(This article was originally posted on blogger.com on Nov 4, 2005)
Of course I have to respond to Peter’s criticism – which is more than welcome – of the merits of string theory, which I listed below.
At the moment we actually have two separate theories; the Standard Model of particle physics and the general theory of relativity. String theory aims at finding a theory, which combines the two, i.e. is a theory of quantum gravity, and which in a certain sense is more “fundamental” and maybe “simpler” than the former two theories taken together. The point is not just that you would “like to” have a theory, which combines the two (and surely, this was not the historical reason for studying string theory, but that is rather irrelevant). There are some obvious reason for why we should try to find a theory of quantum gravity; I’ll just list some of them: 1) because of singularity theorems (originally formulated by Hawking and Penrose). It follows from general relativity, that singularities in space-time are unavoidable, so general relativity actually predicts its own breakdown in the sense, that the theory does not apply to the singularities themselves; 2) because we should be able to understand the initial conditions in cosmology: cosmology is incomplete if its beginning cannot be described in physical terms; 3) because of the evolution of black holes: black holes radiate with a temperature proportional to Planck’s constant (the Hawking temperature). To understand the final evaporation, a full theory of quantum gravity is needed; (in this respect, black holes are important testing grounds for a quantum theory of gravity); 4) because we would like to have unification of all interactions: all non-gravitational interactions have so far been successfully accommodated into the quantum field theory framework (i.e. in the Standard Model); 5) because of the inconsistency of an exact semi-classical theory: all attempts to construct a fundamental theory where a classical gravitational field is coupled to quantum fields have failed up to now – –and there is an obvious reason for this; if matter is quantum-mechanical, then the energy-momentum tensor in Einstein’’s equation is not a c-number, but an operator and should presumably be replaced by its expectation value. Then we end in turmoil, since the metric depends nonlinearly on the state of the matter system 6) because of the avoidance of divergences: String theory (and to a certain extend also loop quantum gravity) provides indications for a discrete structure at smaller scales, and therefore the emergence of a natural cutoff at small distances.
Peter is of course correct in saying, that what I said below (the list of credits of string theory) has been said many times before. That, however, does not make it less true -or more correct for that matter. Science should not be like politics, philosophy (or intelligent design), where you can discuss the pros and cons without ever reaching a “final” conclusion. So, I’ll defend some (for now, only a few) of the comments I made below.
My statement about the “reproduction” of the Standard Model in string theory was clearly not precise enough.
The Standard Model in itself contains a lot of unexplained assumptions; a gauge group (SU(3) x SU(2) x U(1)), some 21 free parameters, the number (three) of fermion generations, the 3+1 dimensions of space-time, chiral fermions in certain representations; gauge bosons, such as the gluons, the W^(+-) and the photon. The masses of these particles have not been computed in string theory (which was also secretly implied in what I was saying since I did not claim, that we understand how supersymmetry should be broken). What we can get is the gauge group and chiral fermions. Let’’s see how this could work: the gluons, for example, are described by a four-dimensional SU(3) Yang-Mills theory -– and this theory is closely related to the U(3) Yang-Mills theory which arises at low energies on the world-volume of three coincident D3-branes. This is because the U(3) gauge theory of nine (3 times 3) interacting gauge fields on three coincident D-branes contains a decoupled U(1) theory – the remaining eight interacting gauge fields define the SU(3) gauge theory (since, locally, U(3) = SU(3) x U(1)). Likewise, the SU(2) x U(1) electroweak Yang-Mills theory can be realized by including two additional coincident D3-branes; these two D-branes should obviously not coincide with the three color D-branes, since otherwise we would get a U(5) Yang-Mills theory. Furthermore, at low energies the gauge group is SU(3) x U(1) and this symmetry breaking is triggered when certain charged scalar fields, the Higgs fields, acquire expectation values. But in order to use string theory to describe the full Standard Model, we must study the matter particles and the charges they carry. Roughly, the fermions are represented as strings ending on the D-brane configurations that carry the gauge bosons. A central property of the Standard Model is that the spectrum of fermions is chiral (i.e. the left- and right-handed particle states do not have the same charges). How can we get chiral fermions?
First of all, in the Standard Model, the electroweak interactions SU(2) x U(1) acts chirally, so the fermions remain massless until the Standard Model gauge group is broken down to SU(3) x U(1), after which the masses are determined by the Higgs sector mentioned above. In the string theory picture, quarks, for example, are simply open strings that have one endpoint on one of the three coincident D-branes mentioned above (so that the color charges are determined by which D-branes the open strings end on) -– and the anti-quarks are simply oppositely oriented open strings. Now, where should the other endpoints of the open strings lie?
For illustrational purposes, I’ll just mention the left-handed quarks. The quark states fall into representations of SU(2) (weak symmetry) and are characterized by their isospin. The state with isospin I=1/2 is an up-quark and the one with I=-1/2 is a down-quark. The D-brane picture is very simple: a u-quark is an open string that begin on one of the two coincident D3-branes mentioned above (which we can call the weak-branes), and end on one of the three coincident D3-branes (or color-branes).
However, with a group of three coincident (color) D3-branes and a parallel group of two coincident (weak) D3-branes as above, our construction is doomed to fail: the spectrum is not chiral since the spectrum contains left-handed and right-handed quarks with the same charges (this can also be seen by noticing, that a string stretching from a color-brane to a weak-brane is massive). So the color D3-branes should actually intersect with the other two coincident weak D3-branes (I will not go into detail with this).
This was just included to give an illustration of how the Standard Model can be incorporated in string theory – but the embedding of the Standard Model in string theory is obviously not unique. There are other ways to obtain a string theory model of the Standard Model; one, for example, includes intersecting D6-branes wrapped on a six-torus T^6 in the Type IIA theory. But of course we should note, that these models are not fully realistic – the breaking of the electroweak symmetry needs to be worked out, the mass parameters and other couplings should also be calculated etc, and we cannot have six extra non-compact dimensions as above, since these other dimensions would be visible.
What about the free parameters of the Standard Models?
In string theory, on the other hand, there are no adjustable dimensionless parameters; the parameters of the Standard Model should come out as vacuum expectation values of certain scalar fields. These values are determined by the correct vacuum – which we don’t know how to find yet (and I don’t think the anthropic principle will help much).
The discussion about the number of dimensions “‘predicted”’ by string theory is rather old, and often also a bit off-mark, I think. Some argue, that string theory started by predicting 26 dimensions (the bosonic string), then ten dimensions (the superstring) and then eleven dimensions (via M-theory). I don’’t think anybody thought of the 26-dimensional theory as a “realistic”’ one, since it does not include space-time fermions. In superstring theory, ten dimensions are required by a vanishing total central charge, which is a mathematical constraint. There are no obvious reasons for why the result should be exactly ten dimensions – in principle, it could have been five or seven. In the Standard Model, the 3+1 dimensions is something, which is an input, and the theory could have been mathematically consistent in, for example, 7+1 dimensions. In M-theory, we have not ten, but eleven dimensions, which can seem strange; however, if we think of M-theory as the strong coupling limit of the Type IIA theory, then the original ten dimensions were just a result of a calculation done in a perturbative superstring theory, so roughly, the string coupling interpolates between a ten-dimensional and an eleven-dimensional theory (that the question about the number of space-time dimensions is subtle, is also something we learn from the AdS/CFT correspondence).
But, sure, string theory (or whatever it is going to be called) is not in any way a final theory and much work has to be done…
Note: More comments relating to Peter’s critique will be posted later 😉