On the non-existence of path integrals

Kevin E. Cahill, Randolph Reeder

Wei Wen

The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mechanics and in QED are neither unified nor directly connected. This suggests the existence of a missing path integral that bridges relativistic action and the Dirac equation at the single-particle level. In this work, we analyze the consistency and completeness of existing path-integral theories and identify a spinor path integral that fills this gap. Starting from a relativistic action written in spinor form, we construct a spacetime path integral whose kernel reproduces the Dirac Hamiltonian. The resulting formulation provides a direct link between the relativistic classical action and the Dirac equation, and it naturally extends the scalar relativistic path integral developed in our earlier work. Beyond establishing this structural connection, the spinor path integral offers a new way to interpret the origin of classical mechanics for the Dirac equation and suggests a spacetime mechanism for spin and quantum nonlocal correlations. These features indicate that the spinor path integral can serve as a unifying framework for existing path-integral approaches and as a starting point for further investigations into the spacetime structure of quantum mechanics.

Osvaldo Civitarese, Manuel Gadella

Job Feldbrugge, Neil Turok

Many interesting physical theories have analytic classical actions. We show how Feynman’s path integral may be defined non-perturbatively, for such theories, without a Wick rotation to imaginary time. We start by introducing a class of smooth regulators which render interference integrals absolutely convergent and thus unambiguous. The analyticity of the regulators allows us to use Cauchy’s theorem to deform the integration domain onto a set of relevant, complex “thimbles” (or generalized steepest descent contours) each associated with a classical saddle. The regulator can then be removed to obtain an exact, non-perturbative representation. We show why the usual method of gradient flow, used to identify relevant saddles and steepest descent “thimbles” for finite-dimensional oscillatory integrals, fails in the infinite-dimensional case. For the troublesome high frequency modes, we replace it with a method we call “eigenflow” which we employ to identify the infinite-dimensional, complex “eigenthimble” over which the real time path integral is absolutely convergent. We then bound the path integral over high frequency modes by the corresponding Wiener measure for a free particle. Using the dominated convergence theorem we infer that the interacting path integral defines a good measure. While the real time path integral is more intricate than its Euclidean counterpart, it is superior in several respects. It seems particularly well-suited to theories such as quantum gravity where the classical theory is well developed but the Euclidean path integral does not exist.

RAMAMURTI RAJARAMAN

Wen, W.

The Feynman path integral plays a crucial role in quantum mechanics, offering significant insights into the interaction between classical action and propagators, and linking quantum electrodynamics (QED) with Feynman diagrams. However, the formulations of path integrals in classical quantum mechanics and QED are neither unified nor interconnected, suggesting the potential existence of an important bridging theory that could be key to solving existing puzzles in quantum mechanics. In this work, we delve into the theoretical consistency, completeness, and integration with established path integral theories, revealing this concealed path integral form. This newly uncovered form not only connects various path integral approaches but also demonstrates its potential in explaining quantum phenomena like the origin of spin and quantum nonlocal correlations. It transcends conventional quantum mechanics, proposing a more profound and fundamental physical principle.