Path integrals in classical theory

L. E. Reichl

Mustafa Türe, Mithat Ünsal

We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.

Martin C. Gutzwiller

Seth Lloyd, Olaf Dreyer

Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration. This paper defines a universal path integral, which sums over all computable structures. This path integral contains as sub-integrals all possible computable path integrals, including those of field theory, the standard model of elementary particles, discrete models of quantum gravity, string theory, etc. The universal path integral possesses a well-defined measure that guarantees its finiteness, together with a method for extracting probabilities for observable quantities. The universal path integral supports a quantum theory of the universe in which the world that we see around us arises out of the interference between all computable structures.

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.