Non-fiction: The Lagrangian in Quantum Mechanics

Overview
Paul Dirac proposes a formulation of quantum mechanics that places the Lagrangian and the classical action at the center of quantum transition amplitudes. He suggests that the quantum amplitude for a system to go from one configuration to another can be expressed in terms of the classical action evaluated along a path, introducing a phase factor e^{iS/ħ} associated with how the system traverses configuration space. The note sketches how successive short-time amplitudes compose to yield a cumulative amplitude that depends on the action of whole trajectories.

Central ideas
Dirac emphasizes the Lagrangian as a natural generator of quantum propagation, rather than the Hamiltonian operator used in canonical quantization. The amplitude to go from an initial coordinate to a final coordinate in a short time interval is taken to be proportional to exp(iLΔt/ħ), where L is the Lagrangian evaluated along the short segment. Composing many such short-time amplitudes and integrating over intermediate coordinates leads to an expression in which the phase is the classical action S = ∫L dt for the entire path.

Technical content
The note derives a composition law for transition amplitudes: the amplitude to go from point A to C via an intermediate point B is obtained by integrating the product of amplitudes A→B and B→C over the intermediate coordinate. By assuming the short-time kernel has the exponential dependence on the Lagrangian, the multiple composition suggests that the full kernel can be written as a limiting product or integral whose phase equals the sum of the short-time actions, i.e., the total action along the path. Dirac points out the close formal relationship between the quantum amplitude and Hamilton's principal function, highlighting how classical generating functions reappear in the quantum phase.

Classical limit and interpretation
The formulation makes the emergence of classical mechanics transparent through stationary-phase reasoning: in the limit ħ → 0, rapidly oscillating phases cause destructive interference among most paths, and constructive interference picks out paths that make the action stationary, recovering the classical equations of motion. This supplies a clear route from quantum superposition to the classical principle of least action. Dirac frames the Lagrangian representation as a bridge between quantum amplitudes and classical variational principles, offering an intuitive picture of quantum evolution as a coherent sum over possible histories weighted by exp(iS/ħ).

Historical significance and legacy
The note served as a crucial conceptual precursor to the path integral formulation developed later by Richard Feynman, who acknowledged Dirac's idea that amplitudes are related to the action as a motivating seed. Although Dirac did not develop a full measure-theoretic construction or the explicit operational rules that characterize later path integrals, the suggestion that quantum mechanics can be built from a Lagrangian-based sum over paths shaped subsequent work in nonrelativistic quantum mechanics, quantum field theory, and semiclassical analysis.

Limitations and later developments
Dirac's sketch leaves several technical issues unresolved: the normalization and prefactor of the kernel, the precise definition of the infinite-dimensional integral over paths, operator-ordering subtleties, and rigorous justification of the limit process. Subsequent developments, most notably Feynman's path integral formalism and semiclassical methods like the van Vleck–Gutzwiller propagator, supplied concrete prescriptions for computing these kernels and clarified the relationship with canonical quantum mechanics. Nevertheless, the central insight that the action governs quantum phases remains a foundational idea influencing perturbation theory, instanton methods, and modern formulations of quantum dynamics.