This 8d space is _a_ Lie algebra, but it's not the e8 Lie algebra, it's the Cartan subalgebra. The set of 240 8-dimensional vectors described in the link is the E8 root system, which lives inside the Cartan subalgebra.
Actually constructing the Lie group E8 is apparently quite complicated, and...
Coin, you're getting closer, but you are confusing E8 (the Lie group) with e8 (its Lie algebra).
E8, the Lie group is a 248 dimensional manifold. It has a multiplication that is NOT always commutative. i.e. a*b != b*a.
e8, the Lie algebra is the 248 dimensional tangent space of E8. It has an...
Actually Loops 07 was back in June (TWF 253 is from June). But you can get some of the audio and slides online.
Also John didn't go to Loops, if you read carefully you will see he is talking about Garrett going to Loops.
Part 1 of a basic introduction to E8 is now posted at http://sigfpe.blogspot.com/
So far it's just an introduction to the concepts of Lie group and Lie algebra, but it is written in a very accessible way.
Christine Dantas compiled a good list of resources on her old blog, http://christinedantas.blogspot.com/ - you have to scroll down to the heading "A basic curriculum for Quantum Gravity".
In particular I recommend:
- Carlo Rovelli's book "Quantum Gravity". This is probably more accessible than...
It does! If it were otherwise I'm not sure anyone would take entanglement entropy seriously as a candidate for the black hole entropy.
The problem is that you get a divergent answer unless there is some kind of minimal length scale for the entropy to depend on.
So you have to put the theory...
No. One can also define what it means for a mixed state to be entangled, but the theory is more complicated because there are several different useful measures of entanglement. For pure states the situation is much simpler, because these measures are all equal to the entanglement entropy.
It's true that by measuring one half of an entangled state you can learn about the other half. But entanglement can't be used to send signals, so there is no violation of causality. In particular, there's no way that someone on the other side of the horizon could send a message back to the...
You're right that the entanglement entropy is the same as the entropy of the thermal atmosphere. But because the entanglement entropy is symmetric, it is also the entropy of the vacuum state restricted to the black hole interior. If we take this point of view, the entanglement entropy counts the...
There really is a very good reason it is called entanglement entropy, which I completely glossed over. It's based on these theorems:
1. A pure state has zero entanglement entropy if and only if it is a product state. So the standard definition that George gave of "not entangled" is equivalent...
Hi Marcus.
The von Neumann entropy is the natural generalization of entropy for quantum systems. It's defined in terms of the density matrix as
S(\rho) = -\text{Tr}(\rho \log \rho)
This formula just means you take the eigenvalues of the density matrix p_1 \ldots p_n and apply the usual Gibbs...
The argument of (1) doesn't follow logically. Hawking's derivation might be incorrect (it might have incorrect assumptions), but that doesn't make the conclusion wrong. The fact that different derivations lead to the same conclusion suggest that Hawking's conclusion was correct.
To the best of...
AP,
If you're still stuck looking for references on noncommutativity in LQG, Ashtekar's very recent review "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions" includes emergence of noncommutative QFT as one of the four advances. Unfortunately this seems to apply...
Thanks Marcus. I'm not a student at Perimeter and don't know the authors, so I haven't heard anything about it. But it's likely that they will speak about it at one of the quantum gravity meetings.