As it seems somewhat remiss to be at least nominally a physicist but to know nothing much about superstrings, I’ve decided to learn a bit of string theory during my lunch breaks. As the theory becomes clearer and I put the pieces together I’ll write about it here. This series will be a little different to my other articles on physics: usually I only write about things that I understand quite thoroughly but here I’m writing about things that I don’t understand very well. I’ll try not to say anything too incorrect but you should consider these articles to be provisional and subject to revision as I learn more. (This series will have at least one thing in common with my other series: it’s likely to peter out…)
When people talk about “string theory” they usually mean “superstring theory”, the quantum theory of relativistic, supersymmetric, interacting strings. The defining characteristic of string theories is that the fundamental objects are not pointlike particles but extended one-dimensional objects. Unlike the more familiar strings of everyday live, these strings have no internal structure. They’re supposed to be fundamental entities in just that same way that quarks or leptons are supposed to be fundamental in the standard model of particle physics. It’s my intention to sneak up on the full superstring theory by starting out with non-quantum, relativistic, non-supersymmetric, non-interacting strings and progressively adding more complexity. (This is the approach in the excellent A First Course in String Theory by Barton Zwiebach, which I’m using as my guide for the first part of the journey.)
There are deep analogies between the theory of non-quantum relativistic strings, classical field theory and the theory of classical particles. Each of these theories can be cast in a “Lagrangian” form, which is all about a quantity called the “action”. Before I describe how this applies to strings (in Part 1) I’d like to spend this installment talking about the two simpler cases. Let’s start by considering the simplest case, that of a classical particle. Consider all the possible paths in spacetime along which such a particle can travel between two fixed events. Associated with the particle is a mathematical gadget called the Lagrangian, which depends on the position and speed of the particle at each event along its path (and potentially on the event itself). The Lagrangian can be added up along the path — remember it’s a path through space and time — to give a quantity called the action. The simplest case is that of a non-relativistic particle, for which the Lagrangian is just the difference between the particle’s kinetic and potential energies.
Given that we can work out the action for any path, the obvious question is: which path does the particle actually follow? The answer is quite delightfully elegant. Consider a path between the initial and final events and an arbitrary but very small “variation” of that path, which takes the original path to another nearby path that has the same starting and ending points. We can work out how much the variation in the path causes the associated action to vary. The actual physical path is one for which an arbitrary very small variation leaves the action unchanged (to first order). From this condition we can deduce so-called “Euler-Lagrange equations”, which are the equations of motion for the particle. (For a non-relativistic particle we find that the equations of motion are just Newton’s laws of motion.)
As well as variations we also care about symmetries. For a physical path, any small variation gives no first-order change in the action. A symmetry, on the other hand, is a transformation that we can apply to the whole system that leaves the action unchanged for any path (and so in particular transforms any physical solution into another physical solution). A particular beautiful theorem called Noether’s theorem links symmetries to conserved quantities. For example, if a system is symmetrical under translations in time then energy is conserved. If it’s symmetrical under translations in space then momentum is conserved. If it has a rotational symmetry then angular momentum is conserved.
Next, let’s consider a classical field theory. Whereas a configuration for a particle is a path from one event to another, a configuration for a field is a value for the field at each event in spacetime. Instead of a Lagrangian, we have a Lagrangian density, which is a quantity that depends on the values of the field and its rate of change in each direction in spacetime at each event, and which we can add up over the whole spacetime to get the action. For a field, a variation changes the field’s value in an arbitrary but small way at each event in the spacetime (or a region of the spacetime subject to some boundary conditions). A physical field configuration is one which is left unchanged to first order by small variations. This condition gives us the “field equations” for the field. Instead of giving a globally conserved quantity, symmetries in field theories give rise to “conserved currents”, which is a technical way to say that there’s some quantity that can flow around in spacetime but can’t be created or destroyed.
The simplest example of this is electromagnetism. In this theory the fundamental field is the Faraday tensor, which contains the more familiar electric and magnetic fields. The Lagrangian density is a sort of inner product of the Faraday tensor with itself. The field equations derived from the condition that the action be unchanged by small changes in the field are the famous Maxwell equations. There’s a local “gauge” symmetry in the theory that leads to a conserved current that describes the local conservation of electric charge. This is all fairly neat and beautiful.
Let’s summarise everything so far in a table:
| Classical particles | Classical fields | |||
|---|---|---|---|---|
| Parameters | Time | t | Spacetime events | x |
| Configuration | Path | x(t) | Field | φ(x) |
| Integrated quantity | Lagrangian | L(x, ∂tx; t) | Lagrangian density | L(φ, ∂μ&phi ;x) |
| Action | S[x(t)] = ∫ L dt | S[φ(x)] = ∫ L d4x | ||
| Euler-Lagrange equations | Newton’s laws | Field equations | ||
| Symmetries give… | Conserved quantities | Conserved currents | ||
In the next part, I’ll actually get to strings!
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I admire your sincere efforts in exploring string theory which can become very involved. Keep me posted on your progress.