Types of quantum field thories
as well as three of the four fundamental forces as currently known, which, somewhat roughly, are
The main ingredient missing from the standard model is the quantum version of the field of gravity. For decades, a large part of theoretical physics has been absorbed with attempts to understand how this last of the known fundamental forces might fit into the picture.
Although there are several approaches to formulate a mathematically precise definition of what a quantum field theory is, there is no rigorous formulation (yet) that comprises the whole standard model.
There is a plethora of attempts and suggestions for variations and generalizations of the standard model into models that are conceptually more satisfying from the point of view of models in theoretical physics.
Shortly after the conception of general relativity, it was observed by Kaluza and Klein that the force of gravity alone may effectively appear – if considered on a spacetime that is a bundle whose fiber has a tiny volume (as meaured by the Riemannian metric) – as the field of gravity coupled to gauge fields on the base of the bundle. For details on this see Kaluza-Klein mechanism.
The huge conceptual simplification that this observation suggested had excited theoreticians early on, but a problem of Kaluza–Klein models is that not only does the “compactified” theory of gravity as if by magic emulate gauge fields, but it also always contains further scalar fields that are not experimentally observed.
For that reason interest in Kaluza–Klein theories had decreased in the middle of the last century. Physics departments saw a major revival of the idea when string theory (see below) gained interest, since that theory necessarily exhibits a Kaluza–Klein mechanism. Incidentally, the problem of the spurious fields – the moduli – was still present in this approach. For more on this see the entry landscape of string theory vacua.
One of the oldest studies of variations of the standard model is the investigation of grand unified theories (GUTs), which are Yang–Mills theories that instead of the standard model gauge group have a bigger gauge group which is however a simple group.
A widespread perception is that some of the conceptual problems with the standard model point to the fact that some basic assumption of 20th century physics on the nature of reality is oversimplified. According to the approach of noncommutative geometry, modeling spacetime as a smooth manifold is an oversimplification that makes itself felt when the quantization of the force of gravity becomes relevant.
In a class of “noncommutative” generalizations of the standard model, spacetime is therefore replaced more generally by a spectral triple that models a possibly “noncommutative space”. One of the more successful approaches in this direction is the Connes–Lott–Chamseddine model. This effectively is a Kaluza-Klein theory (see above), but with the crucial difference that the fiber in the KK-picture is a highly non-classical non-commutative space, whose classical dimension is that of a point, but whose intrinsic dimension is 6. (This is incidentally the same value of the internal dimension as suggest by string theory.)
For more on this see
A more drastic theoretical modifications to the standard model is proposed in the context of string theory, where the entire concept of quantum field theory is proposed to be refined by something else. As opposed to GUTs, this approach at least suggests a way in which also the fourth remaining force field of gravity could be incorporated into the picture.
There are various other variations, too.
| | gravity | electroweak and strong nuclear force | fermionic matter | scalar field | | field content: | vielbein field | principal connection | spinor | scalar field | | Lagrangian: | scalar curvature density | field strength squared | Dirac operator component density | field strength squared + potential density | | | | | | |
A general review is in
For an Lab dictionary (work in progress) of the language of mathematicians and physicists see:
There are tons of textbooks about the standard model, so any recommendation is hopelessly biased. The following textbook is a short and relativly easy introduction that nevertheless covers a lot of ground:
For further references see quantum field theory and the Wikipedia entry