{"id":13330,"date":"2019-11-28T07:51:08","date_gmt":"2019-11-28T15:51:08","guid":{"rendered":"http:\/\/www.preposterousuniverse.com\/blog\/?p=13330"},"modified":"2020-11-26T07:50:06","modified_gmt":"2020-11-26T15:50:06","slug":"thanksgiving-14","status":"publish","type":"post","link":"https:\/\/preposterousuniverse.com\/blog\/2019\/11\/28\/thanksgiving-14\/","title":{"rendered":"Thanksgiving"},"content":{"rendered":"\n

This year we give thanks for space<\/a>. (We’ve previously given thanks for the Standard Model Lagrangian<\/a>, Hubble’s Law<\/a>, the Spin-Statistics Theorem<\/a>, conservation of momentum<\/a>, effective field theory<\/a>, the error bar<\/a>, gauge symmetry<\/a>, Landauer’s Principle<\/a>, the Fourier Transform<\/a>, Riemannian Geometry<\/a>, the speed of light<\/a>, the Jarzynski equality<\/a>, and the moons of Jupiter<\/a>.)<\/p>\n\n\n\n

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Via Wikipedia<\/a><\/figcaption><\/figure><\/div>\n\n\n\n

Even when we restrict to essentially scientific contexts, “space” can have a number of meanings. In a tangible sense, it can mean outer space<\/a> — the final frontier, that place we could go away from the Earth, where the stars and other planets are located. In a much more abstract setting, mathematicians use “space” to mean some kind of set with additional structure, like Hilbert space or the space of all maps between two manifolds. Here we’re aiming in between, using “space” to mean the three-dimensional manifold in which physical objects are located, at least as far as our observable universe is concerned.<\/p>\n\n\n\n

That last clause reminds us that there are some complications here. The three dimensions we see of space might not be all there are; extra dimensions<\/a> could be hidden from us by being curled up into tiny balls (or generalizations thereof) that are too small to see, or if the known particles and forces are confined to a three-dimensional brane<\/a> embedded in a larger universe. On the other side, we have intimations that quantum theories of gravity imply the holographic principle<\/a>, according to which an N-dimensional universe can be thought of as arising as a projection of (N-1)-dimensions worth of information. And much less speculatively, Einstein and Minkowski taught us long ago that three-dimensional space is better thought of as part of four-dimensional spacetime<\/a>.<\/p>\n\n\n\n

Let’s put all of that aside. Our everyday world is accurately modeled as stuff, distributed through three-dimensional space, evolving with time. That’s something to be thankful for! But we can also wonder why<\/em> it is the case.<\/p>\n\n\n\n

I don’t mean “Why is space three-dimensional?”, although there is that<\/a>. I mean why is there something called “space” at all? I recently gave an informal seminar on this at Columbia, and I talk about it a bit in Something Deeply Hidden<\/a><\/em>, and it’s related in spirit to a question Michael Nielsen recently asked on Twitter, “Why does F=ma<\/em>?<\/a>” <\/p>\n\n\n\n

Space is probably emergent rather than fundamental<\/a>, and the ultimate answer to why it exists is probably going to involve quantum mechanics, and perhaps quantum gravity in particular. The right question is “Why does the wave function of the universe admit a description as a set of branching semi-classical worlds, each of which feature objects evolving in three-dimensional space?” We’re working on that<\/a>! <\/p>\n\n\n\n

But rather than answer it, for the purposes of thankfulness I just want to point out that it’s not obvious that space as we know it had to exist, even if classical mechanics had been the correct theory of the world. <\/p>\n\n\n\n

Newton himself thought of space as absolute and fundamental. His ontology, as the philosophers would put it, featured objects located in space, evolving with time. Each object has a trajectory, which is its position in space at each moment of time. Quantities like “velocity” and “acceleration” are important, but they’re not fundamental — they are derived from spatial position, as the first and second derivatives with respect to time, respectively. <\/p>\n\n\n\n

But that’s not the only way to do classical mechanics, and in some sense it’s not the most basic and powerful way. An alternative formulation is provided by Hamiltonian mechanics<\/a>, where the fundamental variable isn’t “position,” but the combination of “position and momentum,” which together describe the phase space<\/a> of a system. The state of a system at any one time is given by a point in phase space. There is a function of phase space cleverly called the Hamiltonian H<\/em>(x,p<\/em>), from which the dynamical equations of the system can be derived.<\/p>\n\n\n\n

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Via Wikipedia<\/a>.<\/figcaption><\/figure>\n\n\n\n

That might seem a little weird, and students tend to be somewhat puzzled by the underlying idea of Hamiltonian mechanics when they are first exposed to it. Momentum, we are initially taught in our physics courses, is just the mass times the velocity. So it seems like a derived quantity, not a fundamental one. How can Hamiltonian mechanics put momentum on an equal footing to position, if one is derived from the other?<\/p>\n\n\n\n

The answer is that in the Hamiltonian approach, momentum is not defined<\/em> to be mass times velocity. It ends up being equal to that by virtue of an equation of motion, at least if the Hamiltonian takes the right form. But in principle it’s an independent variable.<\/p>\n\n\n\n

That’s a subtle distinction! Hamiltonian mechanics says that at any one moment a system is described by two quantities, its position and its momentum. No time derivatives or trajectories are involved; position and momentum are completely different things. Then there are two equations telling us how the position and the momentum change with time. The derivative of the position is the velocity, and one equation sets it equal to the momentum divided by the mass, just as in Newtonian mechanics. The other equation sets the derivative of the momentum equal to the force. Combining the two, we again find that force equals mass times acceleration (derivative of velocity).<\/p>\n\n\n\n

So from the Hamiltonian perspective, positions and momenta are on a pretty equal footing. Why then, in the real world, do we seem to “live in position space”? Why don’t we live in momentum space?<\/p>\n\n\n\n

As far as I know, no complete and rigorous answer to these questions has ever been given. But we do have some clues, and the basic principle is understood, even if some details remain to be ironed out.<\/p>\n\n\n\n

That principle is this: we can divide the world into subsystems that interact with each other under appropriate circumstances. And those circumstances come down to “when they are nearby in space.” In other words, interactions are local<\/em> in space. They are not<\/em> local in momentum. Two billiard balls can bump into each other when they arrive at the same location, but nothing special happens when they have the same momentum or anything like that.<\/p>\n\n\n\n

Ultimately this can be traced to the fact that the Hamiltonian of the real world is not some arbitrary function of positions and momenta; it’s a very specific kind of function. The ultimate expression of this kind of locality is field theory — space is suffused with fields, and what happens to a field at one point in space only directly depends on the other fields at precisely the same point in space, nowhere else. And that’s embedded in the Hamiltonian of the universe, in particular in the fact that the Hamiltonian can be written as an integral over three-dimensional space of a local function, called the “Hamiltonian density.”<\/p>\n\n\n\n

\"H = \int \mathcal{H}(\phi, \pi) \, d^3x,\"<\/p>\n\n\n\n

where \u03c6 is the field (which here acts as a “coordinate”) and \u03c0 is its corresponding momentum.<\/p>\n\n\n\n

This represents progress on the “Why is there space?” question. The answer is “Because space is the set of variables with respect to which interactions are local.” Which raises another question, of course: why are interactions local with respect to anything<\/em>? Why do the fundamental degrees of freedom of nature arrange themselves into this kind of very specific structure, rather than some other one?<\/p>\n\n\n\n

We have some guesses there, too. One of my favorite recent papers is “Locality From the Spectrum<\/a>,” by Jordan Cotler, Geoffrey Penington, and Daniel Ranard. By “the spectrum” they mean the set of energy eigenvalues of a quantum Hamiltonian — i.e. the possible energies that states of definite energy can have in a theory. The game they play is to divide up the Hilbert space of quantum states into subsystems, and then ask whether a certain list of energies is compatible with “local” interactions between those subsystems. The answers are that most<\/em> Hamiltonians aren’t compatible with locality in any sense, and for those where locality is possible, the division into local subsystems is essentially unique. So locality might just be a consequence of certain properties of the quantum Hamiltonian that governs the universe.<\/p>\n\n\n\n

Fine, but why that Hamiltonian? Who knows? This is above our pay grade right now, though it’s fun to speculate. Meanwhile, let’s be thankful that the fundamental laws of physics allow us to describe our everyday world as a collection of stuff distributed through space. If they didn’t, how would we ever find our keys?<\/p>\n","protected":false},"excerpt":{"rendered":"

This year we give thanks for space. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, the Spin-Statistics Theorem, conservation of momentum, effective field theory, the error bar, gauge symmetry, Landauer’s Principle, the Fourier Transform, Riemannian Geometry, the speed of light, the Jarzynski equality, and the moons of Jupiter.) Even when we restrict […]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28],"tags":[],"class_list":["post-13330","post","type-post","status-publish","format-standard","hentry","category-science"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13330","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/comments?post=13330"}],"version-history":[{"count":4,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13330\/revisions"}],"predecessor-version":[{"id":13674,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13330\/revisions\/13674"}],"wp:attachment":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/media?parent=13330"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/categories?post=13330"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/tags?post=13330"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}