Worldgraph approach to amplitudes

String theory uses the first-quantized method (quantum mechanics) to derive scattering amplitudes. External states are considered as vertex operators inserted on the worldsheet generated by an internal string and the amplitudes are calculated perturbatively by calculating vacuum expectation values (vevs) of these vertex operators on worldsheets with different topologies. This approach is different from the common approach of particle theories in which the second-quantized method (quantum field theory) is adopted to calculate amplitudes. A natural question to ask is whether there exists a first-quantization formalism for particles that gives particle amplitudes to all orders. This thesis presents the recent research in answering this question.In the first-quantized approach for particles, amplitudes are considered as the vevs of vertex operators inserted on different graphs generated by an internal particle. I refer to these graphs as worldgraphs and first-quantized approach for particles as worldgraph approach. To evaluate these vevs, vertex operators for several external states and Green functions on different worldgraphs are needed. In this thesis, various vertex operators are considered and a general method to obtain scalar Green functions on different worldgraphs is obtained. Some examples of the worldgraph approach to amplitudes in scalar theory and Yang-Mills theory are presented.