I’ve never been familiar enough with group-theory stuff to memorize the names (which, warning, also might mean that it will take you a lot of time to write a sufficiently-dumbed-down version), but the internet suggests ~Iso(n,1) is related to… the Minkowski metric? I would be flabbergasted to learn that something so specific-to-our-universe was relevant to this toy mathematical contraption.
I thank you for your effort! I am currently missing a lot of the mathematical background necessary to make that post make sense, but I will revisit it if I find myself with the motivation to learn!
It looks to me like James Camancho is wrong, as I pointed out in my comment there. Regardless if he is correct or not, it is in my opinion a terrible explanation if you only know basic quantum mechanics, and also not good even if you have a fair bit more background.
Here is my relatively short explanation of the argument:
James Camancho is arguing that because the symmetries of relativity in 1D must act on quantum states in a simple way. Namely, you should be able to split into subspaces, where in each subspace you just multiply by for angle (giving the angle in the symmetry group) and constant depending only on the subspace. He then says that must be rational to have a finite number of rules.
I disagree—nobody said the cellular automaton’s rules had to be symmetry transformations of the space. Additionally, nobody said the space is 1D—the dimension of the cellular automata you are talking about is just that you have a 1D “tape” of states, not that the states themselves are wavefunctions in 1D. For example, if you were imagining qubits implemented as spin states, then that’s definitely not 1D wavefunctions.
Even if I’m missing something and he’s right anyways, you should note that this would only be about what you can implement in our universe. A different physics could have different symmetries, and thus different representations. Some variant symmetries and representations are even proposed extensions to the current settled science!
More details:
The physical laws of the universe are invariant under certain symmetries, and studying the implications of this fact is quite important to modern physics. If it’s a symmetry then it should be invertible; and the observation probabilities |<x|y>|^2 for normalized states must be the same. In other words if we take two states, normalize, and compute |<x|y>| we should get the same thing before and after transforming. But then by Wigner’s theorem there is a unique (up to global phase scale) unitary or antiunitary transformation (as in, <Ux|Uy> = <x|y> - or for antiunitary it’s the conjugate) that gives the same transformation.[1] It’s an easy exercise to show that a surjective unitary transformation between inner product spaces must be linear (otherwise I think it must be antilinear). Regardless, we have what’s called a projective unitary representation, meaning our symmetries act on projective space by unitary operators.
A representation is a group acting on a vector space via linear transformations.
Alright, now we can look at the projective representations of the symmetries that relativity implies (called the Poincare group, which has translations (and time translations) and rotations along with Lorentz boosts). Details in footnote[2].
The projective representations of the 1D version of the Poincare group’s universal cover are apparently of the simple form mentioned earlier.
This is usually written in ‘projective space’, which means you consider two vectors the same when they differ only by a nonzero scale factor—because wavefunctions that differ by a global scale factor aren’t physically different.
We have a problem—projective representations are annoying. Well, if we basically look at the derivative we get a linear algebra thing, which can be easily deprojectivized. Then, we can basically integrate. However, the derivatives only capture the local structure, and will thus “not see” any global problems—for example, the circle and the real line both have the same tangent space at the origin, but the circle ‘curves’ in. If we just integrate, we’ll get the real line. In general we’ll get the version of the circle without any ‘loops’ that can’t be shrunk down continuously to a point. That is, we want the universal cover, which is the simply connected (meaning all loops can be shrunk) space that locally looks like a bunch of copies of the symmetries. For example, the universal cover of the circle is a ‘helix’ that projects down to the circle.
I’ve never been familiar enough with group-theory stuff to memorize the names (which, warning, also might mean that it will take you a lot of time to write a sufficiently-dumbed-down version), but the internet suggests ~Iso(n,1) is related to… the Minkowski metric? I would be flabbergasted to learn that something so specific-to-our-universe was relevant to this toy mathematical contraption.
I wrote up my explanation as its own post here: https://www.lesswrong.com/posts/LpcEstrPpPkygzkqd/fractals-to-quasiparticles
I thank you for your effort! I am currently missing a lot of the mathematical background necessary to make that post make sense, but I will revisit it if I find myself with the motivation to learn!
It looks to me like James Camancho is wrong, as I pointed out in my comment there. Regardless if he is correct or not, it is in my opinion a terrible explanation if you only know basic quantum mechanics, and also not good even if you have a fair bit more background.
Here is my relatively short explanation of the argument:
James Camancho is arguing that because the symmetries of relativity in 1D must act on quantum states in a simple way. Namely, you should be able to split into subspaces, where in each subspace you just multiply by
I disagree—nobody said the cellular automaton’s rules had to be symmetry transformations of the space. Additionally, nobody said the space is 1D—the dimension of the cellular automata you are talking about is just that you have a 1D “tape” of states, not that the states themselves are wavefunctions in 1D. For example, if you were imagining qubits implemented as spin states, then that’s definitely not 1D wavefunctions.
Even if I’m missing something and he’s right anyways, you should note that this would only be about what you can implement in our universe. A different physics could have different symmetries, and thus different representations. Some variant symmetries and representations are even proposed extensions to the current settled science!
More details:
The physical laws of the universe are invariant under certain symmetries, and studying the implications of this fact is quite important to modern physics. If it’s a symmetry then it should be invertible; and the observation probabilities |<x|y>|^2 for normalized states must be the same. In other words if we take two states, normalize, and compute |<x|y>| we should get the same thing before and after transforming. But then by Wigner’s theorem there is a unique (up to global phase scale) unitary or antiunitary transformation (as in, <Ux|Uy> = <x|y> - or for antiunitary it’s the conjugate) that gives the same transformation.[1] It’s an easy exercise to show that a surjective unitary transformation between inner product spaces must be linear (otherwise I think it must be antilinear). Regardless, we have what’s called a projective unitary representation, meaning our symmetries act on projective space by unitary operators.
A representation is a group acting on a vector space via linear transformations.
Alright, now we can look at the projective representations of the symmetries that relativity implies (called the Poincare group, which has translations (and time translations) and rotations along with Lorentz boosts). Details in footnote[2].
The projective representations of the 1D version of the Poincare group’s universal cover are apparently of the simple form mentioned earlier.
This is usually written in ‘projective space’, which means you consider two vectors the same when they differ only by a nonzero scale factor—because wavefunctions that differ by a global scale factor aren’t physically different.
We have a problem—projective representations are annoying. Well, if we basically look at the derivative we get a linear algebra thing, which can be easily deprojectivized. Then, we can basically integrate. However, the derivatives only capture the local structure, and will thus “not see” any global problems—for example, the circle and the real line both have the same tangent space at the origin, but the circle ‘curves’ in. If we just integrate, we’ll get the real line. In general we’ll get the version of the circle without any ‘loops’ that can’t be shrunk down continuously to a point. That is, we want the universal cover, which is the simply connected (meaning all loops can be shrunk) space that locally looks like a bunch of copies of the symmetries. For example, the universal cover of the circle is a ‘helix’ that projects down to the circle.