How does the idea of entropy relate to something like E8

[Jimster41]

I get that a Lie group like E8 is smoothly differentiable. But as I understand it, it has dimension and structure. Does it support the Idea of a phase space, and an entropy measure for that phase space, or at least for regions of it?

 

[fzero]

Mathematically, for a manifold to admit a phase space structure, Hamiltonian system, etc., it must be symplectic. On a closed symplectic manifold, the cohomology class $[\omega]$ of the symplectic form is nonzero. The group $E_8$ is compact and closed, but $H^2(E_8)=0$ (see http://www.mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0021.0024.ocr.pdf), so $E_8$ itself is not a symplectic manifold. Therefore, we don't expect to be divide coordinates into positions and momenta, define a Hamiltonian and use the definition of entropy you seem to have in mind.

The cotangent bundle of any manifold is symplectic. We could take position variables to label a point in $E_8$ and the cotangent vectors at each point as momenta. I'm not sure whether this is what you had in mind.

 

[Jimster41]

One specific question to clarify whether I am getting cohomology at all: A space with only non-commuting operators has no cohomology class? Or a space with any non-commuting operators has no cohomology class. Or is that non-sense.So as I understand it the SM has a set of complementary properties (non-commuting observable operators). They are pair-wise, fixed, and bounded in number. I have been associating that list with the "Superselection rules".

As I understand it the E_8 is meant to be a more complete, coherent, structurally consistent version of that. Like a SUSY? But that may be missing something major.

The cotangent bundle of any manifold is symplectic. We could take position variables to label a point in E8E_8 and the cotangent vectors at each point as momenta. I'm not sure whether this is what you had in mind.

If I understand what you are saying, one could lay out the E_8 as list of property pairs with the QM superselection rules that enforce complementarity - pairing them, and constraining values. This 2d space would then be symplectic, with non zero cohomology over all values, and would represent the rule-space of classical states for E_8 - as "interfaced to" or organized by QM uncertainty and superselection.

If that makes any sense at all, it's exactly what I was thinking...

 

[fzero]

One specific question to clarify whether I am getting cohomology at all: A space with only non-commuting operators has no cohomology class? Or a space with any non-commuting operators has no cohomology class. Or is that non-sense.

Cohomology is a topological property that can be defined in various ways, but here I am using the concept of de Rham cohomology, which is based on differential forms. On the other hand, I can define noncommuting operators with any topology (even on Euclidean spaces). If you wanted to narrow the definition of "noncommuting operator" to tangent vectors or something, even an abelian group like a torus can have nonzero cohomology classes in every dimension. I don't think there is any sort of existence argument of the type you are suggesting.

So as I understand it the SM has a set of complementary properties (non-commuting observable operators). They are pair-wise, fixed, and bounded in number. I have been associating that list with the "Superselection rules".

As I understand it the E_8 is meant to be a more complete, coherent, structurally consistent version of that. Like a SUSY? But that may be missing something major.

Well, in particle physics, we have symmetry groups and the particles can have nonzero charges under the groups. These charges then become labels that we can use to distinguish elementary particles from one another. The term superselection usually means something a bit different; like the inability for a state of zero net charge to evolve into a state with nonzero net charge.

If $E_8$ were a natural symmetry, then we would also label the elementary particles by their charges under $E_8$. That's a very different thing from trying to define a phase space on $E_8$ though.

If I understand what you are saying, one could lay out the E_8 as list of property pairs with the QM superselection rules that enforce complementarity - pairing them, and constraining values. This 2d space would then be symplectic, with non zero cohomology over all values, and would represent the rule-space of classical states for E_8 - as "interfaced to" or organized by QM uncertainty and superselection.

If that makes any sense at all, it's exactly what I was thinking...

The cotangent bundle isn't symplectic because we are adding in cohomology. The cotangent bundle is not compact, so those cohomology classes are no longer required to exist.

 

[Jimster41]

Thanks for taking the time to try and explain some of that. Though it is fun to try, I'm not able say I have any foothold. It is just plain over my head. I don't have the vocabulary for it. Wish I did.

No pain, no gain I guess.
If I were going to try to workout on a few key concepts/terms any recommendations?

I've always found the sort of zoological diversity of the SM to be so overwhelming. I like the E8 and similar because I can picture one of those cool mandala-like graphics as a mental placeholder for "the structure, pattern, symmetry that dictates what the fundamental physical building blocks can be, and how they can change, combine and interact" But I've not found anything that methodically decomposes one of those diagrams via a key -explaining what the colors, shapes, lines etc mean. I have some hope that if I could find such a thing it would help.

And when I imagine such a schematic, in that naive way, it seems plausible that one could use it to calculate if not the states of some reality building processes, then at least a set of probability biases, or flows, critical points, sinks, etc.

And I got here wondering if it were possible to separate out a classical deterministic system from a set of hypothetical hidden QM variables using such a paradigm. In other words how does one of those diagrams organize/reflect the classical QM boundary problem?

Then the question occurred to me today, how does an Ising lattice model relate to one of those things?

 

[fzero]

I don't know your physics/math background, but if you've got a solid understanding of physics at the level of the Feynman lectures, you might be able to understand some of Georgi's Lie algebras in particle physics. I don't think there's much there on $E_8$, but it would provide an important base for future studies.