So "quantum information" is a bit of a pop science buzz word it sounds like. It's unfortunate that I've been trying to puzzle it out then.
However, some good came from this discussion and I think you all for your time. Specifically, the mention of Weyl quantization (by others) and symmetry...
I can't seem to wrap my head around the notion of conservation of quantum information. One thing that might help that is if someone can tell me what the associated symmetry is. For example, phase symmetry leads to conservation of electric charge according to Noether's theorem; a fact that...
Suppose a black hole isn't sucking in any new material. Then it is doomed to evaporate due to Hawking radiation and become smaller and smaller over time. Is there anything left when it's done evaporating?
I've been studying the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds but have almost zero background in physics. The WRT of a 3-manifold is closely related to the Chern-Simons (CS) invariant via the volume conjecture. My question is, what does the CS invariant of a 3-manifold...
Well, add the point at infinity to the real line and you are in business. I have no idea if that makes any physical sense though. The space S^2xS^1 is the complement of the unknot with no framing. The associated invariants are easy to compute. Unfortunately nothing interesting falls out as I...
I want to understand the topology of a black hole so that I can think about how (or if it's even possible) to compute its Witten-Reshetikhin-Turaev invariant.
I'm not sure about the physics term so maybe I should have stuck with the math. By cross section, I mean one of the boundaries of a cobordism between two 3-manifolds.
Can a black hole be presented as a Heegaard decomposition or as the complement of a knot?
I'll try and elaborate: If I understand correctly, the cross section of spacetime near a black hole can be thought of topologically as a manifold. What manifold is it? Can the manifold be decomposed?
If I understand the theory of quantum invariants of 3-manifolds correctly (possibly I don't), TQFTs on different presentations of closed 3-manifolds produce different values. However, the same quantum invariants (Reshetikhin-Turaev invariants for example) are produced on a closed manifold...
I stumbled onto the answer to my own question. I'm sufficiently motivated now. Anyway, the Reshetikhin-Turaev Invariant of a 3-manifold obtained from surgery on a link in #S^3# are colored jones polynomials of the link. Roughly (very roughly), calculate the Reshetikhin-Turaev Invariant of a...
The Reshetikhin-Turavev construction comes with an invariant that is sometimes called the Reshetikhin-Turaev Invariant. I'm currently attempting to wrap my head around this construction but was hoping for a sneak peak to help motivate me. My question is, what does the Reshetikhin-Turaev...
https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf
I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space ##Y## and a spacetime ##X## the author talks about the associated Quantum Hilbert Spaces...
That's quite possible. So then, what are characteristic numbers, how are characteristic numbers related to Dijkgraaf-Witten theory and what physical quantity (if any) do they correspond to in the real world?
The only type of characteristic numbers I'm aware of come from representation theory...
I like where your intuition is going with that. Unfortunately I would need to study the theorem of Noether you mention more extensively before even beginning to consider what you suggest. I do understand enough to see why you might suggest that though.
I don't think I articulated my question very well. Let me try again: The number of principal G bundles of a manifold is a topological invariant. What I would like to know is, does that invariant correspond to any physical quantity?
I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold.
My question is for the Physicists in the room, why do you want to know the number of...
I worked my way through this paper
http://www.math.harvard.edu/theses/senior/lee/lee.pdf
as part of a mathematics reading project and believe I have a fairly good understanding of the material. There is virtually no physics in this paper yet we seem to arrive at Dijkgraaf-Witten Theory quite...
I'm referring to ##g## in two different contexts which is what I think made my question unclear. So, let ##g## be the metric from GR and let ##d## be the metric from math class. Does defining ##g## on a differentiable manifold automatically induce ##d##?
As an aside, this is interesting to me...
You're right, I suppose I should make that precise. The metric topology I'm referring to has as open sets
##B_r(m_0)=\{m_0\in M : g(m_0,m)<r\ \forall m\in M \text{ where } r>0\}##
Is it fair to say, when talking about spacetime with a given metric, it would be redundant to state that the associated set has the metric topology placed on it. In other words, let ##M## be a set, ##O## the metric topology, ##\nabla## a connection, ##g## a metric, and ##T## be the direction of...
I was watching this video on Abstract Algebra and the professor was discussing how at one point a few mathematicians conjectured the special orthogonal group in ##\mathbb{R}^3## mod the symmetries of an icosahedron described the shape of the universe (near the end of the video).
My question is...
So I cleaned my solution up a little based on the comments but I also worked through mathwonks solution which is much nicer. It used some mathematics that were a little beyond where I'm at currently but I managed to get a handle on those mechanics by "reading ahead."
Anyway, if anyone is...
Thanks mathwonk and fresh_42 for the great comments. I've read them and will rework this problem shortly. Seems like it needs a lot of work though so it might take a day or two.
I'm trying to learn Category Theory; this isn't homework or anything. I've attached a problem from the text "Basic Homological Algebra" by Osborne and I show my attempt at a solution. My solution doesn't seem exactly correct and I state why in the attachment as well. Can someone take a look...
That's along the lines of what I was guessing. I suppose an idea for a paper would be to carry this line of thought out and make it precise. I suppose I would have to learn some GR which seems rather daunting.
At the risk of sounding ignorant I'd like to propose a question to someone well versed in Homological Algebra and General Relativity. I'm starting to study the tensor product functor in the context of category theory because I'm interested in possibly doing a paper on TQFT for a directed...
I have a question about Automorphisms. Please check the following statement for validity...
An automorphism of a group should map generators to generators. Suppose it didn't, well then the group structure wouldn't be preserved and since automorphisms are homomorphisms this would be a...
We were studying the Icosahedral (or dodecahedral which ever you prefer) group or equivalently the rotational symmetries of the Icosahedron.
I should elaborate on my question with a little more specifics using an example. In one step he claimed all elements of order 3 are conjugate and to...
My professor was proving that the Dodecahedron is isomorphic to ##A_5## and in the process utilized the stabilizer (which one can intuit ) of an edge, vertex or face to determine the conjugacy class (which is hard to intuit) of elements of the same order. This seems like a valuable skill but I...
This isn't homework, I'm just trying to refresh my memory on cyclic groups.
My question is, in this problem solution, how does ##{\sigma_i}^m=1## follow from ##\sigma_i## being disjoint?
http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf [Broken]
Please look at problem 2 on page 39 of the problems/solutions linked above.
I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of...
Here's a link to a professor's notes on a contour integration example.
https://math.nyu.edu/faculty/childres/lec12.pdf
I don't understand where the ##e^{i\pi /2} I## comes from in the first problem. It seems like it should be ##e^{i\pi}## instead since ##-C_3## and ##C_1## are both on the real...
OK thanks, just to be clear, are you saying I did parts (a) and (b) correct?
EDIT:
Actually, I clean my answer part (a) up a little bit below. It's not a significant change but it should make my reasoning more clear.
---------------
This is true. By hypothesis, there exists ##z_n\in...
Homework Statement
In each case, state whether the assertion is true or false, and justify your answer with a proof or counterexample.
(a) Let ##f## be holomorphic on an open connected set ##O\subseteq \mathcal{C}##. Let ##a\in O##. Let ##\{z_k\}## and ##\{\zeta_k\}## be two sequences...
Sure. The books introduction to meromorphic functions is scattered throughout the text and relies on a separate section on the Riemann Sphere to build an intuition for ##\hat{\mathbb{C}}##. The instructor doesn't like the use of the Riemann Sphere and told the class he would skip that section...
Agreed. For example, for (a), it's not difficult to show that there is a finite set around each pole and then use the fact that the union of finite sets are countable.
I'm looking for an additional introduction to meromorphic functions that includes this theorem. Any suggestions?
Hi,
In my textbook the following theorem is designated "Proposition 3.4.2 part (vi)". There are 6 parts total in the overall theorem. I'll just type the part I'm interested in below. My question is, is there a more standard name for this theorem? I would like to find an additional...
Sorry, wasn't trying to be rude. Your help was much appreciated and during the process I learned a lot. However, this solution is from the professor and was revealed before I was able to finish the problem using your strategy. For the sake of completeness I thought I should post it.
Yes, the...
Here's an outline to a solution:
##f## and ##g## holomorphic implies ##\frac{f}{g}(z)=\sum_{n=0}^{\infty}c_n(z-a)^n## is holomorphic for ##|z-a|<r##
By the product rule for power series
##f(z)=\frac{f}{g}(z)g(z)=\sum_{n=0}^{\infty}[\sum_{j=0}^{n}c_{j}b_{n-j}](z-a)^n##
Then by uniqueness of the...
This is probably obvious but just to be clear does ##f_{2N}=\sum_{n=0}^{2N}a_nz^n=a_0+a_1z^n+\cdots + a_{2N-1}z^{2N-1}+a_{2N}z^{2N}=f_N+\sum_{n=N+1}^{2N}a_nz^n## ?
Also, I'm not 100% clear on what you mean by a "grid". However, I wrote out each term of ##\sum_{j=0}^{p}c_jb_{p-j}## thinking it...