The claim made by PBS Spacetime that the path integral formulation of quantum mechanics (QM) is both mathematically equivalent to and more powerful than other formulations is a common perspective among physicists. However, it is important to note that different formulations of quantum mechanics serve different purposes and have different advantages in various contexts. Let's address your points one by one:
Mathematical Equivalence: The path integral formulation, pioneered by Richard Feynman, is mathematically equivalent to the more traditional wave function formulation, known as the Schrödinger equation or the Heisenberg picture. This means that both formulations can yield the same results and describe the same physical phenomena. The path integral formulation provides an alternative way of calculating probabilities and transition amplitudes by summing over all possible paths, rather than solving differential equations as in the wave function formulation.
Power for Quantum Field Theory (QFT): The path integral formulation is particularly well-suited for describing quantum field theory (QFT), which extends quantum mechanics to incorporate fields and interactions. QFT is an essential framework for studying particle physics, high-energy phenomena, and the behavior of quantum fields. The path integral formulation naturally lends itself to calculations in QFT, making it a powerful tool in that context. The Feynman diagrams commonly used in QFT calculations are a direct consequence of the path integral formulation.
Default Formulation: The choice of the "default" formulation in physics often depends on historical reasons and practical considerations. The wave function formulation, based on differential equations, was the original formulation developed by Schrödinger and Heisenberg. It provided significant insights and was easier to grasp initially. As a result, it became the mainstream framework for quantum mechanics. The path integral formulation was introduced later by Feynman and gradually gained popularity, particularly in the context of quantum field theory.
However, it's worth noting that the wave function and path integral formulations are mathematically equivalent, so any quantum mechanical problem can be solved using either approach. The choice of formulation often depends on the nature of the problem, personal preferences, and the mathematical techniques that are most convenient for specific calculations.
In summary, the path integral formulation of quantum mechanics is considered more powerful in the context of quantum field theory, and it provides a complementary perspective to the wave function formulation. While the path integral formulation is highly valuable, both formulations have their strengths and are used depending on the specific requirements of the problem at hand.