Schwinger's and Feynman's versions of quantum field theory are not inherently adversarial but rather complementary approaches to understanding and calculating quantum phenomena. Both Schwinger and Feynman made significant contributions to the development of quantum field theory and introduced different formalisms and techniques to describe and calculate quantum processes.
Schwinger's approach, known as the Schwinger action principle or the operator formalism, emphasizes the use of operator equations of motion to derive and study quantum field theory. Schwinger's formalism is based on the Hamiltonian formulation of quantum mechanics and employs the concept of creation and annihilation operators to describe the dynamics of particles and fields.
Feynman, on the other hand, introduced the path integral formulation of quantum mechanics and quantum field theory. The Feynman path integral approach is based on summing over all possible paths of particles to calculate transition probabilities and expectation values. Feynman diagrams, which represent the contributions of different particle interactions, are a key tool in this formalism.
While Schwinger's and Feynman's formalisms have different mathematical representations and calculational techniques, they ultimately describe the same underlying physical reality. In fact, their approaches are often used in combination, with each providing valuable insights and techniques in different contexts. Feynman diagrams, for example, are widely used to calculate scattering amplitudes in quantum field theory, while Schwinger's operator formalism is often employed in the study of quantum field theory in curved spacetime and the calculation of vacuum polarization effects.
Overall, Schwinger's and Feynman's versions of quantum field theory are more complementary than adversarial. They offer different perspectives and mathematical tools that are often used in combination to explore and understand the intricate behavior of particles and fields in the quantum realm.