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2

Intuitive answer: Keep in mind that in three dimensions you can have point (no dimension) and line (1D) defects. If you mean line defects, you're right, $2\pi$ line defects are unstable (although $2\pi$ point defects are stable). In a 2D nematic, only point defects are possible and you're also right: a $2\pi$ disclination in a 2D nematic is stable (in the ...


5

The "topological" in "topological order" and the "topological" in "topological insulator" have different meanings. The 'topological' in topological order means 'robust against ANY local perturbations'. The "topological" in "topological insulator" means 'robust against some local perturbations that respect certain symmetry'. In fact the properties of ...


1

As the causal future of $p$ is the set of points joined to $p$ by timelike or null curves, and the constant path $\gamma(t) = p$ joining $p$ to $p$ itself has vanishing tangent vector and hence is a null curve (though a rather silly one), $p \in J^+(p)$, and so, $S \subset J^+(S)$.


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I'm fairly sure I got it. The causal future $J^+(p)$ of a point $p$ is defined as the set of all points $q$ connected by a future pointing timelike or null curve to $p$. I think the secret lies in that this is a closed set in Minkowski spacetime. To see this, we see that the curves connecting the points in $J^+(p)$ are the timelike curves (negative length) ...


0

Comments to the question (v3): The possible quantum states depend on the topology of spacetime. For instance, the momentum in the extra (fifth) direction of a 5-dimensional Kaluza-Klein theory is quantized/discrete variable if the extra (fifth) dimension is a compact circle $S^1$, but a continuous variable if the extra (fifth) dimension instead is a ...


1

OP comments as an example of what the question is about: Let us consider the case of an electron confined to a curved surface. Does the geometry of the background have any consequences for the state space? A simple answer can be given for ordinary QM: A (scalar, i.e. spin-0) particle moving in one-dimension has state space $L^2(\mathbb{R})$, a particle ...


2

We are not entirely sure what OP's question (v4) is asking, but here are some comments: I) The Dirac belt trick demonstrates that the Lie group $SO(3)$ of 3D rotations is doubly connected, $$ \pi_1(SO(3))~=~\mathbb{Z}_2. $$ II) As for the title question Are spinors somehow connected to spacetime? one answer could be: Yes, in the sense that the mere ...


0

You might equally well ask, "How does the physical belt in the Dirac trick sense the topology?" This question is, when you think about it, no less mysterious than yours. The answer, by experiment, is that it simply does. And ultimately, if something transforms "compatibly" with the Lorentz group, or with $SO(3)$, then there is really only a one-bit question ...


-3

Here is a simple thought experiment to help visualize the shape of a Big Bang universe. All directions point back in time. Theoretically if one could see far enough back in time, one is looking towards the Origin, a single point, the only point we all have in common. Therefore all directions ultimately point toward the Origin, which in some sense can be ...


0

However, I think there are always possibilities that different structures of universe may also give rise to this redshift. Is it true? The redshift is on average the same in every direction and depends on the distance, so if it came not from cosmic expansion but from "structure" (matter, energy distribution) that would mean that we were in the cosmic ...



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