New answers tagged topology
3
votes
Accepted
Is the electroweak gauge group a semidirect product?
Note that the group monomorphism $$U(1)_Q\quad \stackrel{i}{\hookrightarrow}\quad G~:=~ N\times U(1)_Y, \qquad N~:=~SU(2)_I,$$ depends on the Weinberg angle, cf. e.g. my Phys.SE answer here. Here the ...
- 187k
1
vote
Accepted
What does really mean to glue the endpoints of a closed string?
The equivalence relation $\sigma\sim\sigma+2\pi$ is just constructing a circle for you: $\mathbb{S}^1 \cong \mathbb{R}/2\pi\mathbb{Z}$.
Instead of using the equivalence class $[\sigma]$ we use a ...
- 3,994
2
votes
Accepted
Self-connected Einstein-Rosen wormhole
The wormhole itself is regarded as the common boundaries of one or two manifolds when identified with each other to glue them together. Therefore the (three-dimensional) hypersurface with $r=r_\mathrm{...
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2
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Quantum Field Theory on non-globally hyperbolic spacetimes?
The difficulty in doing QFT on non-globally hyperbolic spacetimes is that, because there is no Cauchy surface, there is prima facie no (well motivated) way to define the canonical commutation ...
2
votes
Accepted
Wormholes as instantons?
If one defines an instanton as a Euclidean solution, then the answer is no, because there are Lorentzian wormholes.
If one restricts to Euclidean wormholes, and stipulates that to be an instanton, it ...
- 12k
2
votes
Accepted
Why don’t black rings exist in 3+1 dimensions?
Here is a naive intuitive argument. Heuristically we should be able to construct a black ring solution $(M^d,g^d)$ on a $d$-dimensional spacetime as follows. First take a black hole solution $(M^{d-1}...
- 187k
2
votes
Accepted
Gauge transformation as transition functions?
The same way you can locally view diffeomorphisms as coordinate changes.
Let $\pi: P\rightarrow M$ be a principal $G$-bundle. As it is well-known, for a PFB, local trivializations and local sections ...
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How is a wormhole (Einstein-Rosen Bridge) different than a tunnel?
While a tunnel in mountain is a "hole" in matter, a wormhole is a "hole" in fabric of spacetime. Although spacetime and matter are closely related via equations of general ...
- 1,053
4
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In what way are Lie groups generated by the basis of their Lie algebra?
Maybe as a simple example, consider $O(2)$ (real $2\times 2$ matrices $M$ with $M^T M=\mathbb{1}$). You can see that $\det M=\pm1$, so the group is disconnected (you cannot have a continuous path from ...
- 3,910
8
votes
Accepted
In what way are Lie groups generated by the basis of their Lie algebra?
The main points are that given a Lie group $G$,
the Lie algebra $\mathfrak{g}=T_eG$ is the tangent space of the identity element $e\in G$.
generators are a basis for the Lie algebra $\mathfrak{g}$.
...
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0
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In what way are Lie groups generated by the basis of their Lie algebra?
Let us consider simply connected Lie groups first. Because all of them are groups, the group property allows every element of the group to be generated by the exponential map on the Lie algebra space ...
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