A unique theory of all forces

In discussing the construction of a consistent theory of quantum gravity unified with the gauge interactions we are naturally led to a string theory. We review its properties and the five consistent supersymmetric string theories in ten dimensions. We finally discuss the evidence that these theories are actually special limits of a unique 11-dimensional theory, called M-theory, and a recent conjecture for its explicit formulation as a supersymmetric Matrix theory. 1 Unification of all interactions The electro-weak and strong interactions are described by the so-called standard model of particle physics. It is a gauge field theory based on the direct product of the group SU(3) colour with coupling constant g3 for the strong interactions and the group SU(2) ⊗ U(1) with coupling constants g2 and g1 respectively for the electro-weak interactions. A non abelian gauge theory is a generalization of quantum electrodynamics (QED) to the case of a non-abelian gauge group and is described by a Lagrangian of the type: L = − 1 4g2 F 2 μν + Ψ̄ (iγ Dμ −m)Ψ ; Dμ = ∂μ + iAμ (1.1) The field Ψ describes the matter fields that in the standard model are the quarks and leptons, while the gauge field Aμ and its field strenght Fμν describe the gauge particles responsible for providing the interaction among the matter particles. In the standard model the gauge fields are the 8 gluons for the strong interactions and the W±, the Z and the photon for the electro-weak interactions.