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27

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 ...


27

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 ...


22

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 ...


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 ...


16

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 ...


15

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 ...


14

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. $$\psi_L \rightarrow e^{i\alpha }\psi_L$$ ...


14

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 ...


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.)


12

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 ...


12

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} ...


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

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 ...


10

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 ...


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

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 ...


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

Here is an overview of quantization methods: http://arxiv.org/abs/math-ph/0405065 Most of this article deals with QM on manifolds.


10

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 ...


9

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 ...


9

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


9

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: ...


8

(1) What is topological degeneracy in strongly correlated systems such as FQH? From Wiki, http://en.wikipedia.org/wiki/Topological_degeneracy : Topological degeneracy is a phenomenon in quantum many-body physics, that the ground state of a gapped many-body system become degenerate in the large system size limit, and that such a degeneracy cannot be lifted ...


8

Regarding your first question, please state it more clearly. Without Hausdorff property, uniqueness property of limits of sequences fails and it implies many bad consequences for several results concerning abstract uniqueness, e.g. of solutions of differential equations on manifolds. Moreover, without Hausdorff you do not have smooth hat functions that are ...


8

This vector potential can be written in every point on the plane except the origin as: $$ A_x = -\frac{\partial \psi}{\partial y}$$ $$ A_y = \frac{\partial \psi}{\partial x} $$ with $$\psi = \frac{1}{2}\mathrm{log}(x^2+y^2)$$ $A$ is not exact, because $\psi$ is singular at the origin. But this means that the magnetic field is zero at every point except ...


8

I believe that Andy and Cumrun didn't want to say that this manifold would have a complex structure modulus in isolation. However, as is clear from the "conifold" setup, the manifold given by $xy-zt=\mu$ is being incorporated into a larger manifold, so this equation only describes the vicinity of some region. When you exploit the fixed asymptotic shape of ...


7

The explicit formula for the diffeomorphism between the affine quadric and the cotangent bundle $T^{*}S^3$ is known. It is given for example in: Hall and Mitchel's Coherent states on spheres (Equation 18). I'll write it here using the same notation of the question for completeness: $\mathbf{z}(\mathbf{x}, \mathbf{p}) = ...



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