5 ms·

Fascinating article. I wonder if this is indeed the same E8 as the Lie group that Garrett Lisi suggested (incorrectly) as a possible structure for a theory of e

by Daneel_ 11y ago

Fascinating article. I wonder if this is indeed the same E8 as the Lie group that Garrett Lisi suggested (incorrectly) as a possible structure for a theory of everything?

andyjohnson0 11y ago
Lisi's proposal was discussed 8 years ago at [0]. A quick scan of [1] and [2] leads me to suspect that it is the same E8 - but I'm not a mathematician so don't go and bet any money on it... [0] https://news.ycombinator.com/item?id=258198 https://news.ycombinator.com/item?id=258198 [1] https://en.wikipedia.org/wiki/E8_(mathematics) https://en.wikipedia.org/wiki/E8_(mathematics) [2] https://en.wikipedia.org/wiki/Leech_lattice https://en.wikipedia.org/wiki/Leech_lattice
- gjm11 11y ago
  Not the exact same mathematical object, but closely related -- it's not at all coincidental that both are called E8. The lattice is really easy to describe. It's a set of points in 8-dimensional space, so each point is specified by 8 numbers. A point is in the lattice if the following conditions hold. (1) Either all its coordinates are integers, or all are half-integers. (2) The sum of the coordinates is an even integer. (And then the densest packing of spheres in 8-dimensional space puts the centre of a sphere at each point. The radius of the spheres is then 1/sqrt(2).) The E8 that features in Garrett Lisi's attempted "theory of everything" is a so-called "Lie group". I'll attempt to sketch the relationship between lattices like the E8 lattice, and "Lie algebras", and Lie groups, and particle physics, but note that all of what follows is really handwavy and sometimes just plain wrong. Sometimes on purpose, probably sometimes by mistake. I'm sorry about any mistakes, but the deliberate lies and sketchiness are necessary to keep this shorter than textbook length. I hope it's at least semi-comprehensible. (Although it's shorter than a textbook, it's also too long for a single HN comment. I'll break it into pieces and make each a reply to its predecessor. HN mods, let me know if this is considered bad form and I'll refrain from doing it in the future.) OK, here goes. PHYSICS AND SYMMETRY OK, let's begin at the physics end. Here are some facts about plain old Newtonian physics. (1) The physical laws make no reference to particular positions in space; equivalently, if we e.g. replace (x,y,z) with (x+a,y+b,z+c) everywhere, for constant a,b,c, then the laws don't change. (2) If you have a system of particles interacting only with one another, the total momentum (sum of mass times velocity) is constant. It turns out that #1 and #2 are sort of equivalent: for a broad class of physical theories, symmetries (like #1, the fact that adding a constant to the coordinates changes nothing) are equivalent to conservation laws (like #2, conservation of momentum). Here's a subtler one: the laws don't pick out preferred times any more than preferred positions, and that fact is equivalent to conservation of energy. That was all in Newtonian physics, but it works more or less the same in relativity and in quantum mechanics. There are some much subtler kinds of symmetry present in our physical laws, and they give rise to subtler conservation laws. For instance, you may be familiar with the notion of "phase" in optics; in a classical picture this corresponds to the exact location/timing of the electromagnetic waves that make up light; in quantum mechanics everything is described by a "wavefunction" whose value at every point in the relevant configuration space is a complex number, and the optical phase of light turns out to correspond to the phase of the photons' wavefunctions. Changing all the phases by the same constant amount leaves every possible observation you could make unaltered, which means it's a symmetry of the physical laws; and this symmetry turns out to correspond to conservation of electric charge. This picture gets more complicated when you throw in the "weak nuclear force", which turns out actually to be kinda the same thing as electricity and magnetism, and hairier still when you throw in the "strong nuclear force". You get symmetries that, e.g., describe exchanging different kinds of particle with one another (kinda), and corresponding conservation laws. GROUPS AND LIE GROUPS If you look at a high enough level of abstraction at what different kinds of symmetry have in common, you get the pure-mathematical notion of a group. The idea, roughly, is that if you have some thing (e.g., a block of metal or a physical theory) and some kind of symmetry (e.g., ways you can move the block of metal around and end up with it looking exactly the same as before; ways you can change some things in the laws of physics and end up with equivalent laws), then all the symmetries of the thing together have a particular kind of mathematical structure. This basically comes down to these rather obvious observations: if you have two operations that leave your thing looking the same then doing one and then the other also leaves your thing looking the same; and undoing one such operation also leaves your thing doing the same. And a mathematical structure that supports "doing one thing and then another", and "undoing a thing", is a group. These symmetries in physics have an important feature: they don't occur in isolation but come in continuously-varying families. E.g., translation-invariance, the (x+a,y+b,z+c) thing, allows any value of a,b,c; you can vary them continuously and get the same results all the time as you do so. This means that the symmetry groups that describe them mathematically have this continuously-varying property, and the mathematical formalization of this gives you the notion of "Lie groups". (Named after a mathematician called Sophus Lie.) Some Lie groups can be built out of other simpler ones, in something a bit like the way that some numbers can be written as the product of other smaller ones. If you break them down as far as possible, you get to the so-called "simple Lie groups", which are kinda like the prime numbers: every Lie group can be built out of simple Lie groups. And it turns out that we can figure out what all the possible simple Lie groups are. There are four infinite families of them called A,B,C,D, and a few special ones that don't fit into the families. The most complicated of these special ones is called E8.
  - gjm11 11y ago
    (Second part. There will be one more.) A BIT MORE PHYSICS Garrett Lisi's attempted "theory of everything" was based around the idea that the pattern of symmetries of the laws of physics is that of the group E8. We don't see all those symmetries in the laws we know, but that isn't necessarily a problem because of a thing called spontaneous symmetry breaking. Imagine a big tank of water. Within that tank (at least away from its edges) no direction is different from any other -- the laws of physics don't have preferred directions, and neither does a bunch of water molecules bouncing off one another at random. This symmetry corresponds to conservation of angular momentum, so e.g. movements of water within the tank should obey that law. OK; now what if we reduce the temperature and the water freezes? Ice crystals, unlike liquid water, have a regular structure and they do have a preferred direction; so some things that happen in our tank full of ice may fail to conserve angular momentum. (Let me be careful about this. Of course the overall laws of physics still satisfy that law, so if we consider the whole contents of the tank C of AM will still hold. But if we treat the ice in the tank as "background" and consider, I dunno, electrons moving around within the ice or something, their total angular momentum may fail to be constant, because of angle-dependent interactions with the ice.) The same kind of thing can happen with the more "abstract" kinds of symmetry: as the universe cools -- and note that the universe now is much much cooler than shortly after the big bang -- it may settle into some less-symmetrical configuration (like our tank of ice), and the asymmetries in the universe may look to use like asymmetries in the laws, even though the real fundamental laws are symmetrical. So, anyway, Lisi's theory is probably wrong, but the mere fact that E8 is bigger than the group of symmetries we actually observe in the universe isn't in itself good reason to disbelieve it. Now let's do (or at least gesture vaguely towards) some more mathematics, with the hope of ending up with some understanding of how the group E8 relates to the lattice E8. LIE ALGEBRAS To describe the relationship between the Lie group E8 and the lattice E8, I need to go via another kind of mathematical object called a Lie algebra. These are in some sense simpler cousins of the Lie groups, but they're harder to describe and harder to explain why you'd care about them. Still, here goes. Let's look at one of the very simplest Lie groups. Take the ordinary euclidean plane -- imagine an infinitely large perfectly flat piece of paper. Fix a point in it; call it O. And consider rotations around O. These form a Lie group; it's sometimes called U(1), never mind why. We can describe it geometrically: it's a circle. (Pick some point that isn't O; call it P. Describe any rotation by saying where P goes to. That point is always on the circle through P with centre O. Conversely, if we pick such a point, there's a rotation that takes P there.) A circle is pretty simple, but you know what's even simpler? A straight line. And if you look only at very small portions of the circle, they look a lot like very small portions of a straight line. This is the fundamental idea behind calculus: all nice things look linear if you look at a small enough scale. And linear things are easier to work with, mathematically. So: suppose we have a Lie group -- that is, a continuously-varying set of symmetries of something. Take a tiny region of it near to the origin; that is, near to the trivial symmetry that does nothing. This tiny region looks like a piece of euclidean space in some number of dimensions, but there's some extra structure on it that describes the relationships between those symmetries -- what happens when you do one followed by another. What does that extra structure look like? It turns out that it's best described by what's called a bracket operation, meaning that given two elements x,y we have another written [x,y] which you should think of as meaning xy-yx. Imagine e.g. that x,y are actually matrices. Matrix multiplication usually doesn't commute, so generally xy-yx is not zero. Here, in some sense the multiplication operation corresponds to the "do one symmetry followed by another" operation on the original Lie group; that no longer exists in our "infinitesimal" view, but the bracket describing what changes when you switch the order of the symmetries still does. Kinda. The bracket operation obeys a number of laws like the so-called Jacobi identity [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0. The details aren't important right now. Anyway, the idea that we end up with when we follow this path and do attend to the details is that of a thing called a "Lie algebra", which describes what small pieces of a Lie group look like, and which in general looks like a euclidean space of some dimension together with a "bracket operation". Corresponding to the Lie group called E8 there is, therefore, a Lie algebra also called E8.
- Sniffnoy 11y ago
  E_8 in that sense is the Lie group E_8. A Lie group has an associated Lie algebra, which has an associated root system, which gives rise to a root lattice. This E_8 is the root lattice. So yes they're very closely related though not the same object.
leecarraher 11y ago
yes it is the same.

by Daneel_ 11y ago

Fascinating article. I wonder if this is indeed the same E8 as the Lie group that Garrett Lisi suggested (incorrectly) as a possible structure for a theory of everything?

andyjohnson0 11y ago

Lisi's proposal was discussed 8 years ago at [0]. A quick scan of [1] and [2] leads me to suspect that it is the same E8 - but I'm not a mathematician so don't go and bet any money on it... [0] https://news.ycombinator.com/item?id=258198 https://news.ycombinator.com/item?id=258198 [1] https://en.wikipedia.org/wiki/E8_(mathematics) https://en.wikipedia.org/wiki/E8_(mathematics) [2] https://en.wikipedia.org/wiki/Leech_lattice https://en.wikipedia.org/wiki/Leech_lattice

gjm11 11y ago
Not the exact same mathematical object, but closely related -- it's not at all coincidental that both are called E8. The lattice is really easy to describe. It's a set of points in 8-dimensional space, so each point is specified by 8 numbers. A point is in the lattice if the following conditions hold. (1) Either all its coordinates are integers, or all are half-integers. (2) The sum of the coordinates is an even integer. (And then the densest packing of spheres in 8-dimensional space puts the centre of a sphere at each point. The radius of the spheres is then 1/sqrt(2).) The E8 that features in Garrett Lisi's attempted "theory of everything" is a so-called "Lie group". I'll attempt to sketch the relationship between lattices like the E8 lattice, and "Lie algebras", and Lie groups, and particle physics, but note that all of what follows is really handwavy and sometimes just plain wrong. Sometimes on purpose, probably sometimes by mistake. I'm sorry about any mistakes, but the deliberate lies and sketchiness are necessary to keep this shorter than textbook length. I hope it's at least semi-comprehensible. (Although it's shorter than a textbook, it's also too long for a single HN comment. I'll break it into pieces and make each a reply to its predecessor. HN mods, let me know if this is considered bad form and I'll refrain from doing it in the future.) OK, here goes. PHYSICS AND SYMMETRY OK, let's begin at the physics end. Here are some facts about plain old Newtonian physics. (1) The physical laws make no reference to particular positions in space; equivalently, if we e.g. replace (x,y,z) with (x+a,y+b,z+c) everywhere, for constant a,b,c, then the laws don't change. (2) If you have a system of particles interacting only with one another, the total momentum (sum of mass times velocity) is constant. It turns out that #1 and #2 are sort of equivalent: for a broad class of physical theories, symmetries (like #1, the fact that adding a constant to the coordinates changes nothing) are equivalent to conservation laws (like #2, conservation of momentum). Here's a subtler one: the laws don't pick out preferred times any more than preferred positions, and that fact is equivalent to conservation of energy. That was all in Newtonian physics, but it works more or less the same in relativity and in quantum mechanics. There are some much subtler kinds of symmetry present in our physical laws, and they give rise to subtler conservation laws. For instance, you may be familiar with the notion of "phase" in optics; in a classical picture this corresponds to the exact location/timing of the electromagnetic waves that make up light; in quantum mechanics everything is described by a "wavefunction" whose value at every point in the relevant configuration space is a complex number, and the optical phase of light turns out to correspond to the phase of the photons' wavefunctions. Changing all the phases by the same constant amount leaves every possible observation you could make unaltered, which means it's a symmetry of the physical laws; and this symmetry turns out to correspond to conservation of electric charge. This picture gets more complicated when you throw in the "weak nuclear force", which turns out actually to be kinda the same thing as electricity and magnetism, and hairier still when you throw in the "strong nuclear force". You get symmetries that, e.g., describe exchanging different kinds of particle with one another (kinda), and corresponding conservation laws. GROUPS AND LIE GROUPS If you look at a high enough level of abstraction at what different kinds of symmetry have in common, you get the pure-mathematical notion of a group. The idea, roughly, is that if you have some thing (e.g., a block of metal or a physical theory) and some kind of symmetry (e.g., ways you can move the block of metal around and end up with it looking exactly the same as before; ways you can change some things in the laws of physics and end up with equivalent laws), then all the symmetries of the thing together have a particular kind of mathematical structure. This basically comes down to these rather obvious observations: if you have two operations that leave your thing looking the same then doing one and then the other also leaves your thing looking the same; and undoing one such operation also leaves your thing doing the same. And a mathematical structure that supports "doing one thing and then another", and "undoing a thing", is a group. These symmetries in physics have an important feature: they don't occur in isolation but come in continuously-varying families. E.g., translation-invariance, the (x+a,y+b,z+c) thing, allows any value of a,b,c; you can vary them continuously and get the same results all the time as you do so. This means that the symmetry groups that describe them mathematically have this continuously-varying property, and the mathematical formalization of this gives you the notion of "Lie groups". (Named after a mathematician called Sophus Lie.) Some Lie groups can be built out of other simpler ones, in something a bit like the way that some numbers can be written as the product of other smaller ones. If you break them down as far as possible, you get to the so-called "simple Lie groups", which are kinda like the prime numbers: every Lie group can be built out of simple Lie groups. And it turns out that we can figure out what all the possible simple Lie groups are. There are four infinite families of them called A,B,C,D, and a few special ones that don't fit into the families. The most complicated of these special ones is called E8.
- gjm11 11y ago
  (Second part. There will be one more.) A BIT MORE PHYSICS Garrett Lisi's attempted "theory of everything" was based around the idea that the pattern of symmetries of the laws of physics is that of the group E8. We don't see all those symmetries in the laws we know, but that isn't necessarily a problem because of a thing called spontaneous symmetry breaking. Imagine a big tank of water. Within that tank (at least away from its edges) no direction is different from any other -- the laws of physics don't have preferred directions, and neither does a bunch of water molecules bouncing off one another at random. This symmetry corresponds to conservation of angular momentum, so e.g. movements of water within the tank should obey that law. OK; now what if we reduce the temperature and the water freezes? Ice crystals, unlike liquid water, have a regular structure and they do have a preferred direction; so some things that happen in our tank full of ice may fail to conserve angular momentum. (Let me be careful about this. Of course the overall laws of physics still satisfy that law, so if we consider the whole contents of the tank C of AM will still hold. But if we treat the ice in the tank as "background" and consider, I dunno, electrons moving around within the ice or something, their total angular momentum may fail to be constant, because of angle-dependent interactions with the ice.) The same kind of thing can happen with the more "abstract" kinds of symmetry: as the universe cools -- and note that the universe now is much much cooler than shortly after the big bang -- it may settle into some less-symmetrical configuration (like our tank of ice), and the asymmetries in the universe may look to use like asymmetries in the laws, even though the real fundamental laws are symmetrical. So, anyway, Lisi's theory is probably wrong, but the mere fact that E8 is bigger than the group of symmetries we actually observe in the universe isn't in itself good reason to disbelieve it. Now let's do (or at least gesture vaguely towards) some more mathematics, with the hope of ending up with some understanding of how the group E8 relates to the lattice E8. LIE ALGEBRAS To describe the relationship between the Lie group E8 and the lattice E8, I need to go via another kind of mathematical object called a Lie algebra. These are in some sense simpler cousins of the Lie groups, but they're harder to describe and harder to explain why you'd care about them. Still, here goes. Let's look at one of the very simplest Lie groups. Take the ordinary euclidean plane -- imagine an infinitely large perfectly flat piece of paper. Fix a point in it; call it O. And consider rotations around O. These form a Lie group; it's sometimes called U(1), never mind why. We can describe it geometrically: it's a circle. (Pick some point that isn't O; call it P. Describe any rotation by saying where P goes to. That point is always on the circle through P with centre O. Conversely, if we pick such a point, there's a rotation that takes P there.) A circle is pretty simple, but you know what's even simpler? A straight line. And if you look only at very small portions of the circle, they look a lot like very small portions of a straight line. This is the fundamental idea behind calculus: all nice things look linear if you look at a small enough scale. And linear things are easier to work with, mathematically. So: suppose we have a Lie group -- that is, a continuously-varying set of symmetries of something. Take a tiny region of it near to the origin; that is, near to the trivial symmetry that does nothing. This tiny region looks like a piece of euclidean space in some number of dimensions, but there's some extra structure on it that describes the relationships between those symmetries -- what happens when you do one followed by another. What does that extra structure look like? It turns out that it's best described by what's called a bracket operation, meaning that given two elements x,y we have another written [x,y] which you should think of as meaning xy-yx. Imagine e.g. that x,y are actually matrices. Matrix multiplication usually doesn't commute, so generally xy-yx is not zero. Here, in some sense the multiplication operation corresponds to the "do one symmetry followed by another" operation on the original Lie group; that no longer exists in our "infinitesimal" view, but the bracket describing what changes when you switch the order of the symmetries still does. Kinda. The bracket operation obeys a number of laws like the so-called Jacobi identity [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0. The details aren't important right now. Anyway, the idea that we end up with when we follow this path and do attend to the details is that of a thing called a "Lie algebra", which describes what small pieces of a Lie group look like, and which in general looks like a euclidean space of some dimension together with a "bracket operation". Corresponding to the Lie group called E8 there is, therefore, a Lie algebra also called E8.
Sniffnoy 11y ago
E_8 in that sense is the Lie group E_8. A Lie group has an associated Lie algebra, which has an associated root system, which gives rise to a root lattice. This E_8 is the root lattice. So yes they're very closely related though not the same object.

leecarraher 11y ago

yes it is the same.