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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 …

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 …

What you seem (to me) to be asking is under which conditions on a Lorentzian manifold its Alexandrov topology not even $T_0$. If that is the case, then it is easy to see that if $(M,g)$ is not chronol …

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 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 …

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 …

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 …

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 …

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 …

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 …

$\DeclareMathOperator\Ann{Ann}\DeclareMathOperator\Tr{Tr}$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 "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 …

In the formulation of QFT using formal functional integrals, as mentioned by Igor in his answer, the Schwinger-Dyson equation becomes an infinite-dimensional differential equation for the partition fu …

If you want a criterion which is not tautological, that is, beyond the very definition of equivalence of *-representations, there are (at least) two situations where there is a criterion for equivalen …