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kilobug comments on Yet more "stupid" questions - Less Wrong Discussion
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I don't see any direct link between determinism and conservation of energy. You can have one or the other or both or none. You could have laws of physics like "when two protons collide, they become three protons", determinist but without conservation of energy.
As for "libertarian free will" I'm not sure what you mean by that, but free will is concept that must be dissolved, not answered "it exists" or "it doesn't exist", and anyway I don't see the link between that and the rest.
You can have determinism without conservation of energy, but I opine that you cannot have conservation of energy (plus the other things that are conserved in our physics) without determinism.
JUST conservation of energy, sure... consider a universe composed of a ball moving at constant speed in random directions.
But conserving everything our physics conserves means you're using our physics. It's not even a hypothetical if you do that.
Suppose you changed electromagnetism to be one over r-cubed instead of r-squared. What conservation law breaks? Or just fiddle with the constants.
Hrm. Well, I suppose that if you change the constants then you're not conserving the same exact things, but they would devolve to words in the same ways. All right, second statement withdrawn.
I'll take the first further, though - you can have an energy which is purely kinetic, momentum and angular momentum as usual, etc... and the coupling constants fluctuate randomly, thereby rendering the world highly nondeterministic.
Ok, but it's not clear to me that your "energy" is now describing the same thing that is meant in our universe. Suppose everything in Randomverse stood still for a moment, and then the electric coupling constant changed; clearly the potential energy changes. So it does not seem to me that Randomverse has conservation of energy.
Hmmm... yes, totally freely randomly won't work. All right. If I can go classical I can do it.
You have some ensemble of particles and each pair maintains a stack recording a partial history of their interactions, kept in terms of distance of separation (with the bottom of the stack being at infinite separation). Whenever two particles approach each other, they push the force they experienced as they approached onto the pair's stack; the derivative of this force is subject to random fluctuations. When two particles recede, they pop the force off the stack. In this way, you have potential energy (the integral from infinity to the current separation over the stack between two particles) as well as kinetic, and it is conserved.
The only parts that change are the parts of the potential that aren't involved in interactions at the moment.
Of course, that won't work in a quantum world since everything's overlapping all the time. But you didn't specify that.
EDITED TO ADD: there's no such thing as potential energy if the forces can only act to deflect (cannot produce changes in speed), so I could have done it that way too. In that case we can keep quantum mechanics but we lose relativity.
I still don't think you're conserving energy. Start with two particles far apart and approaching each other at some speed; define this state as the zero energy. Let them approach each other, slowing down all the while, and eventually heading back out. When they reach their initial separation, they have kinetic energy from two sources: One source is popping back the forces they experienced during their approach, the other is the forces they experienced as they separated. Since they are at their initial separation again, the stack is empty, so there is zero potential energy; and there's no reason the kinetic energy should be what it was initially. So energy has been added or subtracted.
The idea of having only "magnetic" forces seems to work, yes. But, as you say, we then lose special relativity, and that imposes a preferred frame of reference, which in turn means that the laws are no longer invariant under translation. So then you lose conservation of momentum, if I remember my Noether correctly.
You got it backwards. The stack reads in from infinity, not from 0 separation. As they approach, they're pushing, not popping. Plus, the contents of the stack are included in the potential energy, so either way you cut it, it adds up. If the randomness is on the side you don't integrate from, you won't have changes.
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As for the magnetic forces thing, having a preferred frame of reference is quite different from laws no longer being invariant under translation. What you mean is that the laws are no longer invariant under boosts.
Noether's theorem applied to that symmetry yields something to do with the center of mass which I don't quite understand, but seems to amount to the notion that the center of mass doesn't deviate from its nominal trajectory. This seems to me to be awfully similar to the conservation of momentum, but must be technically distinct.
Yes. Then as they separate, they pop those forces back out again. When they reach separation X, which can be infinity if you like (or we can just define potential energy relative to that point) they have zero potential energy and a kinetic energy which cannot in general be equal to what they started with. The simplest way of seeing this is to have the coupling be constant A on the way in, then change to B at the point of closest approach. Then their total energy on again reaching the starting point is A integrated to the point of closest approach (which is equal to their starting kinetic energy) plus B integrated back out again; and the potential energy is zero since it has been fully popped.
Yes, you are correct. Still, the point stands that you are abandoning some symmetry or other, and therefore some conservation law or other.
Not necessarily. Consider the time-turner in HPMOR. You could have physics which allow such stable time loop, with no determinism on which loop among the possible ones will actually occur, and yet have conservation of energy.
As I mentioned a few times, HPMoR time turners violate general relativity, as they result in objects appearing and disappearing without any energy being extracted from or dissipated into the environment. E.g. before the loop: 1 Harry, during the loop: 2 Harries, after the loop: 1 Harry.
Yes, but you could very well think about something equivalent to the time-turner that exchange matter between the past and the present, instead of just sending matter to the past, in a way that keeps energy conservation. It would be harder to use practically,but wouldn't change anything to "energy conservation" vs "determinism" issues.
Don't forget that to fully conserve energy, you have to maintain not only the total mass, but also the internal chemical potentials of whatever thing you're shifting into the past and its gravitational potential energy with respect to the rest of the universe. I think you'll have a hard time doing this without just making an exact copy of the object. "Conservation of energy" is a much harder constraint than is obvious from the three words of the English phrase.
I don't see that as a theoritecal problem against a plausible universe having such a mechanism. We could very well create a simulation in which when you timetravel, the total energy (internal from mass and chemical bounds, external from gravity and chemical interaction with the exterior) is measured, and exchanged for exactly that amount from the source universe. If we can implement it on a computer, it's possible to imagine a universe that would have those laws.
The hard part in time-turner physics (because it's not computable) is the "stable time loop", not the "energy conservation" part (which is computable).
Yep.