/sci/ - Science & Math

>>11438676

>>11411214
Weinberg

>>11261198
1. I have not.
2. The principal seminal volume of texts is Methods of Mathematical Physics by Simon and Reed, however this might be beyond what you are looking for. For exposition into the Green function method and distributions in general I'd suggest you looking into texts on PDEs and optimization, such as Schwarz or Brezis. Strocchi also has some good stuff on such things in his Non-Perturbative QFT text.
>As I understand, mathematical physics itself has different definitions
In general it means the study of the mathematical aspects of physical theories, but this itself already spans several if not all mathematical disciplines. It's hard for people who's been working on one end of math-phys to look into the opposite end.

>>11098944
But anon, physics and math are one and the same.

>>10959689
http://pirsa.org/
It's public-access.

>>10092712
There are in general two approaches to QM
1. Geometric quantization via classical mechanics and symplectic geoemtry (Woodhouse, Guillemin-Sternberg, Marsden-Weinstein).
2. C*-algebraic and constructive QM via local operator algebras, e.g. GNS representation (>>10092747, Haag, Strocchi-Swieca).
Both have their pros and cons,
1. Geometric quantization makes explicit the classical intuitions behind QM and the implementation of gauge invariance is straightforward via the moment map. In addition, the topological effects (i.e. A-B effect, Dirac monopole quantization, etc.) are almost immediate. However, such "nice" properties rely on some pretty stringent equivariant cohomological conditions, and the continuum limit (i.e. QFT) is not obvious.
2. Operator algebras and the GNS representation is a purely C*-algebraic construction, namely it is completely independent of whether you're in a QM or a QFT setting. In addition, the QFT you obtain automatically satisfies all the Streater-Wightman axioms and the Euclidean reconstruction theorem can be obtained with relative ease. However, it is not clear how gauge bundle structure actually affects the operator algebra, which makes gauge symmetry elusive; and the topology begins to intertwine with the algebraic geometry, which is a massive headache.
These two approaches have been proven (Zhang-Baez) to be equivalent in free theories via the particle-wave duality theorem, however, but it is not clear how to bridge the two in terms of their pros can cons.

>>10046191
"Just" boundary conditions doesn't constitute an entire field theory on the boundary, and "just" boundary conditions doesn't tell us anything about what the theory is in the bulk. AdS/CFT is a much stronger condition in which bulk anti-gravity theory is equivalent to CFT on the boundary.
There has been evidence for uTQFT/CFT correspondence in all but d = 4 (d = 2 is the Chern-Simons/conformal WZW correspondence proven in Kohno). However a CFT is encoded in unitary metaplectic categories, which have much more structure than bordism categories that encode TQFTs. If it were "just" boundary conditions then that'd entail boundary conditions encoding more information than the bulk theory, which is obviously wrong to even any undergrad.
So no. It's not "just" boundary conditions.

>>9534288
>what do you mean by 'use Grassmann numbers to represent the fermion operators?
Have you been listening at all? Grassmann functions are representations of fermionic wavefunctions in the coordinate basis.
>It makes every definition that then follows (Berezin integration, differentiation of Grassmann numbers, "Taylor expanding" Grassmann valued functions) look completely artificial.
Ask yourself "why are c-number functions not artificial?" The way those numbers work is also artificially constructed to suit our own understanding of the world.
>When Grassmann numbers and the integration functional are properly defined AND motivated, there are hardly any issues.
They are, and you should keep your issues to yourself before you've understood them.
Also you'll have to excuse me for the late replies. I'm currently organizing a workshop on topological field theory at California.

>>9461291
>perturbative aQFT has been under the radar of most people for quite some time, but the interest is rapidly increasing (at least this is true for the mathematicians, not sure if physicists like that kind of approach)
I don't think most physicists are even aware of regular AQFT, though this definitely seem interesting. What category in arxiv would perturbative AQFT go under?
>special Kähler stuff & affine diffgeo
Loop spaces are important in geometric quantization as well as CFT (they facilitate [math]U(1)[/math] gauge invariance), and nice Kähler structures can be put on them to study their geometric properties. And I believe affine diff geo could be used to study connections on moduli spaces which can be used to investigate the existence of wavefunctions and the like (e.g. existence of projectively flat Hitching connection on Verma modules [math]\Rightarrow[/math] existence of conformal blocks).

>>9439125
"This" as in the OP or "this" as in what I said? It should be obvious if you meant the former but it's not directly related to what I said if you meant the latter. Also the picture is a joke so you don't nee to take it so seriously.
>>9439190
>Regulating a theory in the IR is probably a well-defined procedure
"Well-defined" only in a very limited number of cases. There are theories (e.g. Yang-Mills with boundaries, AdS/CFT, non-Abelions on non-compact manifolds, chiral QCD, etc.) where the thermodynamic limit breaks internal symmetries that generate possibly unwanted Goldstone modes. The treatment of anomalies on a mathematically rigorous ground has only been done for TQFTs, CFTs and Yang-Mills, as far as I'm aware.
>In terms of feeling good about the foundations of field theory, I think realistically it will not matter.
Part of the reason why people find divergent S-matrices back in the 1950's is because they didn't know what Borel convergence is. Haag's theorem poses a problem in our fundamental understanding of what a QFT is (though there had been some ad hoc solutions such as considering families of Hamiltonians instead https://arxiv.org/abs/1501.05658)), but this is still unsatisfactory.
>Non-existence of the interaction picture is less of a problem (in my opinion) than the fact we use asymptotic series to compute finite quantities.
Except we understand fundamentally what goes wrong in the latter but not in the former.

>>9396187
Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.

>>9372869
https://arxiv.org/abs/1705.02240
The cobordism hypothesis can be proved, so where are the proofs of this homotopy "hypothesis"?
Answer: there are none, because cobordism hypothesis is the cornerstone of something concrete (i.e. TQFT) while this homotopy hypothesis is the cornerstone of absolute algebraic wank.

>>9347758
If you really do wish that then you'd be reading and studying instead of posting on /sci/.

>>9317140
Yeah. I also miss /yys/.

>>9280335
Depends on what you mean by relevant. There are very popular experimental research being done that uses little to no math and very unpopular theoretical research being done that uses very advanced elegant math.
At the end of the day research should be done for your own satisfaction. If you're trying to get into research for someone else's sake or approval then you won't last long.

>>9234801
Quantum field theory and topology by Schwarz.

>>9104542
>>9104605

>>9051158
Lmao somebody actually made a physics general the absolute madman.

I've just finished a MSc, and I'm heading for a PhD soon, been reading Turaev and Moore & Seiberg on the relationship between TQFTs and CFTs on the side. There is a direct correspondence (made explicit by metaplectic UMCs a la Nayak) between Friedan-Shenker operator algebra-based RCFTs and topological 2-DRMFs, and there are generalizations of this correspondence to that between the sequence [math]SU(2)_k[/math] of Chern-Simon TQFTs and (in general irrational) CFTs as illustrated in Kohno via the Witten invariant.
I'm thinking if there are ways to generalize this further to investigate the topological (i.e. non-perturbative) contents of TQFTs with more general simple Lie groups as gauge groups, and perhaps even establish a general framework for AdS/CFT given some proper geometric data is imposed on the TQFT side.

>>9032439
And of course I made typos
[math]G_1(z) = \Phi(X,Y)G_2(1-z)[/math],
[math]\zeta_{k_1\dots k_n}(z) = \sum_{m_1<\dots <m_n}\frac{z^{m_n}}{m_1^{k_1}\dots m_n^{k_n}}[/math].
>>9032510
What kind of research is that anon? Please don't say number theory.

>>9017830
>what are you studying this summer?
Currently reading through Turaev's chapter on 2-dimensional modular functors, which basically dresses kindergarten paper arts-and-crafts in really complicated algebraic language.
>did you read any interesting problems, theorems, proofs, textbooks, or papers recently?
An interesting object described in Kohno is the Vassiliev invariant [math]v:\mathscr{K}\rightarrow \mathbb{C}[/math], where [math]\mathscr{K} = \{S^1,S^3\}[/math] is the space of knots. These polynomial knot invariants form affine vector spaces [math]V_m[/math] that can be used to approximate the cohomology space [math]H^0(\Sigma,\mathbb{C})[/math], where [math]\Sigma = \{f \in \mathscr{K} \mid f ~\text{singular}\}[/math], and in turn the cohomology space [math]H^0(\mathscr{K}\setminus\Sigma,\mathbb{C})[/math] by Alexander duality, as [math]m \rightarrow \infty[/math].
This looks a lot like Wentzl's limit. If this really is just another representation of Wentzl's limit then I'd be able to find a really concrete correspondence between unitary TQFTs and CFTs.

https://arxiv.org/abs/1707.00282
Might extend CFT techniques to supersymmetry.
>>9012966
Sounds interesting. I await your progress.