direction. Now crank through the algebra of special relativity to see what this looks like to an observer watching the train go by, and take the limit where the train’s velocity goes to the speed of light.)

I haven’t done the math (though it would really make a great homework problem), but my intuition is that since the photon is moving at the speed of light, the relativistic length contraction effects will force the electric and magnetic field vectors to lie on a plane perpendicular to its motion.

Regarding Laughlin’s book—this is a good way of visualizing scalar particles (like the Higgs). What the analogy doesn’t capture are fields with higher spin, since air pressure doesn’t have any intrinsic “arrow” associated with it. A good way of visualizing the wave function (or field) of a vector particle would be a map of wind velocities—at every point on the map there’s an arrow that represents the wind direction and speed at that point. I have not heard of any satisfactory visualization of a spin-1/2 field. 🙂

Best,
Flip

Oh, very good! I can vaguely form attachments between sundry phys&math I have picked up earlier.

As something I momentarily stumbled on was the difference between spin direction and spinning direction in the photon figure. I figure as a non-native english speaker my language is still ‘too flat’.

But as always a good description brings more questions than it answers. Why are matter particles (fermions) non-vectorial? Interestingly here, fields break classical intuition and moves momentum from association with matter to fields (specifically those that mediate interactions). But that is a “half explanation” why it can be so, not why it is so or has to be so.

And, bringing in a rephrasing of chirality of spin as the inverse number of turns needed to bring the phase back to itself, why would nature be unable of “larger degrees” of symmetry? (I.e. a spin-3 particles phase rotated 1/3 turn would get the same value unless I’m mistaken. 6 times as many rotations as fermions, who needs 2 turns. But such a field has no “conserved current” I take it, there is no such particle symmetry.) Interesting.

• Hello Torbjörn! Thanks for the questions. I’m not sure if I can come up with satisfactory answers. Here’s an attempt:

1. Your first question is really: why are particles with half-integer spin fermions, while particles with integer spins are bosons. Fermions are matter particles, by which I mean they obey the Pauli exclusion principle. Bosons are force particles, by which I mean they do not obey the Pauli exclusion principle. You can have arbitrarily many bosons in the same quantum state, but only one fermion per quantum state. This is known as the Spin Statistics Theorem, and unfortunately I don’t know a good proof of the top of my head. It’s certainly true for known particles, and I believe rigorous proofs exist, but I am not the right person to try to extract the main idea of the proof. It seems to rely on the very important minus sign when you rotate a half-integer spin particle by 360 degrees… I suppose one just has to relate this rotation to properties of the wavefunction that you know based on the Pauli exclusion principle.

2. I think the conserved current argument is the main reason that people don’t think about these fields. (There are other issues regarding “renormalizability,” but that is perhaps a more formal objection.) The argument isn’t terribly deep, but it does require a little more mathematical machinery than I wanted to bring up on this blog. The main idea is that any local interaction is encoded in a Lagrangian which is a scalar object (or, depending on your preferred level of rigor, a four-form). This means that all of the Lorentz indices have to be contracted. If you have a spin-3 field, there are no objects you can construct from the Standard Model (or Standard Model + GR) that could absorb the indices of the spin-3 field. (You can probably cook up some counter-arguments at this level, but then we just end up going deeper down the rabbit hole and I’ll quickly reach the point where I’m no longer qualified to say anything.)

I should probably mention that higher ‘spin’ particles *can* be realized in nature as bound states, mesons and atoms are good examples. Atoms can have very high angular momentum, but this comes from the orbit of different parts of the bound state, not from ‘intrinsic’ spin angular momentum.

Cheers,
Flip

• Wouter

hi Flip,
I admire your willingness to communicate with those not fully schooled in the art. Too few specialists bother to do that. Thanks.
Your references to “too many degrees of freedom” in a four-vector reminds me of an other eye-opener: describing 3D-rotations with quaternions. This avoids the gimbal lock problem inherent to Euler-angle rotations. Any similarities, or just in my head?
Another “waw-effect” is the mysterious relation between “double groups” with identity = 4 pi rotation versus fermions-alias-nouns, and then “normal groups” with identity= 2 pi rotation versus bosons-alias-verbs. Very cute!
How surprising that 2pi vs. 4pi rotation can actually be demonstrated by putting an object on the palm of your hand, and swinging it round over your head (through 2 pi) to end up with your arm twisted along its axix; and then rotating further (aside, not above the body) through again 2 pi, to arrive again at the starting position. Even if it’s not the ‘real’ (sic) mathematical 4 pi rotation, it’s still cool.

Wouter.

• Hi Wouter, thanks! There is, indeed a connection to quaternions, though it’s not obvious to me if there’s any ‘deep’ significance. In jargon: the generators of SL(2,C) obey the same algebra as quaternions, and these are the things which act on spinors when you perform a Lorentz transformation. In less jargon: there are indeed quaternionic objects in the mathematical expressions we use.

Regarding your example: yes! There’s a math textbook somewhere that has a sequence of photographs demonstrating this, though I think it is most effective when a lecturer pulls it off with a full glass of water. 🙂

Thanks for the comment,
Flip

• Lei Zhang

Hi Flip!

Thanks for your nice blog. I have two naive questions.

I have been confused with the spin property of photos for a long time.

In classical physics, the polarisation of the photo(or EM wave) is visualised by the direction of Electric(E) and Magnetic(B) fields as you showed in this article. But in QFT picture, the polarisation of the photo is described by 4-vector potential–A. And A is also regarded as the spin of the photo.

1) My first question is that what’s the connection between the polarisation definition of E&B with A? Can we still use the classical picture to understand the polarisation of the photo?

2) My second question is that: the high energy photos can converted to electron-positron (e-e+) pairs in the Coulomb field. I understood that the decayed e-e+ pairs can be used to measure the polarisation of the photo.
My I ask how the polarisation (or spin A) defined in QFT characterise the decayed e-e+? Can we still use E and B to understand it?

Looking forward to you explanation.

Cheers.

Lei

• David George

Thank you Flip, the analogy with length contraction (to total flatness at c?) in SR does give a picture of the “plane wave” description. (I have read the “light clock” derivation of gamma, but my last physics class was in junior high school fifty years ago, so sometimes I feel like an imposter among “real” physicists.) Regarding the visualization via a map of wind velocities, could that map be treated as if it had been created out of an earlier map of changes in “horizontal air pressure” (i.e. wind?), so that a wave could appear out of a sequence of pressure changes over time? I realize analogies have a limit, but it seems something like that could be done.