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Hot answers tagged topology

50

All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of the standard lore deals just with the local and infinitesimal aspects -- the perturbative aspects_ and fiber bundles play little role there. But they are the all-important structure that govern the global -- the non-perturbative -- aspect. Bundles are the ...

33

We need to be precise about the phrase the size of the universe. Specifically I'm going to take it to mean the maximum possible separation between any two points. In an infinite universe two points can be separated by an arbitrarily large distance, so if the maximum distance between two points is finite this means the universe must not be infinite. The ...

32

Topology is of fundamental importance even to systems in classical mechanics. The configuration space (or phase space) of a generic classical mechanical system is a manifold and manifolds are topological spaces with some extra structure (e.g. a smooth structure in the case of smooth manifolds). At the very start of any classical mechanics problem, you need ...

26

Basically, this is related to the theory of Riemann surfaces. Such surfaces arise in complex analysis when one tries to extend the complex plane to make multi-valued functions well-defined. Take a simpler example first. Consider the function $f(z) = \sqrt{ z}$. This function is not single valued since $f( e^{2\pi i } z ) = - f(z)$. The usual way this is ...

24

I'm not going to provide a full answer here, because I don't know the answer, but I want to give some statements that illustrate quite nicely the kind of problems one would face when determining topology of anything: We know spacetime is a manifold. That means, locally, it looks just like $\mathbb{R}^4$. That's already a bummer. We can't do jack at one ...

20

Just one additional point that I haven't seen mentioned above: if the space-time has non-trivial fundamental group, it won't be seen by an observer at infinity. This is the content of the Topological Censorship Theorem. The implication is that for an asymptotically flat space-time, any interesting topology will be hidden behind the event-horizon. The proof ...

18

There are a bunch of questions here. Let me try to take them in order: Is it possible that our Universe has the feature that if you travel far enough you return to where you started? Yes. The standard Big-Bang cosmological model is based on the idea that the Universe is homogeneous and isotropic. One sort of homogeneous spacetime has the geometry of a 3-...

18

In the presence of massless chiral fermions, a $\theta$ term in can be rotated away by an appropriate chiral transformation of the fermion fields, because due to the chiral anomaly, this transformation induces a contribution to the fermion path integral measure proportional to the $\theta$ term Lagrangian. $$\psi_L \rightarrow e^{i\alpha }\psi_L$$ $${\... 17 As far as I understand it there are essentially two ways in which you can study quantum mechanics on a manifold with some curvature. Classically speaking these two ways lead to the same physics but in a quantum mechanical approach they are distinct. The first approach is to think of a particle moving "freely" through three-dimensional space, but subject to ... 17 Let me first answer your second questions about the physical intuition behind fiber bundles: Fiber bundles ( with compact structure groups) describe internal degrees of freedom such as spin and isospin just as manifolds describe translational degrees of freedom. For example, (a non-trivial fibre bundle is needed to describe the rotation of a neutral spinning ... 15 This claim is simply wrong. The flat hyperplane is of course infinite, but non trivial topologies can be flat and still finite. The simplest example is the 3-torus, but there are even the Klein bottle and the Hantzsche-Wendt manifold. See for example page 27 of Janna Levin - Topology and the Cosmic Microwave Background, which show you ten different closed ... 14 First a warning: I don't know much about either algebraic topology or its uses of physics but I know of some places so hopefully you'll find this useful. Topological defects in space The standard (but very nice) example is Aharonov-Bohm effect which considers a solenoid and a charged particle. Idealizing the situation let the solenoid be infinite so that ... 14 Euler density is simply the integrand in 2n dimensions of the integral that is equal to the Euler characteristic. The Euler characteristic may be written as the integral of the following Euler density in 2n dimensions:$$E_{2n} = \frac{1}{2^n} R_{i_1 j_1 k_1 l_1} \dots R_{i_n j_n k_n l_n} \epsilon^{i_1 j_1 \dots i_n j_n} \epsilon^{k_1 l_1 \dots k_n l_n} $... 14 Well, the simplest case is that some topologies of spacetime may only allow a particular class of metrics. But unfortunately, it usually requires the knowledge of the metric at every point to be quite certain. Here's a few thing we can probably assume about the spacetime manifold : All the usual jazz about manifolds in general relativity (paracompactness,... 14 The bible for the mathematical formulation of classical Mechanics, namely Foundations of Mechanics by Abraham and Marsden, defines a hamiltonian system as a triple$(M, \omega, X_H)$where$(M, \omega)$is a symplectic manifold, and$X_H$is the Hamiltonian vector field corresponding to a hamiltonian function$H:M\to\mathbb R$. Now, are there typically any ... 13 Our spacetime cannot be unorientable. That's because the laws of physics describing our spacetime are not left-right symmetric. We say that they break the P-symmetry (parity) or that they are "chiral" (derived from a Greek word for the hand which is either left or right.) For example, a left-handed neutrino would turn into a right-handed neutrino if you ... 13 Very loosely speaking the reasoning is this. Imagine a two band system in which the fermi sea has one filled band with Chern number$n$and another system with$N$filled bands but also with Chern number$n$. Physically they have the same topological properties (for example the same Hall conductance, edge states and so on), but cannot be deformed ... 13 No, because a Lorentz transformation is continuous with a continuous inverse. While an open ball is not mapped to itself, it is mapped to some other open set, in an invertible way. (That a Lorentz transformation is continuous of course follows from that it is linear.) 11 I don't know the answer, but your intuition is right on -- the fact that the equations are local doesn't mean that there cannot be a constraint on the topology of a global solution. For example, in Euclidean signature,$R_{ij} = g_{ij}$immediately implies that the scalar curvature is positive, which in turn leads to topological constraints. If the four-... 11 the answer to the Magellan question for our Universe is actually "No". If the Universe were a static 3-sphere, as Ted Bunn suggests, then you could "swim" around the Universe just like Megallan around the Earth. But an important fact about our Universe is that its size is changing: the size can't stay constant just like the apple can't sit in the middle of ... 11 I was always told that to find whether or not a field is conservative, see if the curl is zero. This is almost always true, but not always true. I have now been told that just because the curl is zero does not necessarily mean it is conservative. Correct! To illustrate what's going on, let's do an example. Conside the following vector field:$$\vec{... 10 Here is an overview of quantization methods: http://arxiv.org/abs/math-ph/0405065 Most of this article deals with QM on manifolds. 10 No. Firstly, weak cosmic censorship can only hold in the generic sense, as there are known examples of nakedly singular space-times. (See, e.g. Christodoulou 1993, and Christodoulou 1999.) Observe in particular that the nakedly singular space-time constructed in the 1993 paper is spherically symmetric with a central axis, and the initial data is ... 10 Orbifolds are spaces of the type$O = M/G$where$M$is a manifold and$G$is a group acting nonfreely on$M$. That is there are fixed points (or more generally submanifolds of this action); i.e., points$x \in M$such that$G.x = x$for all$G$. These fixed points are called singular points, they have the property that geometrical objects (such as the ... 10 You can't do calculus on a manifold that is not Hausdorff and paracompact. If you can't do calculus, doing physics with field equations is pretty pointless. 10 The phase space is a symplectic manifold, so any manifold$\mathcal{M}$that admits a closed nondegenerate 2-form is a possible phase space. Now, what is necessary (or sufficent) for admitting such a form? First, as you mention,$\mathcal{M}$must be even-dimensional. Second,$\mathcal{M}$must be orientable. Why? Because orientability is equivalent to ... 10 As has been discussed in many questions around here (e.g. here), relativity tells us only about local properties and behavior of a space-time. There are some exceptions when we make global assumptions - if we have a space of globally and strictly constant positive curvature, non-trivial topology is imminent because the space has to be the 3-sphere$\mathbb{S^...

10

My favorite reference for these sorts of things that straddle physics and geometry is Frankel's "The geometry of physics". In the chapter on harmonic forms, you will find what he refers to simply as "Hodge's Theorem". It's a little more general than you need, because it applies to general $p$-forms, and you only need functions (0-forms). So I'll ...

9

Since only a fraction of the universe is actually observable, I am not sure if it possible to answer this question.

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