Hamilton Revised: The Action Principle for Initial Value Problems

TL;DR

This paper develops a variational action principle for initial value problems in classical mechanics, derived from quantum mechanics, revealing new insights into path fluctuations and initial conditions in the classical limit.

Contribution

It introduces a rigorous derivation of the action principle for initial value problems from the Schwinger-Keldysh formalism, clarifying the roles of forward and backward paths and their fluctuations.

Findings

Both Keldysh paths have classical paths and fluctuations.

Minus paths propagate backwards in time and are zero in the classical limit.

Implications for non-holonomic constraints and gauge-dependent quantum field theories.

Abstract

We present the variational action principle for initial value problems in classical, conservative-force point particle mechanics. We rigorously derive this formulation by taking the classical limit of the Schwinger-Keldysh expression for the time dependence of the expectation value for operators in quantum mechanics. We clarify the connection between the variation of the position and the variation of the velocity of a particle when implementing Hamilton's Principle in deriving the Euler-Lagrange Equations. We show that both the plus and minus Keldysh paths (of the average and difference of the forward/backward paths) have classical paths and fluctuations -- unlike the common perception that the minus path provides the fluctuations around the single classical solution given by the plus path -- and that the fluctuations of both paths are crucial for the correct normalization of the classical limit. The classical limit yields "initial conditions" and equations of motion for the minus paths such that the unique classical solution for the minus paths is that they are identically zero, and, fascinatingly, that the minus paths' solution propagates backwards in time; thus one does not need to set the minus paths to zero by hand when taking the classical limit of the Schwinger-Keldysh formalism. We note implications for the classical and quantum mechanics of non-holonomic constraints and quantum field theories with gauges dependent on the derivatives of the fields.