The Feynman path integral (PI) and Schrödinger’s equation (SE) are completely equivalent formulations of QM in the sense that they give the same time evolution of an initial state. They have exactly the same information content. It’s true that you can derive SE from the PI, while the reverse derivation isn’t very natural. On the other hand, the PI is mathematically completely non-rigorous (roughly, the space of paths is too large) while SE evolution can be made precise.
Practically, the PI cannot be used to solve almost anything except the harmonic oscillator. This is a serious handicap in QM, since SE can be used to solve many problems exactly. But in quantum field theory, all the calculations are perturbations around harmonic oscillators, so the PI can be very useful.
Many physicists would agree that the PI is more “fundamental” because it’s gives insight into QFT and theoretical physics. But the distinction is largely a matter of taste.
They’re only syntactically equivalent. Their semantics are completely different. In my opinion, Feynman’s semantics is objectively correct regarding the ‘literal path’ of a particle through spacetime. Given we don’t officially know their paths, but we do know their end destinations (wave equation), we can figure all possible paths and have the practically impossible paths cancel each other out: leaving only the probable literal paths of a particle complete with a graph of their trajectories. Schrodinger’s equation is far behind semantically. I think Feynman’s path integrals are superior.
You’re drawing a philisophical distinction based on a particular ontology of the wavefunction. As simpler version arises in classical electromagnetism: we can integrate out the charges and describe the world entirely as an evolving state of the E&M field with the charges acting as weird source terms, or we can do the opposite and integrate out the E&M field to get a theory of charges moving with weird force laws. These are all equivalent descriptions in that they are observationally indistinguishable.
The Feynman path integral (PI) and Schrödinger’s equation (SE) are completely equivalent formulations of QM in the sense that they give the same time evolution of an initial state. They have exactly the same information content. It’s true that you can derive SE from the PI, while the reverse derivation isn’t very natural. On the other hand, the PI is mathematically completely non-rigorous (roughly, the space of paths is too large) while SE evolution can be made precise.
Practically, the PI cannot be used to solve almost anything except the harmonic oscillator. This is a serious handicap in QM, since SE can be used to solve many problems exactly. But in quantum field theory, all the calculations are perturbations around harmonic oscillators, so the PI can be very useful.
Many physicists would agree that the PI is more “fundamental” because it’s gives insight into QFT and theoretical physics. But the distinction is largely a matter of taste.
They’re only syntactically equivalent. Their semantics are completely different. In my opinion, Feynman’s semantics is objectively correct regarding the ‘literal path’ of a particle through spacetime. Given we don’t officially know their paths, but we do know their end destinations (wave equation), we can figure all possible paths and have the practically impossible paths cancel each other out: leaving only the probable literal paths of a particle complete with a graph of their trajectories. Schrodinger’s equation is far behind semantically. I think Feynman’s path integrals are superior.
You’re drawing a philisophical distinction based on a particular ontology of the wavefunction. As simpler version arises in classical electromagnetism: we can integrate out the charges and describe the world entirely as an evolving state of the E&M field with the charges acting as weird source terms, or we can do the opposite and integrate out the E&M field to get a theory of charges moving with weird force laws. These are all equivalent descriptions in that they are observationally indistinguishable.