Timeline for QFT and mathematical rigor
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 13, 2021 at 18:28 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Added remarks, small rephrasings, typos corrected
|
| May 27, 2021 at 17:42 | comment | added | Abdelmalek Abdesselam | @Blazej: See this article by Magnen, Rivasseau and Sénéor projecteuclid.org/journals/… and references therein to work of Balaban and Federbush. | |
| May 27, 2021 at 17:27 | comment | added | Blazej | Pedro you mention that Euclidean Yang-Mills theory for $d=4$ is constructed in finite volume, do you have a reference for that? | |
| May 27, 2021 at 14:39 | comment | added | Pedro Lauridsen Ribeiro | @AbdelmalekAbdesselam whoa. That's impressive. Thanks for the important news! | |
| May 27, 2021 at 14:37 | comment | added | Pedro Lauridsen Ribeiro | (continued) In this case, strictly speaking one is really no longer talking about Hilbert spaces, but about the more general Krein spaces. For those, one can still define a topology and a notion of completeness (which may be defined for any topological vector space, since completeness depends only on the uniform structure associated to a vector topology), but the procedure is far more complicated. The book by Bognár I quoted above studies this matter in depth. | |
| May 27, 2021 at 14:35 | comment | added | Abdelmalek Abdesselam | About $\phi^4$ in $d=4$, the problem is not that open anymore. Confirming RG based predictions by physicists, Aizenman and Duminil-Copin have proved triviality in an article soon to appear in Annals of Math annals.math.princeton.edu/articles/17711 | |
| May 27, 2021 at 14:17 | comment | added | Pedro Lauridsen Ribeiro | A complex (resp. real) Hilbert space is a complex (resp. real) vector space $V$ endowed with an inner product, that is, a positive definite Hermitian (resp. symmetric) sesquilinear (resp. bilinear) form $\langle\cdot,\cdot\rangle$ whose associated norm $\|x\|=\sqrt{\langle x,x\rangle}$ makes $V$ a complete normed vector space. In finite dimensions completeness is automatic. Physicists tend to abuse language when they speak of inner products, sometimes including indefinite but still nondegenerate Hermitian (resp. symmetric) sesquilinear (resp. bilinear) forms into the above definition. | |
| May 27, 2021 at 13:06 | comment | added | JustWannaKnow | @Pedro, do you know if this is what is usually assumed to "define" Hilbert spaces in physics books? I've heard several times "this Hilbert space has an indefinite inner product" or even "A Hilbert space is a vector space with a positive-definite inner product" but I never understood this assertions. So actually the difference is between Hilbert spaces and Kreub spaces? | |
| May 27, 2021 at 12:35 | comment | added | Pedro Lauridsen Ribeiro | One has the classic book by J. Bognár, Indefinite Inner Product Spaces (Springer-Verlag, 1974). | |
| May 27, 2021 at 7:49 | comment | added | lalala | Do you know any good introductory material to Krein spaces? | |
| May 27, 2021 at 6:04 | comment | added | Jules Lamers | @PedroLauridsenRibeiro Thanks a lot for this summary of the state of the art for some of the prototypical examples of QFT | |
| May 27, 2021 at 6:01 | comment | added | Pedro Lauridsen Ribeiro | @JulesLamers As a rule, if the theory has been completely constructed and satisfies the Euclidean version of the Wightman axioms - the so-called Osterwalder-Schrader axioms - one can recover the theory in Lorentzian signature, which will then satisfy the Wightman axioms. This cannot be done if one only knows the Euclidean theory in finite volume, as it's the case for QED in $d=3$ and pure Yang-Mills in $d=4$. In the case of non-interacting models, the construction is done directly in Lorentz signature if the lack of interaction is know a priori. This was not the case for $\phi^4$ in $d>4$. | |
| May 27, 2021 at 5:56 | comment | added | Pedro Lauridsen Ribeiro | (continued) Massless QED in $d=2$ (Schwinger model) is equivalent to a free field theory and hence is not difficult to construct. Massive QED in $d=3$ has been recently constructed (2020) also in finite volume by Dimock. The case of $d=4$ QED is also open - both QED and $\phi^4$ in $d=4$ are believed to be trivial, but this is based on information coming from partial resummation of the perturbative series, so it's hard to tell for sure at our current state of knowledge. | |
| May 27, 2021 at 5:51 | comment | added | Jules Lamers | @PedroLauridsenRibeiro Thanks. Does this apply for Lorentzian as well as Euclidean signature? | |
| May 27, 2021 at 5:49 | comment | added | Pedro Lauridsen Ribeiro | @JulesLamers $\phi^4$ models have been constructed for $d=2,3$ and $d>4$ space-time dimensions, and are known to satisfy the Wightman axioms. However, for $d>4$ the interaction doesn't survive after non-perturbative renormalization, i.e. the theory is free (trivial). The case $d=4$ is still open. Pure Abelian gauge theories are non-interacting and are not hard to constructed rigorously, apart from the fact that in covariant gauges the field acts on a Krein space, as mentioned above. Pure non-Abelian Yang-Mills is known (I think) for $d=2$, but for $d=4$ only in finite volume. | |
| May 27, 2021 at 5:41 | comment | added | Jules Lamers | Do you know if $\phi^4$ theory (scalar field with quartic potential) fits in any axiomatic approach? And abelian (say Maxwell, with $G=U(1)$) gauge theory? And, more ambitiously, nonabelian gauge theory (with $G=SU(2)$, say)? | |
| May 26, 2021 at 23:42 | vote | accept | JustWannaKnow | ||
| May 26, 2021 at 23:42 | comment | added | JustWannaKnow | again thank you for the answer! This was really enlightening and I feel happy to understand a little more by having such contact with experts on the subject! | |
| May 26, 2021 at 23:26 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Small rephrasing
|
| May 26, 2021 at 23:17 | history | answered | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |