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3

(1) Yes, take ${\cal H} = L^2(\mathbb R, dx)\oplus L^2(\mathbb R, dx)$ and thereon $\left(X (\psi, \phi)\right)(x,y) := (x\psi(x),y\phi(y))$. We have $\sigma(X)=\sigma_c(X)$ and the degeneracy is just $2$. (2) Yes, use the example (1) with a countably infinite copies of $L^2(\mathbb R, dx)$ and use the Hilbertian direct sum of Hilbert spaces. (There are ...

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I think you make it sound much more mysterious than it is. The relativistic distribution function is $$f_p = \frac{1}{(2\pi)^3}\exp(-(\mu-u\cdot p)/T)\,$$ where $u_\alpha$ is the 4-velocity of the fluid, $p_\alpha$ is the 4-momentum of the particle, $T$ is temperature, and $\mu$ is the chemical potential. This is sometimes called the Juttner ...

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There are attempts to use nonstandard analysis (e.g., Albeverio) or Colombeau algebras, but these haven't been developed very far. I haven't seen anything in terms of surreal numbers, but they may probably substitute for the infinitesimals in nonstandard analysis.

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Here is a purely geometrical way to think about this Edward says it is possible to cut a wedge out of a flat spacetime and glue the edges together. So in my mind this looks like a paper cone. A cone is flat precisely because it can be created by rolling up a flat sheet. Rolling up preserves the metric on the interior of the sheet (not on the boundaries ...

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What came to be called "discrete torsion" is simply the data that makes the B-field gerbe be equivariant over the orbifold. This was clarified by Eric Sharpe, see the references here: Eric Sharpe, Discrete Torsion and Gerbes I (arXiv:hep-th/9909108) Discrete Torsion and Gerbes II (arXiv:hep-th/9909120) Discrete Torsion, Quotient Stacks, and String ...

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I can only answer the mathematical part of your question (or make a stab at it). We could say that by describing a space as an orbifold, the singularities are taken care of by somehow declaring them to be under control. Where a manifold is a topological space that may be very complicated, but locally looks very nice, namely like $\mathbb R^n$, an orbifold ...

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