Julian Schwinger and Richard Feynman made significant contributions to the development of quantum field theory (QFT), but they approached the subject from different perspectives. Here are some key differences between Julian Schwinger's formulation of QFT and Richard Feynman's:
Mathematical Formalism: Schwinger developed his formulation of QFT using the operator-based approach, known as the Schwinger-Tomonaga formalism. He treated fields and particles as operators acting on a Hilbert space. His approach focused on the mathematical structure of the theory and employed techniques from quantum mechanics.
Feynman, on the other hand, introduced a pictorial representation of particle interactions known as Feynman diagrams. His formulation, called the Feynman diagrammatic approach, emphasized graphical calculations using a set of intuitive rules. Feynman diagrams allowed for a more visual and intuitive understanding of particle interactions and calculations of scattering amplitudes.
Propagators: Schwinger's formulation emphasized the concept of propagators, which are mathematical objects that describe the propagation of particles and fields in space and time. He introduced the Schwinger proper-time method, which involves integrating over all possible paths of particles. This method allowed for the calculation of scattering amplitudes and the description of particle interactions.
Feynman's formulation also made use of propagators but in a different way. He introduced the concept of the Feynman propagator, which is a fundamental element in Feynman diagram calculations. The Feynman propagator relates the initial and final states of particles involved in an interaction and is represented as a mathematical expression involving the mass, energy, and momentum of the particles.
Vacuum Structure: Schwinger's formulation focused on the role of the vacuum state in QFT. He introduced the concept of vacuum polarization, which refers to the creation and annihilation of particle-antiparticle pairs in the vacuum due to the presence of external fields. Schwinger's approach allowed for a detailed analysis of the vacuum's response to external perturbations and played a crucial role in understanding phenomena like the Lamb shift and the anomalous magnetic moment of the electron.
Feynman's formulation also considered the role of the vacuum state, but it emphasized a different aspect known as vacuum fluctuations. Feynman diagrams naturally incorporated virtual particle-antiparticle pairs appearing and annihilating in the vacuum, contributing to the overall behavior of physical interactions. Feynman's approach provided insights into phenomena like the Casimir effect and quantum electrodynamic corrections.
These are just a few of the key differences between Schwinger's and Feynman's formulations of QFT. While Schwinger focused on mathematical formalism, propagators, and vacuum polarization, Feynman introduced graphical calculations through Feynman diagrams, emphasized vacuum fluctuations, and brought a more intuitive approach to understanding particle interactions. Both formulations made significant contributions to the development of quantum field theory and have had a lasting impact on theoretical physics.