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The dependence on AC through the use of Zorn's lemma in the proof of the Choquet-Bruhat–Geroch theorem on the existence of a maximal globally hyperbolic development for solutions of the Einstein equat …

By itself, a (Yang-Mills) instanton is a classical concept. It is a solution of the classical Yang-Mills equations (considered on a manifold with a Riemannian, rather than a Lorentzian, metric), such …

Here is an argument whose conclusion is that it is unlikely that exotic 4d manifolds are physically important in general relativity. For physical reasons (mainly causality, stability, and determinism) …

Let me add a few comments that might fill in some gaps not covered by other answers. When would it be safe to say that gravity has been successfully quantized? A quantum theory consists of an alge …

Option (1) Use the definition $(\omega \otimes S) \wedge (\eta \otimes S) = (\omega \wedge \eta) \otimes (S\otimes T)$ of the wedge product for Lie algebra valued forms. Define Lie bracket and Killing …

If I read your updated question correctly, you are asking whether people have considered non-linear modifications of quantum mechanics in order to accommodate interacting QFTs. I'm sure someone, somew …

Willie's answer is of course correct. But the part that I think is most interesting in the physical context is only briefly mentioned in point 3. of the last paragraph. Let me expand on that. The rel …

Reed & Simon, Methods of modern mathematical physics I: Functional analysis (Academic Press, 1980): Chapter VIII, Section 3, Theorem VIII.6 (combined with property (b) of a projection-valued measure, …

In the form (1), if you compute the variation $\delta S / \delta x(t) = E(t)$, you find that $E(t) = E(x(t),\dot{x}(t), \ddot{x}(t) ,t)$ is a local/differential expression (the value of $E(t)$ does no …

I think from the physical point of view, the question should be reversed. Symplectic and presymplectic manifolds naturally occur in physics because such geometric structures naturally follow from a va …

Both proofs in the question are of the following type. Suppose you want to prove a certain identity among numbers $X$, say $f(X)=0$. If you can find another function $g(X,Y)$ such that $f(X)=g(X,Y)$ f …

Classical field theories (Lagrangian variational principles) sometimes come in families. The families may be finite dimensional, or also infinite dimensional. One could even take the family to consist …

There is lots of work going on in mathematical relativity (and more generally in non-linear hyperbolic PDEs) on trying to establish global non-linear stability around interesting exact solutions. The …

You are describing what is commonly known as second quantization (as you probably already realize). In a nutshell, the main mathematical statement behind second quantization is the following: the alge …

1.) Yes. In the commutative case, this is the statement of the Riesz representation theorem (any linear functional on $C(K)$ is an integral against a measure, which has to be positive by the positivit …