Lecture 18 Jan

Lecture Notes

• st20080118

Synopsis

Today we talked about the equations of motion and boundary conditions for a classical relativistic string, in an arbitrary worldsheet coordinate system.

We found that the only two boundary conditions for open strings physically consistent with the classical equations of motion are

1. Dirichlet (can be imposed on any or all spatial coordinates), where open string endpoints are stuck to particular constant values, OR
2. Free Endpoint, where open strings leak no momentum off their ends.

We then noted that static gauge, a partial fixing of worldsheet coordinate invariance, is physically simple to understand because it aligns target space time with worldsheet time.

We next introduced the notion of the s parametrization, in which the worldsheet spatial coordinate is chosen such that ds=|dX|
where X are the spatial guys. In this particular choice, we found that the Nambu-Goto action simplified a great deal, to the point where it looked a lot like a string generalization of the familiar geometric action for the point-particle. In this s parametrization and in static gauge, we found out two physically interesting facts: the open string endpoints move perpendicular to the string and at the speed of light.

String Theory Strikes Back 

Please also read an excellent article by Prof. Michael Dine of UCSC in this month’s edition of Physics Today magazine [local PDF file]. In it, Dine explains brilliantly why string theory is very relevant at the dawn of the LHC era, and he also brilliantly debunks critics of the field (like popular-book criticism emanating from the general direction of Lee Smolin of the Perimeter Institute and some blogs…). N.B: this is a must-read article for anyone interested in the relevance of string theory to physics.

P.S.: Regarding Gary’s question:

Note: Gary’s in-class observation was right – it looks like choosing Dirichlet BCs tells us that our wee D-brane can’t move. But don’t let that fool us – if we wanted to kick our D-brane a little to see how it’d react, we’d obviously need to have a mechanism of kicking the D-brane! In other words, just like if we wanted to make a point particle move (say, under an electric field), then physically we’d need the kick of energy-momentum to come from somewhere… and so adding that into the story results in (inevitably!) extra term(s) in the effective action for the whole system. The action for the whole system would then include the D-brane of interest plus whatever we’re kicking it it with. This would be just like adding the electromagnetic coupling to the geometric action for the free particle so we can study interactions of particles with EM fields.

The Nambu-Goto action we’ve used for calculation so far is for free strings living in flat spacetime. We can and will handle more complex cases later on. Worldsheet reparametrization invariance (conformal symmetry) is the fundamental quantum symmetry principle; the equations of motion are consistency conditions. If there is more than just a free string living in flat spacetime in the picture, then more terms will naturally be there in the low-energy effective action. Quantum consistency is what dictates the low-energy effective action for strings, D-branes and stuff they couple to, not inspired guessing.

Once we’ve developed [light-cone] quantization technology, we’ll see very clearly that D-branes most definitely can move; they gravitate and do other stuff besides. The deep quantum consistency of string theory is what tells us how to write down the correct equations for D-branes (in motion or whatever) and their interactions with open and closed strings.

Homework thought exercise from a student question:

Can an open string have Free-Endpoint boundary conditions on one end and Dirichlet on the other? Is there anything inconsistent about that?