At a human level, the counts for each type of atom is basically always conserved too, so it’s not just a question of why not momentum but also a question of why not moles of hydrogen, moles of carbon, moles of oxygen, moles of nitrogen, moles of silicon, moles of iron, etc..
I guess for momentum in particular, it seems reasonable why it wouldn’t be useful in a thermodynamics-style model because things would woosh away too much (unless you’re dealing with some sort of flow? Idk). A formalization or refutation of this intuition would be somewhat neat, but I would actually more wonder, could one replace the energy-first formulations of quantum mechanics with momentum-first formulations?
> could one replace the energy-first formulations of quantum mechanics with momentum-first formulations?
Momentum is to space what energy is to time. Precisely, energy generates (in the Lie group sense) time-translations, whereas momentum generates spatial translations. So any question about ways in which energy and momentum differ is really a question about how time and space differ.
In ordinary quantum mechanics, time and space are treated very differently: t is a coordinate whereas x is a dynamical variable (which happens to be operator-valued). The equations of QM tell us how x evolves as a function of t.
But ordinary QM was long-ago replaced by quantum field theory, in which time and space are on a much more even footing: they are both coordinates, and the equations of QFT tell us how a third thing (the field ϕ(x,t)) evolves as a function of xand t. Now, the only difference between time and space is that there is only one dimension of the former but three of the latter (there may be some other very subtle differences I’m glossing over here, but I wouldn’t be surprised if they ultimately stem from this one).
All of this is to say: our best theory of how nature works (QFT), is neither formulated as “energy-first” nor as “momentum-first”. Instead, energy and momentum are on fairly equal footing.
I suppose that’s true, but this kind of confirms my intuition that there’s something funky going on here that isn’t accounted for by rationalist-empiricist-reductionism. Like why are time translations so much more important for our general work than space translations? I guess because the sun bombards the earth with a steady stream of free energy, and earth has life which continuously uses this sunlight to stay out of equillbrium. In a lifeless solar system, time-translations just let everything spin, which isn’t that different from space-translations.
Ah, so I think you’re saying “You’ve explained to me the precise reason why energy and momentum (i.e. time and space) are different at the fundamental level, but why does this lead to the differences we observe between energy and momentum (time and space) at the macro-level?
This is a great question, and as with any question of the form “why does this property emerge from these basic rules”, there’s unlikely to be a short answer. E.g. if you said “given our understanding of the standard model, explain how a cell works”, I’d have to reply “uhh, get out a pen and paper and get ready to churn through equations for several decades”.
In this case, one might be able to point to a few key points that tell the rough story. You’d want to look at properties of solutions PDEs on manifolds with metric of signature (1,3) (which means “one direction on the manifold is different to the other three, in that it carries a minus sign in the metric compared to the others in the metric”). I imagine that, generically, these solutions behave differently with respect to the “1″ direction and the “3” directions. These differences will lead to the rest of the emergent differences between space and time. Sorry I can’t be more specific!
Why assume a reductionistic explanation, rather than a macroscopic explanation? Like for instance the second law of thermodynamics is well-explained by the past hypothesis but not at all explained by churning through mechanistic equations. This seems in some ways to have a similar vibe to the second law.
At a human level, the counts for each type of atom is basically always conserved too, so it’s not just a question of why not momentum but also a question of why not moles of hydrogen, moles of carbon, moles of oxygen, moles of nitrogen, moles of silicon, moles of iron, etc..
I guess for momentum in particular, it seems reasonable why it wouldn’t be useful in a thermodynamics-style model because things would woosh away too much (unless you’re dealing with some sort of flow? Idk). A formalization or refutation of this intuition would be somewhat neat, but I would actually more wonder, could one replace the energy-first formulations of quantum mechanics with momentum-first formulations?
> could one replace the energy-first formulations of quantum mechanics with momentum-first formulations?
Momentum is to space what energy is to time. Precisely, energy generates (in the Lie group sense) time-translations, whereas momentum generates spatial translations. So any question about ways in which energy and momentum differ is really a question about how time and space differ.
In ordinary quantum mechanics, time and space are treated very differently: t is a coordinate whereas x is a dynamical variable (which happens to be operator-valued). The equations of QM tell us how x evolves as a function of t.
But ordinary QM was long-ago replaced by quantum field theory, in which time and space are on a much more even footing: they are both coordinates, and the equations of QFT tell us how a third thing (the field ϕ(x,t)) evolves as a function of x and t. Now, the only difference between time and space is that there is only one dimension of the former but three of the latter (there may be some other very subtle differences I’m glossing over here, but I wouldn’t be surprised if they ultimately stem from this one).
All of this is to say: our best theory of how nature works (QFT), is neither formulated as “energy-first” nor as “momentum-first”. Instead, energy and momentum are on fairly equal footing.
I suppose that’s true, but this kind of confirms my intuition that there’s something funky going on here that isn’t accounted for by rationalist-empiricist-reductionism. Like why are time translations so much more important for our general work than space translations? I guess because the sun bombards the earth with a steady stream of free energy, and earth has life which continuously uses this sunlight to stay out of equillbrium. In a lifeless solar system, time-translations just let everything spin, which isn’t that different from space-translations.
Ah, so I think you’re saying “You’ve explained to me the precise reason why energy and momentum (i.e. time and space) are different at the fundamental level, but why does this lead to the differences we observe between energy and momentum (time and space) at the macro-level?
This is a great question, and as with any question of the form “why does this property emerge from these basic rules”, there’s unlikely to be a short answer. E.g. if you said “given our understanding of the standard model, explain how a cell works”, I’d have to reply “uhh, get out a pen and paper and get ready to churn through equations for several decades”.
In this case, one might be able to point to a few key points that tell the rough story. You’d want to look at properties of solutions PDEs on manifolds with metric of signature (1,3) (which means “one direction on the manifold is different to the other three, in that it carries a minus sign in the metric compared to the others in the metric”). I imagine that, generically, these solutions behave differently with respect to the “1″ direction and the “3” directions. These differences will lead to the rest of the emergent differences between space and time. Sorry I can’t be more specific!
Why assume a reductionistic explanation, rather than a macroscopic explanation? Like for instance the second law of thermodynamics is well-explained by the past hypothesis but not at all explained by churning through mechanistic equations. This seems in some ways to have a similar vibe to the second law.