Confinement in 3+1 Dimensions

In this thesis we focus on two important problems of modern physics: the phenomenon of confinement in non-Abelian field theories and the unitarity of theories with higher derivatives. In the first part we describe an effective theory of a scalar field, motivated by some features expected in the low energy theory of gluodynamics in 3+1 dimensions. The theory describes two propagating massless particles in a certain limit, which we identify with the Abelian QED limit, and has classical string solutions in the general case. The string solutions are somewhat unusual as they are multiply degenerate due to the spontaneous breaking of diffeomorphism invariance. Nevertheless, all solutions yield an identical electric field and have the same string tension. We conclude the first part by further investigating the Abelian limit of the model presented and constructing a Lagrangian with a four-derivative kinetic term and demonstrate that, despite the seeming nonlinearity of the theory, it is equivalent to a theory of a free photon. In the second part we start by giving a simple discussion of ghosts, unitarity violation, negative norm states, and quantum vs classical behavior in the simplest model with a four-derivative action - the Pais-Uhlenbeck oscillator. We also point out that the normalizable “vacuum state” (in the sense defined below) of this model can be understood as a spontaneous breaking of the emergent conformal symmetry. We provide an example of an interacting system that couples the “particle” and “ghost” degrees of freedom and nevertheless remains unitary on both the classical and quantum levels. The rest of the second part focuses on the analysis of conformal gravity in translationally invariant approximation, where the metric is taken to depend on time but not on spatial coordinates. We find that the field mode, which in perturbation theory has a ghostlike kinetic term, turns into a tachyon when nonlinear interaction is accounted for. The kinetic term and potential for this mode have opposite signs. Solutions of nonlinear classical equations of motion develop a singularity in finite time determined by the initial conditions