- for a compound system, the individual elements follow worldline geodesics, with the other particles factored out into a "potential" / as force terms in E-L equation. These represent how 1 particle, when distorted from its non-interacting trajectory, trades off against the other particles distorting from their OWN free trajectories. It's analogous to how heat energy flows between two systems to maximize their joint entropy; here, the particles' trajectories distort each other via forces to minimize their joint spacetime arc length. (As measured, presumably, in ANY reference frame)
- yes, in any specific context, the Lagrangian represents HOW those things trade off, in the same way that in any specific thermodynamics scenario the microstate structure can tell us d(entropy)/d(energy) and let us compute the equilibrium state.
- but how to complete the analogy to thermodynamic equilibrium--what is the "temperature", exactly, what is "heat"? In a QFT context, the interaction is ITSELF a particle, and has its own contribution to the Lagrangian and action, but how to think about it classically?
It has always seemed to me that this line of thinking is the most natural way to express "Stationary Action".
I'm saying:
- it appears that the "stationary action" is the same as "worldline follows a geodesic of the spacetime metric" (https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechan...). I first heard this from a particle physics professor in undergrad.
- for a compound system, the individual elements follow worldline geodesics, with the other particles factored out into a "potential" / as force terms in E-L equation. These represent how 1 particle, when distorted from its non-interacting trajectory, trades off against the other particles distorting from their OWN free trajectories. It's analogous to how heat energy flows between two systems to maximize their joint entropy; here, the particles' trajectories distort each other via forces to minimize their joint spacetime arc length. (As measured, presumably, in ANY reference frame)
- yes, in any specific context, the Lagrangian represents HOW those things trade off, in the same way that in any specific thermodynamics scenario the microstate structure can tell us d(entropy)/d(energy) and let us compute the equilibrium state.
- but how to complete the analogy to thermodynamic equilibrium--what is the "temperature", exactly, what is "heat"? In a QFT context, the interaction is ITSELF a particle, and has its own contribution to the Lagrangian and action, but how to think about it classically?
It has always seemed to me that this line of thinking is the most natural way to express "Stationary Action".