The universe has many conserved and approximately-conserved quantities, yet among them energy feels “special” to me. Some speculations why:
The sun bombards the earth with a steady stream of free energy, which leaves out into the night.
Time-evolution is determined by a 90-degree rotation of energy (Schrodinger equation/Hamiltonian mechanics).
Breaking a system down into smaller components primarily requires energy.
While aspects of thermodynamics could apply to many conserved quantities, we usually apply it to energy only, and it was first discovered in the context of energy.
I guess the standard rationalist-empiricist-reductionist answer would be to say that this is all caused by the second point combined with some sort of space symmetry. I would have agreed until recently, but now it feels circular to me since the reduction into energy relies on our energy-centered way of perceiving the world. So instead I’m wondering if the first point is closer to the core.
Sure, there are plenty of quantities that are globally conserved at the fundamental (QFT) level. But most most of.these quantities aren’t transferred between objects at the everyday, macro level we humans are used to.
E.g. 1: most everyday objects have neutral electrical charge (because there exist positive and negative charges, which tend to attract and roughly cancel out) so conservation of charge isn’t very useful in day-to-day life.
E.g. 2: conservation of color charge doesn’t really say anything useful about everyday processes, since it’s only changed by subatomic processes (this is again basically due to the screening effect of particles with negative color charge, though the story here is much more subtle, since the main screening effect is due to virtual particles rather than real ones).
The only other fundamental conserved quantity I can think of that is nontrivially exchanged between objects at the macro level is momentum. And… momentum seems roughly as important as energy?
I guess there is a question about why energy, rather than momentum, appears in thermodynamics. If you’re interested, I can answer in a separate comment.
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.
The best answer to the question is that it serves as essentially a universal resource that can be used to provide a measuring stick.
It does this by being a resource that is limited, fungible, always is better to have more of than less of, and is additive across decisions:
You have a limited amount of joules of energy/negentropy, but you can spend it on essentially arbitrary goods for your utility, and is essentially a more physical and usable form of money in an economy.
Also, more energy is always a positive thing, so that means you never are worse off by having more energy, and energy is linear in the sense that if I’ve spent 10 joules on computation, and spent another 10 joules on computation 1 minute later, I’ve spent 20 joules in total.
Cf this post on the measuring stick of utility problem:
Agree that free energy in many ways seems like a good resource to use as a measuring stick. But matter is too available and takes too much energy to make, so you can’t spend it on matter in practice. So it’s non-obvious why we wouldn’t have a matter-thermodynamics as well as an energy-thermodynamics. I guess especially with oxygen, since it is so reactive.
I guess one limitation with considering a system where oxygen serves an analogous role to sunlight (beyond such systems being intrinsically rare) is that as the oxygen reacts, it takes up elements, and so you cannot have the “used-up” oxygen leave the system again without diminishing the system. Whereas you can have photons leave again. Maybe this is just the fungibility property again, which to some extent seems like the inverse of the “breaking a system down into smaller components primarily requires energy” property (though your statements of fungibility is more general because it also considers kinetic energy).
The universe has many conserved and approximately-conserved quantities, yet among them energy feels “special” to me. Some speculations why:
The sun bombards the earth with a steady stream of free energy, which leaves out into the night.
Time-evolution is determined by a 90-degree rotation of energy (Schrodinger equation/Hamiltonian mechanics).
Breaking a system down into smaller components primarily requires energy.
While aspects of thermodynamics could apply to many conserved quantities, we usually apply it to energy only, and it was first discovered in the context of energy.
I guess the standard rationalist-empiricist-reductionist answer would be to say that this is all caused by the second point combined with some sort of space symmetry. I would have agreed until recently, but now it feels circular to me since the reduction into energy relies on our energy-centered way of perceiving the world. So instead I’m wondering if the first point is closer to the core.
Sure, there are plenty of quantities that are globally conserved at the fundamental (QFT) level. But most most of.these quantities aren’t transferred between objects at the everyday, macro level we humans are used to.
E.g. 1: most everyday objects have neutral electrical charge (because there exist positive and negative charges, which tend to attract and roughly cancel out) so conservation of charge isn’t very useful in day-to-day life.
E.g. 2: conservation of color charge doesn’t really say anything useful about everyday processes, since it’s only changed by subatomic processes (this is again basically due to the screening effect of particles with negative color charge, though the story here is much more subtle, since the main screening effect is due to virtual particles rather than real ones).
The only other fundamental conserved quantity I can think of that is nontrivially exchanged between objects at the macro level is momentum. And… momentum seems roughly as important as energy?
I guess there is a question about why energy, rather than momentum, appears in thermodynamics. If you’re interested, I can answer in a separate comment.
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.
The best answer to the question is that it serves as essentially a universal resource that can be used to provide a measuring stick.
It does this by being a resource that is limited, fungible, always is better to have more of than less of, and is additive across decisions:
You have a limited amount of joules of energy/negentropy, but you can spend it on essentially arbitrary goods for your utility, and is essentially a more physical and usable form of money in an economy.
Also, more energy is always a positive thing, so that means you never are worse off by having more energy, and energy is linear in the sense that if I’ve spent 10 joules on computation, and spent another 10 joules on computation 1 minute later, I’ve spent 20 joules in total.
Cf this post on the measuring stick of utility problem:
https://www.lesswrong.com/posts/73pTioGZKNcfQmvGF/the-measuring-stick-of-utility-problem
Agree that free energy in many ways seems like a good resource to use as a measuring stick. But matter is too available and takes too much energy to make, so you can’t spend it on matter in practice. So it’s non-obvious why we wouldn’t have a matter-thermodynamics as well as an energy-thermodynamics. I guess especially with oxygen, since it is so reactive.
I guess one limitation with considering a system where oxygen serves an analogous role to sunlight (beyond such systems being intrinsically rare) is that as the oxygen reacts, it takes up elements, and so you cannot have the “used-up” oxygen leave the system again without diminishing the system. Whereas you can have photons leave again. Maybe this is just the fungibility property again, which to some extent seems like the inverse of the “breaking a system down into smaller components primarily requires energy” property (though your statements of fungibility is more general because it also considers kinetic energy).
Thinking further, a key part of it is that temperature has a tendency to mix stuff together, due to the associated microscopic kinetic energy.