# Determinism and frame-relativity

It's a well known fact that classical mechanics isn't a deterministic theory if you only include the positions and masses of various particles as part of the initial conditions. You also need to include the velocities/momenta of the various particles, which are formulated in terms of time-derivatives. This is somewhat surprising because it is natural to think that velocities/momenta aren't really intrinsic to a time, they are just time-derivatives of position.

This trend is also present in quantum physics, where (for example) the Hamiltonian of a physical system needs to be specified for the laws of physics to be deterministic. The Hamiltonian is equivalent to the total energy, where the total energy depends on other "time-derivative properties" like momentum.

Another strange feature of these "extra" quantities that are needed for determinism to work is that they are frame-relative (unlike mass/charge, for example). The velocity (and therefore momentum and kinetic energy) are all frame-relative. Not all time-derivative quantities are frame relative. For example, acceleration is frame-independent in classical physics. But, acceleration doesn’t help in delivering determinism. Surprisingly, the time-derivative quantities that give us determinism are frame-relative ones.

Question: Is it generally true that these kind of "time-derivative" properties that need to be included for determinism to work are always sensitive to an inertial frame? Or, can determinism be formulated in terms of time-derivative properties that not frame-relative? I am interested in this question both in the case of classical mechanics and quantum mechanics.

• Why are you concerned that the velocity is frame-relative when the position is as well? – G. Smith Jun 13 '20 at 21:36
• Good point. I was thinking that in spacetime theories like GR (or “Galilean space-time” for Newtonian mechanics) there are objective frame independent facts about the spacetime manifold and the fundamental fields that live on that manifold. So perhaps we can ultimately understand “positions” in a frame independent way. I just find it surprising that some time-derivative facts are frame independent (such as acceleration in Galilean space-time) and some aren’t, but the ones that are needed for determinism are frame relative. – Taro Jun 13 '20 at 21:45
• Acceleration is frame-independent under a Galilean boost, but it isn’t under a Lorentz boost. And it isn’t frame-independent under rotations. – G. Smith Jun 13 '20 at 21:51
• That's right, so I guess this puzzle is better motivated in non-relativistic physics where Galilean transformations are what matters? In any case, if these temporal derivatives are inevitably frame-relative, I would take that to be an answer to my question. The answer would be "no", the time-derivative-quantities that are necessary for determinism are inevitably frame-relative. – Taro Jun 13 '20 at 21:54

• You can formulate the deterministic time-evolution laws of classical mechanics as first-order partial differential equations and thus, you wouldn't need to specify any time derivatives of quantities that specify the state of the system at a given instant of time. This is exactly what Hamiltonian formalism does. A state of the system at a given instant of time is given by $$(q,p)$$ where $$q$$ are canonical coordinates and $$p$$ are canonical momenta. Both $$\dot{p}$$ and $$\dot{q}$$ are then determined by the equations of motion which are given by the Hamilton's equations.