The Feynman path integral formulation is a mathematical framework in quantum mechanics that allows us to calculate the probability amplitudes of quantum events. It is a powerful tool that has been successfully applied in various areas of physics, particularly in quantum field theory and quantum electrodynamics.
While it is true that Feynman path integrals involve summations or integrations over a large number of possible paths, including paths that are classically forbidden or appear to diverge, the formulation still yields meaningful and accurate predictions. This apparent contradiction can be understood in the context of mathematical techniques known as renormalization and regularization.
Renormalization is a mathematical procedure that allows us to remove divergences and obtain finite, meaningful results from calculations involving Feynman path integrals. It involves redefining certain quantities, such as the mass or charge, to absorb the divergent terms that arise in the calculations. This procedure preserves the physical predictions of the theory while eliminating the infinities that appear in intermediate steps.
Regularization is another technique employed in Feynman path integrals to deal with divergences. It involves introducing a parameter or cutoff that regulates the integrals and prevents the divergence from occurring. The cutoff is then taken to infinity at the end of the calculation, resulting in finite and physically meaningful results.
Both renormalization and regularization are mathematically rigorous procedures that have been extensively studied and justified within the framework of quantum field theory. They provide a consistent way to handle the apparent divergences that arise in Feynman path integrals and ensure that the predictions of the theory agree with experimental observations.
It is important to note that while the Feynman path integral formulation has been highly successful in making predictions and describing the behavior of quantum systems, the underlying mathematical foundations of quantum field theory, including the rigorous justification of Feynman path integrals, are still subjects of ongoing research in theoretical physics.