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The reason is that there is no mathematically rigorous construction of any interacting quantum field theory in four space-time dimensions to this date. Because of that, one has not been able so far to …

Quoting the first two paragraphs of V. I. Arnol'd, On teaching mathematics, Uspekhi Mat. Nauk 53 (1998) 229-234, translated to English in Russian Math. Surveys 53 (1998) 229-236 (a transcription may a …

As Abdelmalek Abdesselam pointed in his comment to the OP, the axiomatic approach to QFT is rather concerned with answering the question "what is a quantum field?". This is stated right at the Preface …

The standard reference for constructive QFT is the classic book by J. Glimm and A. Jaffe, Quantum Physics: a Functional Integral Point of View (2nd. ed., Springer-Verlag, 1988). It is certainly more t …

There is the construction of the C${}^*\!$-algebra of canonical anticommutation relations (CAR's), which is actually somewhat easier than the construction of free bosonic fields: given any complex pre …

For Lorentzian manifolds, the conformal completion need not be compact. A typical example is the universal covering of the $d$-dimensional anti-de Sitter space-time (the maximally symmetric solution o …

$\DeclareMathOperator\Ann{Ann}\DeclareMathOperator\Tr{Tr}$My answer is somewhat complementary to Nik Weaver's, and admitedly more focused on Question 2 since I have nothing more to add to the latter r …

The signature convention $(−,+,\cdots,+)$ is more commonly used in General Relativity and Lorentzian geometry because of the desire among their practicioners to make a closer parallel to Riemannian ge …

I believe the crucial point behind the physical significance of Lebesgue measure as opposed to Jordan measure boils down to the issue of completeness, as Gerald Edgar remarked in part (2) of his answe …

A small complement to Abdelmalek Abdesselam's answer: on the rigorous, non-perturbative side, there is also a recent (originally two-part, now turned into three-part) exposition by Jonathan Dimock, av …

One must remark that derivatives in Sobolev spaces are usually taken in the sense of distributions: given $k\in\mathbb{N}_0=\{0,1,2,\ldots\}$, $H^k(\mathbb{R}^n)$ is the space of tempered distribution …

Judging by your notation, I reckon you are getting the background for your questions from the Appendix to Section IX.8 of the book by M. Reed and B. Simon, Methods of Modern Mathematical Physics II: F …

There is a fundamental misunderstanding in your translation of Hamilton's formalism to classical field theory, which pertains to the proper identification of dynamical variables. In classical mechanic …

The answer to both questions is no. This is due to two facts: The Klein-Gordon two-point distribution $\omega_2(x,y)=\langle\Omega,\phi(x)\phi(y)\Omega\rangle$ in $\mathbb{R}^4$, where $\Omega_1=\Ome …

The "infinite tensor product" picture may be useful as a sort of concrete image of the state space of a quantum field theory, but in practice is rarely used because of the technical difficulties it br …