QG Seminar: Hamilton Revisited: The Action Principle for Initial Value Problems (Will Horowitz, University of Cape Town)

QG Seminar

Speaker: Will Horowitz (University of Cape Town)

Title: Hamilton Revisited: The Action Principle for Initial Value Problems
 

Abstract:

We present the variational action principle for initial value problems in classical point particle mechanics.

We rigorously derive this formulation by taking the classical limit of the Schwinger-Keldysh/closed time path/in-in 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 rotated Schwinger-Keldysh coordinates have classical paths and fluctuations—unlike the common perception that the minus coordinates are the fluctuations around the single classical solution given by the plus coordinate—and that the fluctuations of both coordinates are crucial for the correct normalization of the classical limit.

The classical limit yields “initial conditions” and equations of motion for the minus coordinates such that the unique classical solution for the minus coordinates is that they are identically zero, and, fascinatingly, that the minus coordinates' solution propagates backwards in time; thus one does not need to set the minus coordinates 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, including the physics of rolling without slipping, field theories with gauges that depend on derivatives (such as the Lorenz, Coulomb, and 't Hooft-Veltman gauges), and the necessary corrections to Poisson brackets and the symplectic structure of phase space.