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54
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6answers
3k views

What is known about the topological structure of spacetime?

General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions: What topological restrictions do Einstein's equations ...
14
votes
4answers
1k views

Shape of the universe?

What is the exact shape of the universe? I know of the balloon analogy, and the bread with raisins in it. These clarify some points, like how the universe can have no centre, and how it can expand ...
20
votes
7answers
2k views

Quantum mechanics on a manifold

In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ...
24
votes
10answers
4k views

Applications of Algebraic Topology to physics

I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most ...
23
votes
4answers
1k views

Is topology of universe observable?

There is an idea that the geometry of physical space is not observable(i.e. it can't be fixed by mere observation). It was introduced by H. Poincare. In brief it says that we can formulate our ...
12
votes
2answers
693 views

Is spacetime simply connected?

As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn. This falls into the ...
12
votes
4answers
1k views

Can spacetime be non-orientable?

This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In ...
15
votes
2answers
377 views

Topology of phase space

Context: From Liouville's integrability theorem we know that: If a system with $n$ degrees of freedom exhibits at least $n$ globally defined integrals of motion (i.e. first integrals), where all ...
21
votes
4answers
1k views

Is there a physical system whose phase space is the torus?

NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure. In an answer to the question What kind of ...
7
votes
2answers
1k views

Book covering Topology required for physics and applications

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
8
votes
3answers
641 views

Is anyone studying how the general topology of spacetime arises from more fundamental notions?

Stephen Wolfram in his book A New Kind of Science touches on a model of space itself based on automata theory. That it, he makes some suggestions about modelling not only the behaviour of matter ...
40
votes
2answers
2k views

Intuitively, why are bundles so important in Physics?

This question probably seems silly and I don't really know if it fits properly here, but the point is the following: I've seem the notion of bundles, fiber bundles, connections on bundles and so on ...
15
votes
2answers
423 views

Global Properties of Spacetime Manifolds

When solving the Einstein field equations, $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi GT_{\mu\nu}$$ for a particular stress-energy tensor, we obtain the metric of the spacetime manifold, ...
15
votes
1answer
513 views

Does the existence of Higgs imply the existence of Magnetic Monopoles?

I am aware that in theories with spontaneous symmetry breaking, Magnetic Monopoles can exist as topological solitons. Can the same be done with the Standard Model gauge group. I am familiar with the ...
30
votes
4answers
3k views

Why does a flat universe imply an infinite universe?

This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all.
11
votes
3answers
433 views

What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
20
votes
1answer
917 views

Why is there no theta-angle (topological term) for the weak interactions?

Why is there no analog for $\Theta_\text{QCD}$ for the weak interaction? Is this topological term generated? If not, why not? Is this related to the fact that $SU(2)_L$ is broken?
7
votes
2answers
587 views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some ...
5
votes
3answers
295 views

What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
6
votes
2answers
200 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
10
votes
1answer
780 views

Topological insulators: why K-theory classification rather than homotopy classification?

I am reading a 2009 paper by Kitaev on K-theory classification of topological insulators. In the 4th page, 1st paragraph in the section "Classification principles", he says, Continuous ...
6
votes
2answers
438 views

Confusion about duality transformation in 1+1D Ising model in a transverse field

In 1+1D Ising model with a transverse field defined by the Hamiltonian \begin{equation} H(J,h)=-J\sum_i\sigma^z_i\sigma_{i+1}^z-h\sum_i\sigma_i^x \end{equation} There is a duality transformation which ...
12
votes
0answers
347 views

Does the existence of instantons imply non-trivial cohomology of spacetime?

Gauge theories are considered to live on $G$-principal bundles $P$ over the spacetime $\Sigma$. For convenience, the usual text often either compactify $\Sigma$ or assume it is already compact. An ...
6
votes
2answers
3k views

How is the topological $Z_2$ invariant related to the Chern number? (e.g. for a topological insulator)

This question relates to the $Z_2$ invariant defined e.g. for topological insulators: Is it correct to relate $Z_2$ = 1 to an odd Chern number and $Z_2$ = 0 to an even Chern number? If yes, is it ...
6
votes
1answer
210 views

Can closed loops evade the spin-statistic theorem in 3 dimensions?

The famous spin-statistics result asserts that there are only bosons and fermions, and that they have integer and integer-and-a-half spin respectively. In two-dimensional condensed matter systems, ...
5
votes
2answers
602 views

Chiral edge state as topological properity of bulk state

As far as I know, quantum hall effect and quantum spin hall effect has chiral edge state. Chiral edge state is usually closely related with delocalization, since back scattering is forbidden. However, ...
3
votes
1answer
573 views

Large gauge transformations

I would like to understand what is the importance of large gauge transformations. I read that these gauge transformation cannot be deformed to the identity, but why should we care about that?
4
votes
1answer
194 views

Why doesn't topological phase transition break any symmetry? Hidden symmetry?

This question may be superficial. However why all people saying this without a proof? Just like the "hidden variables" assumption in quantum mechanics, can one disproof that there is no hidden ...
7
votes
1answer
812 views

geometry inside the event horizon

I'm trying to understand intuitively the geometry as it would look to an observer entering the event horizon of a schwarszchild black hole. I would appreciate any insights or corrections to the above. ...
6
votes
2answers
252 views

Basic question on the Aharonov-Bohm effect

I have a very basic question on the Aharonov-Bohm effect. Why is the curve integral $\oint_\Gamma {A}\cdot d{r}$ non-zero ? $\Gamma$ is the "difference" of both paths $P_1$ and $P_2$. If the ...
5
votes
1answer
299 views

Topology and Majorana bound states

I'm working at the moment on Majorana Bound states and their topological properties. Now I have a question about it. The Altland-Zirnbauer symmetry classes says us how many topological different ...
5
votes
1answer
329 views

What does it mean to “wrap” a D-brane around some manifold?

I am getting quite confused with this terminology when I read the papers. Like while constructing the near horizon $AdS_3$ in the $D1-D5$ system one considers $IIB$ on $R^{1,4}\times M^4 \times S^1$ ...
3
votes
0answers
266 views

Is it mathematically possible or topologically allowable for cutouts, or cavities, to exist in a 3-manifold?

A few weeks back, I posted a related question, Could metric expansion create holes, or cavities in the fabric of spacetime?, asking if metric stretching could create cutouts in the spacetime manifold. ...
3
votes
1answer
418 views

Is a preferred reference frame of the universe the old aether?

About two years ago I posted a question about a symmetrical twin paradox: Here. Recently a new answer was posted and an intense discussion ensued: Here. One of the points discussed concerns a ...
1
vote
1answer
271 views

Defining Euclidean global AdS

How does one see that that the Euclidean AdS is the same as the hyperbolic space at the same dimension ie $EAdS_n = \mathbb{H}_n = SO_0(n,1)/SO(n)$? Or is this to be seen as the definition of ...
6
votes
3answers
881 views

Aharonov-Bohm Effect and Flux Quantization in superconductors

Why is the magnetic flux not quantized in a standard Aharonov-Bohm (infinite) solenoid setup, whereas in a superconductor setting, flux is quantized?
5
votes
3answers
608 views

Does spacetime in general relativity contain holes?

Are there physical models of spacetimes, which have bounded (four dimensional) holes in them? And do the Einstein equations give restrictions to such phenomena? Here by holes I mean ...
5
votes
3answers
514 views

Curvature of Conical spacetime

Inspired by: Angular deficit The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
4
votes
1answer
177 views

How does nature prevent transient toroidal event horizons?

How does nature prevent transient toroidal event horizons?.. and does it really need to? Steps to construct a (transient) toroidal event horizon in a asymptotically flat Minkowski spacetime: take a ...
3
votes
1answer
40 views

Can one make a synthetic dimension “curl around” into a cylinder?

A really cool recent proposal, Synthetic Gauge Fields in Synthetic Dimensions. A. Celi et al. Phys. Rev. Lett. 112, 043001 (2014), arXiv:1307.8349, shows how you can simulate a synthetic ...
3
votes
2answers
439 views

Excluding big bang itself, does spacetime have a boundary?

My understanding of big bang cosmology and General Relativity is that both matter and spacetime emerged together (I'm not considering time zero where there was a singularity). Does this mean that ...
1
vote
2answers
104 views

Coset space and transitiviy

I have a question regarding coset space or homogeneous space $SO(n+1)/SO(n)$ which is simply $S^n$. I need some intuition regarding this result. As everyone knows that for a simple case of ...
0
votes
1answer
550 views

Does the universe have an edge/boundary/barrier? If yes, what is at the edge? [duplicate]

My question related kind of to asking what the shape of the universe is. Say a hypothetical alien civilization built an faster-than-light spaceship. If they keep flying would they end up where they ...