I forgot to mention another classic example: the Yukawa theory! In this theory spin-1/2 particles interact by exchanging a spin-0 particle. This was the very first theory of the strong nuclear force: Yukawa had protons and neutrons interact by exchanging a massive spin-0 “meson”. This force would be attractive, so it could hold nuclei together.

Yukawa proposed his theory in 1935. In 1936 experimenters looking at cosmic rays found a particle of roughly the right mass! Hurrah! But it turned out to be a spin-1/2 particle that didn’t interact strongly at all, prompting I. I. Rabi’s famous quip: “Who ordered that?”

This was the muon, the first of the ‘second generation’ of particles. The existence of three generations is still a total mystery.

Considerably later, in 1947, they found the pion, which fits the bill quite nicely. And Yukawa won the Nobel prize.

They originally called the muon the ‘mu meson’, due to this mixup, but now we say it’s not a meson at all.

The theory of protons and neutrons attracting by exchanging spin-0 pions is still a reasonable approximation in some situations, for example in nuclei and certain layers of a neutron star. And Heisenberg’s development of this theory led to the introduction of ‘isospin SU(2)’, which eventually led Yang and Mills to ‘SU(2) gauge theory’ and the Yang–Mills equations. For example, there are really three pions—positive, negative and neutral—transforming in the adjoint representation of SU(2), just like a gauge boson should.

It’s a wonderful example of how ideas that don’t quite work can still be fruitful. In fact it often feels like Nature is trying to make science easy by obeying not just a single ‘theory of everything’ but a beautifully nested set of better and better approximate theories, starting with ones that use easy math and leading up to more sophisticated ones—sort of like those computer games that have different ‘levels’.

]]>As a mathematician used to thinking of the word “odd” (in the context of spin) as referring to odd-degree components of super-representations, I was at first confused: photons are bosons, where the relevant representations live in even-degree parts, right? But then I thought, no wait, John knows this stuff backwards and forwards, he’d never get mixed up about anything like that. I.e., you really did mean odd spin (as in odd *integer*, not some half-integer)! And then realized I had no idea what to make of the assertion!

There is a kind of lame intuition I have that “bosons attract, fermions repel” in the sense that e.g. two electrons as fermions cannot occupy the same state. I wonder if there’s some twisted way in which a similar idea is at work here… but maybe I should try first to read these references.

]]>Could you explain this a bit more?

Unfortunately I don’t have an intuitive understanding of why this is true: for me it’s just a calculation that I’ve watched people do.

You can see the calculation here:

• Warren Siegel, *Fields*, page 184.

However, it may be better to start with this puzzle: “How can virtual particles be responsible for attractive forces in the first place?” There’s a nice discussion of that here:

• Matt McIrvin, Some frequently asked questions about virtual particles, Physics FAQ.

He starts as follows:

The most obvious problem with a simple, classical picture of virtual particles is that this sort of behavior can’t possibly result in attractive forces. If I throw a ball at you, the recoil pushes me back; when you catch the ball, you are pushed away from me. How can this attract us to each other? The answer lies in Heisenberg’s uncertainty principle.

Suppose that we are trying to calculate the probability (or, actually, the probability amplitude) that some amount of momentum, p, gets transferred between a couple of particles that are fairly well- localized. The uncertainty principle says that definite momentum is associated with a huge uncertainty in position. A virtual particle with momentum p corresponds to a plane wave filling all of space, with no definite position at all. It doesn’t matter which way the momentum points; that just determines how the wavefronts are oriented. Since the wave is everywhere, the photon can be created by one particle and absorbed by the other, no matter where they are. If the momentum transferred by the wave points in the direction from the receiving particle to the emitting one, the effect is that of an attractive force.

The moral is that the lines in a Feynman diagram are not to be interpreted literally as the paths of classical particles. Usually, in fact, this interpretation applies to an even lesser extent than in my example, since in most Feynman diagrams the incoming and outgoing particles are not very well localized; they’re supposed to be plane waves too.

The uncertainty principle opens up the possibility that a virtual photon could impart a momentum that corresponds to an attractive force as well as to a repulsive one. But you may well ask what makes the force repulsive for like charges and attractive for opposite charges! Does the virtual photon know what kind of particle it’s going to hit?

It’s hard even for particle physicists to see this using the Feynman diagram rules of QED, because they’re usually formulated in a manner designed to answer a completely different question: that of the probability of particles in plane-wave states scattering off of each other at various angles. Here, though, we want to understand what nudges a couple of particles that are just sitting around some distance apartâ€”to explain the experiment you may have done in high school, in which charged balls of aluminum foil repel each other when hanging from strings. We want to do this using virtual particles. It can be done.

And then he does it. He does it using lots of simplifying approximations that might be disturbing if one isn’t used to computing these things. But those are all okay. The unfortunate part is this: he doesn’t highlight the role of the photon’s spin. So, you can’t read his argument and see what would change if that spin were even rather than odd.

*However*, if you read Matt’s story you’ll see that in the end, it all depends on some constructive and/or destructive interference, which in turn depends on some signs. So, you can probably at least *imagine* that if we changed the spin of the virtual particle from odd to even, some signs would change and we’d get a force where ‘like charges attract’ (namely gravity, with ‘charges’ being masses), rather than one where like charges repel.

If you take Matt’s essay, and Warren Siegel’s calculation, and look at them together for a long time, some intuitive explanation should emerge. But I haven’t done this yet. Siegel tends to do calculations very rapidly, which is why his book is only 885 pages long. So, the crucial factor of where is the spin of the particle carrying the force, seems to pop out as if by magic!

Feynman would be able to give a nice verbal explanation.

]]>Could you explain this a bit more?

]]>This point of view is quite fruitful; in particular, it implies all of tensor calculus and differential geometry. Things like tensor densities, exterior and covariant derivatives, etc., all have a natural description in terms of representations of the diffeomorphism group, simply because these concepts make sense; by definition, things that make sense transform as representations, or as morphisms between representations, under the group of coordinate transformations.

In one dimension we can say more, because the diffeomorphism algebra is also known as the centerless Virasoro algebra. In this case we encounter the standard problems with general-covariant theories – e.g., the absense of local observables – but this defect can be remedied by removing the prefix “centerless”. So my idea was to generalize the Virasoro extension to the diffeomorphism algebra in higher dimension, and in particular to 4D spacetime. I did find (together with a number of mathematicians) the relevant extensions and off-shell representations, but then I failed to make sense of on-shell representations that would be needed to make the connetion to physics.

Anyway, I recently started my own blog (which I am already neglecting), and wrote up a couple of posts on the subject, starting with this.

]]>We can linearize Einstein gravity, quantize that, and get a theory of massless spin-2 particles: gravitons. These still behave differently than the massless spin-1 particles in QED, namely photons.

By the way, the fact that photons have odd spin while gravitons have even spin can be seen as the reason why like charges repel while like masses attract!

(We can try to make like charges attract by taking the fine structure constant to be negative, but as explained this gives a very nasty theory, which is still not quantum gravity.)

Still, there are interesting parallels between gravity and electromagnetism. In particular, we can take Einstein’s theory of gravity, linearize it, and then chop the gravitational field into ‘gravitoelectric’ and ‘gravitomagnetic’ parts, that are somewhat analogous to the electric and magnetic field. These fields obey equations that look a bit like Maxwell’s equations! Check it out:

• Gravitoelectromagnetic field equations, Wikipedia.

However, be careful: the talk page shows that the Wikipedia editors are pretty confused about what precise assumptions are used to derive the gravitoelectromagnetic field equations from general relativity! Certainly you need to linearize about a flat solution, but you may also need more.

I think this book would help clarify things:

• Ignazio Ciufolini and John Archibald Wheeler, *Gravitation and Inertia*, Princeton U. Press, Princeton, 1995.

If I use the Quantum Electrodynamic to evaluate dynamic of the particles, then could the gravitational effects be calculated using the Quantum Electrodynamic? If it is all true for macroscopic object, it could be true for quantum objects? Or is this too simple? ]]>