D = 3 string theory review and closed string spectrum

Abstract

String theory is a framework in which all matter and force particles are mathematically represented by tiny vibrating strings. One of the most remarkable aspects of the theory is that it is a theory of quantum gravity, in string theory gravity emerges as in the scope of the closed string spectrum. Another quite intriguing property of string theory is the fact that it "dictates" the necessity of a specific space-time dimension, namely the critical dimension, in order for the theory to preserve the Lorentz invariance. It is exactly this aspect that the thesis will build up to and offer another way out other than the renowned 26-dimensional space-time. The thesis, as is customary, will start with a brief investigation of relativistic point particle. The reason laying behind this is that the string case will be treated in a very similar fashion. The relativistic point particle action and the equations of motion will be calculated. The action of the point particle can be generalized to the p-brane action, which is simply the action of a p-dimensional membrane. By making use of the generalized action, the action of the string will then be examined. In the scope of the thesis, the focus will be on the free bosonic strings. The string motion is represented by the worldsheet of the string, the 2-dimensional space-time surface which the string sweeps throughout its motion. The parametrization will be made by specifying the string coordinates by $\sigma$ and a time parameter $\tau$. The string action is derived by considering the area of the worldsheet and is called the Nambu-Goto action, from which will be moved on to obtain the conjugate worldsheet momentum in the Hamiltonian formalism. That conjugate momentum will give rise to two constraints, which then will give rise to the final form of the action that will be examined. The action contains a constant $T$, analogous to the mass $m$ in the relativistic point particle case, which is called the string tension. The equations of motion will again be derived and then the open string boundary conditions will be analysed: Dirichlet and Neumann boundary conditions. For the closed string, the periodicity condition will be introduced together with the reparameterization invariance of the string action. After all of the above-mentioned calculations, the conserved currents and charges will be calculated and after that, a switch to the light-cone coordinates will be made. It will be the spin part of the Noether charge that will be used to check the Lorentz invariance of the theory later on. Then, the string wave equation will be calculated. Moving on from the wave equation, the Fourier mode expansion will be written. Following the calculation of the conserved currents, the mass-shell condition will be derived. The canonical quantization procedure will then be conducted and it will appear that the oscillator modes of the pre-calculated mode expansion will correspond to the annihilation and creation operators when the quantization is made. All other operators will be quantized as well. Finally, it will be possible to check the Lorentz invariance of the theory by looking at the commutation relations of Lorentz charges. This will be equivalent to examining only the commutators of the spin parts of the Lorentz charges and requiring them to be equal to zero will give rise to the critical dimension of D = 26. But also, it will be shown that for the special case of D = 3 this commutation relation also vanishes i.e. preserves the Lorentz invariance. At last, the focus can be directed on the spectrum of the D = 3 theory. Even more specifically, to the D = 3 closed string. The Poincaré invariants will be calculated and then the level-matching condition will be shown. It will then be possible to obtain the states corresponding to different levels in terms of the creation and annihilation operators from before. Eventually, a set of calculations will be conducted to find the spins of different levels and then we will end up with a set of numbers that depend on the normal-ordering constant $a$. After examining the final results, it will be apparent that the spectrum gives rise to anyonic states at some levels regardless of the choice of $a$, states which has spin $s$ where $2s$ is not an integer. An effort to evaluate this result will be made and further areas for possible contributions will be discussed.