What Connections Are There Between Number Theory And Partial Differential Equations?

What does it look like to solve an undergraduate physics problem with string theory?

• How do you get from the equations of string theory and what they describe to something like the static electric field of a charged rod? I would love to see someone show which limits you take and how it stops being quantum mechanical as you get Maxwell's equations, go from 1 string to a large number of electrons (strings still or not? I don't know.), etc.

• Answer:

  The type of questions that string theory tries to answer are very different in nature than the ones you'd look at in intro physics. Let's look at the problem that you mention - that of the electric field due to a charged rod. In order to solve this problem in intro physics you need to make some assumptions. First of all you must assume that there is such a thing as charge and that this actually produces something that we call an electric field. These are not obvious in any way and in order to understand where these concepts come from, one must study electromagnetism in great detail. In particular one writes down something called the action of EM. This is essentially a function that associates a real number with each possible configuration of the EM field. For example, if you set the EM field to zero identically everywhere in space (and time), you find that this number is zero. If instead you put in a point charge and set the electric field to its usual value you get some nonzero number. Now, as you vary the EM field, the action changes its value and it's by minimizing this number that one finds the particular field configurations that can occur in nature (there are infinitely many minima). To be complete, the action for the EM field is [math]S = -\frac{1}{4}\int d^4 x F_{\mu\nu}F^{\mu \nu}[/math] Now, one may ask where this particular action comes from to begin with. Why does it take on the mathematical form that it does? These are deep questions and at first it seems like there are many other actions that one could write down that would be perfectly fine too. In some sense it's the goal of theoretical physics in general to understand why the formulas look the way they do. In the case of EM it turns out that there's a deep symmetry at play called gauge symmetry (in fact, to be precise it's not a symmetry at all but rather a redundancy in the mathematical description but that's not super important at this point). If you demand that the action of EM respects this symmetry, it turns out that EM is essentially unique! This is a major step forward in understanding why EM behaves the way it does, but a natural question still remains: why should we have this gauge symmetry to begin with? This is where quantum mechanics comes in. As it turns out, on a microscopic scale, the EM field is actually made up of tiny particles called photons. The step from the description in terms of electric and magnetic fields to that of photons is called quantization and the resulting theory is called quantum electrodynamics and is an example of a quantum field theory. If one starts with the action above and asks what kind of properties these photons must have, one finds that they must be massless and that they must be spinning around their own axis by a certain amount (spin 1). An amazing result is that the reverse argument holds too! In other words if there are particles that are massless and spin about their axis by this same amount, they MUST give rise to electromagnetism!! More precisely they must give rise to a theory with a gauge symmetry but that's sort of beside the point. So, to recap we have now learned that the assumptions underlying the intro physics solution to the charged rod problem can be explained by the simple assumption that there exists a massless spin 1 particle. This is a huge simplification! Instead of these arbitrary looking equations that you're presented with in intro physics we have boiled it down to the existence of a spin 1 massless particle! Now, there's of course a natural question to ask: why should there be such a thing as a massless spin 1 particle? This is where well established physics sort of ends and where string theory begins. String theory makes the assumption that the only thing that really exists are these tiny vibrating strings and that depending on how they vibrate, they effectively take on different properties such as mass and spin. By making what seems to be an even simpler assumption about nature, string theory actually allows you to derive the existence of this kind of particle. More precisely there is a particular way that a string can vibrate that makes it take on zero mass and spin 1. So, as you see the point of string theory is not to directly study the kinds of problems involving charged rods etc but rather to explain where the assumptions underlying the physics of those problems come from. You can read more about these topics here if you'd like: Action: http://en.wikipedia.org/wiki/Action_(physics) Quantum Electrodynamics: http://en.wikipedia.org/wiki/Quantum_electrodynamics Gauge Symmetry: http://en.wikipedia.org/wiki/Gauge_symmetry