Hmm ... "string theory", maybe it should be classified under meta-physics?
I haven't read Brain Greene's book. But I have been hearing about strings since about 1988, from people of the likes of Gary Horowitz (with whom I did a postdoc later), Ed Witten, David Gross, Strominger etc.
Early strings: In relativity, the action (which is later extremized to find the equations governing its motion) for a free particle is simply the path integral of the infinitesimal distance ds. So Nambu-Gottu proposed that the action governing the motion of a string is its surface area. Now if you just extremize this the solutin would lead to the collapse of the string to a point, so you add in a linear tension term. The resulting solutions to the Eqn of Motion have the usual oscillatory modes etc.
Now, fundamental theories are field theories and particles are thier quanta. So far so good. But when you put in interacting fields (EM field (photons) and Dirac field (charged particles)) and quantize them you can't do it exactly. So you do perturbations about the 0 interaction, or the vaccuum. It turns out that even QED is perturbatively "non-renormalizable", sort of like a divergent series. However, unlike a divergent series, it turns out that removing certain infinities and adding the perturbation term by term does lead to converging answers that agree with experiment to astounding degree of 10^-12 etc!
So why are interacting particle field theories singular? Hand waving, it goes like this, any paricle interaction involves a node in spacetime (the "collision" or disassociation) where 3 worldlines come together. Now this is no longer a "line" and leads to trouble. Also, there are (solved) causality problems and reference frame problems for multiple interactions. However, a string interaction is visualized as a smooth manifold, the trousers topology - imagine two circles coming closer and merging to form one: while the sequence of two dimensional slices of this event appear to have a singularity (non-hausdorff ness or a cusp formation) the spacetime manifold (surface) is smooth!
So lots of excitement, particularly for string field theory. Claims and expectations:
- string interactions are just smooth 2 dimensional surfaces
- lack of point interactions means should be renormalizable
- critical dimensions for string FT to be consistent, either 26 or 10, lots of yummy math, extra dimensions are compact manifolds, could be beautiful Calabi-Yau manifolds, topological invariants etc.
- Then things got even better, solutions to string field equations (which themselves arise from requiring SFT to be “perturbatively renormalizable”) include general relativity with matter, physical matter field equations etc. Hence the claim it a TOE – Theory of everything!
- Lots of connections between different areas of math discovered, by Witten and others.However,
- No 2D surface admits a non-singular, non-degenerate Lorentzian metric everywhere. Metric = infinitesimal distance tensor, which is the geometry of the manifold. Lorentzian means one spacelike and one timelike dimension, I.e, one positive and one negative eigenvalue. In other words, somewhere on that trouser there is a singularity. This is not very important.
- “renormalizable” means “perturbatively renormalizable”. What this means is one that the background spacetime in which the strings live is flat (why should that be so?) and this flat metric is used to setup a measure of distance for the perturbative expansions. Now: no one has actually done any perturbative calculations in full string theory, unlike in QED. Two, assuming a flat background spacetime is senseless for a self-proclaimed TOE (either GR is assumed and hence the string tension will cause curvature, or the background metric is a parameter whose value should be determined by the string EOM – see later. Three, a TOE should be exact, not perturbative, about some again arbitrarily chosen homogeneous solution.
- In the perturbative expansion, in order for the first order term to be finite, certain conditions have to be met. These are the string field equations. One solution to the string field eqn is that the background metric used satisfy Einstein's field eqn for the metric, G mu nu = 8 Pi Tmu nu. However, there are other solutions, how many (not “countably how many”, rather how many dimensions and are these countable)? One doesn't even know the structure of the solution space. Other than knowing that GR is right, there is no way to pick that particular solution and hence claim that GR arises from string theory. Also, all this can only take place in 26 (or 10 or 11) dimensions of the background spacetime. Then one is left with explaining the apparent 4 dimensionality of the physical world – how do the extra dimensions compactify, why are they small and not say large. Note that the extra dimensions are physical (like a very tightly rolled up sheet of paper appears to be 1D but is actually 2D on the small scale), not extra dimensions in the fibre-bundle sense (where the extra dimension is just a property holder, say e.g. a sphere to represent the spin of a point particle).
- See above. So string theory is not just a TOE, it is a TOEPUCI+ – Theory Of Everything Possible U Can Imagine and then some.To counter this and patrt of the previous criticism, let me make an analogy: suppose that immediately after Maxwell's EM and the Lorentz transformations somebody (the analog of a string theorist) said I have this wonderful theory of a thing called a metric that explains everything, I have an action principle (very important in physics then and now), I extremize it, the eqn says the ricci curvature of this metric thing is 0. A solution is that the world is flat! See, I have proved the world is flat. The analog of Lee or myself (in a very small way) or Woit would say, wait a minute, but there are numerous other solutions to your “metric equation” which say the metric is curved (what we now call the weyl curvature can be non-zero), how do you pick your flat metric from all the other solutions without knowing in advance that the world is flat? The answer is that you can't, the world is not flat and the metric theory is correct! So might String theory!
- There have been a series of String bandwagons, all but the most recent few started by Ed Witten: matrix theory, conformal field theory, String BHs (most notably the CGHS black hole - a 2 D mini-micro nanomodel and symmetry reduction of string theory then with the sign of the exponent in the potential term arbitrarily reversed because everything else is too hard to solve), M-theory, membrane theory, Maldacena etc. The lifetime of each of these (as measured by publications) is about 5-6 years. Every two years there is a big string conference, some major breakthrough is announced, stringers dance the macarena in the aisles, the fever subsides, it proves a physics dead end and one moves on.
- Now, contrary to Voit's claim that ST is “not even wrong”, a Weyl-square theorist whose seminar I attended in 1996 pointed out that string theory does actually make one very concrete testable prediction, and that never in the history of science has a theory been so wrong! Roughly speaking, in a non-perturbatively quantized ST, the string tension (actually stress) would be quantized, its quanta would be the planck tension (appropriate dimensional combination of G, c and h), which will also be the leading contributor to the cosmological constant (which is an energy density and has the same dimensions as tension/ area). There are strong astrophysical observational upper limits on the value of the CC, the planck tension is greater by about 40 orders of magnitude!Now, I have written much less than I could write, but already much more than you are interested in reading, so I'll just sto