The group-theoretical basis of the theory of the massless relativistic string

Abstract

Utilises previous work on the Weyl group (i.e. Poincare group plus dilations), whode unitary irreducible representations describe off-mass-shell relativistic particles, to give a group-theoretically sound basis to the (classical and quantum) theory of the massless relativistic string in d dimensions in both the centre-of-mass gauge and the light-cone gauge. The author defines the centre-of-mass oscillator variables alpha mmu (P)=L-1(P)nu mu alpha mnu and shows that in the centre-of-mass gauge, alpha m0(P) approximately=0, the oscillators alpha mi(P) satisfy a fully constrained off-mass-shell Dirac bracket of the same form as found by Bardeen, Bars, Hanson and Peccei (1976) for d=2. In the quantum theory covariant solution, the author explicitly constructs all the states up to N=2. The author constructs another set of non-covariant oscillators Ammu (P) by applying an infinite D+M0(d-1) transformation to alpha mmu (P). There are two different light-cone gauges given by Am+(P) approximately=0 and Am+(P)/(P2)1/2 approximately=0, which for P2 approximately=0 are first class and second class respectively. For P2not approximately=0 they are both second class, and the fully constrained off-mass-shell (Weyl group) Dirac brackets close correctly only for P2 approximately=M2 and Am-(P) approximately=0. The author emphasises that the fundamental difference between the centre-of-mass and light-cone gauges is that, in the light-cone gauges, the constraints Ln approximately=0 are dilatation non-invariant. In the quantum theory elimination-of-variables solution, the author explicitly constructs the eigenstates of the principal spin invariant up to N=3, and shows that they are off-mass-shell continuous spin particles which form spin multiplets only for p2=(N-1)/ alpha ' and p=26. In the quantum theory covariant solution, the author explicitly constructs all the states up to N=2, and shows that for p2=(N-1)/ alpha ' and d=26 the N=2 transverse states are equal to the centre-of-mass gauge physical states plus null states, a result conjectured to be true for all N>or=2.