Analyticity and Acausality of Nonlocal Quantum Field Theory

In this dissertation, we study nonlocal quantum field theories obtained by delocalizing the fields in the interaction terms of some local Lagrangian. The delocalization kernels in momentum space are exponentials of entire functions, thus no additional mass poles are introduced, but they ensure UV finiteness of loop integrals of Feynman diagrams. The transition matrix is defined in the Euclidean regime, followed by analytic continuation of the external momenta. Because of the delocalization kernels, manipulations valid in local theories are prohibited in the nonlocal theories, including Wick rotation and dispersion relation. We study the analyticity structure of the transition matrix of such theories at the perturbative level, using the Landau equations. The result shows that singularities that occur in local theories are also present in the nonlocal theories, except for singularities of the second type. The Cutkosky rule is then derived using Cutkosky's original argument, and the unitarity condition follows as an application of the rule. We then quantify the acausality present in such nonlocal theories using the Bogoliubov causality condition (BCC). First we study the acausality of the effective propagator, which is shown to be exponentially suppressed for large space-time separation. Mellin transform is introduced to help find the asymptotic behavior of acausality. Next we prove that in local theories, for each diagram, the BCC equation amounts to a chain of retarded propagator between two vertices; while in nonlocal theories, the chain consists of $\tilde\Delta_R$, which is the retarded propagator delocalized by the nonlocal kernel and is therefore no longer causal. The acausal amplitudes are then integrated over wave packets to make experimental predictions. We show that for a nonlocality scale of order Plank's length, the acausality will be experimentally immeasurable. Along the way, by extending the BCC to unitarity equations, we show that the theory defined directly in Minkowksi space is not unitary. This reaffirms that the theory has to be defined in the Euclidean regime and then analytically continued