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Now, suppose I am not interested in QFT (in the sense that I don't want to quantize a classical field) but, instead, I want to study many body quantum mechanics. Tough luck! :-) These are mathematic …

I believe all of these topics and more are also covered in Dereziński, Jan; Gérard, Christian, Mathematics of quantization and quantum fields, Cambridge Monographs on Mathematical Physics. Cambridge: …

These are quite orthogonal conditions. To start, one is a global condition, while the other is a local one. Every point has a small enough neighborhood that is strongly causal (even globally hyperbol …

AFAIK, people have not spent much time formalizing Wightman-style axioms for QFT in a category framework. On the other hand, categories and functors have been essential elements in formulating algebr …

If you require that the matrix $C$ in \eqref{2} preserves the involutions \eqref{3}, then you get the consequence $\tilde{\gamma}^\mu = C C^* \tilde{\gamma}^\mu (C C^*)^{-1}$, which then implies that …

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 …

Maybe you already encountered such maximal surfaces in the context of General Relativity. Still, the one application of spacelike maximal surfaces that I am aware of is as special kinds of initial dat …

A good way to meet your future adviser (besides already being located at their institution) is to go to conferences or workshops on the topic that you are interested. Referring to the topics of intere …

Trying to recover as much of the topology/geometry from the causal order as possible has been studied quit a bit since the early paper of Hawking et al that you cite. A quick summary of my understandi …

Although the focus of the original question was on conformal compactification, a necessary step along the way is an introduction of double-null coordinates that are regular on the horizons and bifurca …

I will leave aside what is meant by "effective field theory" in a purely mathematical context and just presume that the question asks whether renormalized interactive perturbative QFT (using formal po …

For a finite dimensional inner product space $(V,\eta)$, $\bigwedge^2 V \cong_\eta \mathfrak{so}(\eta) \subset \operatorname{End}(V) \cong V\otimes V^* \cong_\eta V\otimes V$. The antisymmetry conditi …

There is a quite detailed pedagogical presentation of both the Einstein-Hilbert and the Palatini variational principles for the Einstein equations in §III.3 Lagrangians for General Relativity of Baez …

If $W \subset V$ is a complemented subspace (you seem to be working in the finite dimensional context, so it always is) and $W'\subset V$ is a complement, this means that you get $W' \cong V/W$ and a …

Since your question is now asking for references, here are a few standard ones. For those who wish to study Lie groups and Lie algebras for the purposes of representation theory (one already mentioned …