Numerology and Gravity

Richard Borcherds wrote a post a little while back in which he remarked that we shouldn’t take the Planck units very seriously, since Newton’s constant G doesn’t have quite the same stature as c and \hbar.

His argument went more or less as follows: The latter two seem to be quite fundamental, but from a modern point of view G (actually \frac{G^{-1}}{8\pi}) is just one of the coupling constants that appears in the effective Lagrangian for gravitational fields

L =  \lambda \sqrt{|g|} + \frac{G^{-1}}{16\pi}R\sqrt{|g|} + \mbox{sub-sub-leading terms}

where the coupling constant \lambda =\frac{\Lambda_c  G^{-1}}{16\pi} is basically the cosmological constant, normalized by 16\pi G.

So, G isn’t privileged in any way. It isn’t the coupling constant of the leading order term; a cursory analysis of the scaling dimensions of \lambda and G would lead us to believe that the cosmological constant term should be dominant. It isn’t even normalized nicely, what with that 8\pi \simeq 25. And while it seems quite sensible to work in units where \hbar = 1 and c = 1, we should be a little more cautious about about the meaning we assign to units where G = 1. What we’re actually doing is identifying the length scale where our non-renormalizable effective field theory description of gravity should break down.

[Updated: The computation that previously appeared here involved one of the classic blunders: not checking your units.  \lambda \simeq 10^{-86} \mbox{GeV}^{\bf{+}6}; that’s a +6, not a -6.  Thanks to Thomas Larsson for pointing that I’m an idiot.]

Back from Maui

If any of you were wondering where we had all gone this past week, there’s a simple answer: Maui! More than half of the contributors to this blog were far far away from the internet at “Subfactors in Maui.” This was a relentlessly laidback conference organized by Vaughan Jones (which means it was actually organized by our very own Scott M), entirely dominated by Berkeleyites; a little internet research shows that every mathematician there either received their Ph.D., did a postdoc, or is a faculty member at Berkeley.

I’m afraid we didn’t really keep up our previous standards of talk blogging, but I’ll plead lack of internet and a somewhat exotic collection of material. I can’t hope to do justice to the talks on subfactors (given by Vaughan Jones, Emily Peters, Dietmar Bisch, and Pinhas Grossman), though they were very interesting to me as someone who’s generally been faking it when it comes to subfactors (though I do REALLY want to know what arithmetically equivalent subfactors are) or Laurent Bartholdi’s talk on automatic groups and subfactors (though maybe I should, considering that I took a class on automatic groups with Laurent 4 years ago when he was at Berkeley). That leaves my talk, which was basically on material I’ve already covered, Scott M’s talk, which I’ll let him cover in his own sweet time, and the Station Q denizens Mike Freedman and Kevin Walker. Continue reading

Help David learn Quantum Field Theory II

I’ve pushed on further in my attempts to learn Quantum Field Theory. (Thank you to everyone who commented on the previous post.) I’ve picked up a second textbook, Ryder’s Quantum Field Theory, whose precision balances Zee’s intuition very well. I don’t have so many questions this time, just ideas which I am imperfectly exploring. Let me try to explain what I learned this weekend, which is how to write down a bunch of massive charged spin-zero particles interacting with an electromagnetic field.

When I first learned electro-magnetism, I thought that it was very inelegant that electrons were particles, with particular positions that change according to the Lorentz force law, while light was a field, with an intensity at every position in space that changes according to Maxwell’s equations. I tried to imagine what a field theory would look like for electrons, by imagining an infinite number of charged particles, with infinitesimal charge, all obeying the Lorentz force laws. At first, I thought I would just make a field which, when integrated over any region of space-time, would give the total charge in that region. Later, I realized that I needed to keep track of the momenta as well and imagined a vector-valued field which, integrated over any region of space-time, gave the total momentum in that region. (This history is viewed through the rose-colored classes of hindsight.) If I kept going this way, I would have invented the (charge density, current) four-vector. This field is usually called J.

Still later, I realized that this wouldn’t work either. Here is the reason. Imagine two particle beams right next to each other, with the same particle density and velocity. The particles in the two beams have the same mass, and opposite charges. Then the J-field would be zero, so we couldn’t distinguish it from just a complete absence of charge. From the perspective of Maxwell’s equations, this is true. Two parallel beams of this sort generate no electro-magnetic field. However, from the perspective of the Lorentz force equation, this is not true. If our two particle beams pass through a transverse electro-magnetic field they will be separated, one curving to the left and the other to the right. Thus, the future value of the J-field can not be predicted from knowing the present value of the J-field and knowing the electric field. At this point, I sort of gave up on the project, figuring that all you could do was to imagine a probability density on the state-space of an electron.

It turns out that there is a much nicer answer. It doesn’t even require any complicated math; it could have been a bonus chapter in the second volume of Feynmann’s lectures.

Continue reading