/sci/ - Science & Math

>>9377382
I second this post.

>>9377382
Cobordisms ---> Vector Spaces

>>9377382
I wear pants every day

>>9377403

>Cobordisms
The fuck is that?

>>9377382
Never seen that kind of pants.

>>9377405
my dick is cobordant with your mom's pussy

>>9377405
Ignoring technicalities, a cobordism between two closed (n-1)-manifolds is an n-manifold-w-boundary, whose boundary is the disjoint union of the two (n-1)-manifolds.

>>9377415
okay i can imagine that...basically like binding the two manifolds together? And how is that related to a vector space?

>>9377415
actually isn't my pic one?

>>9377423
https://www.youtube.com/watch?v=Bo8GNfN-Xn4
he explains

>>9377432
thanks anon

>>9377423
A n-dim. TQFT is a symmetric monoidal functor from the category of n-cobordisms to the category of vector spaces.

The category of vector spaces consists:

- vector spaces
- linear maps between them

The category of n-cobordisms consists of:

- closed (n-1)-manifolds
- cobordisms between them

So a n-dim. TQFT associated to every closed (n-1)-manifold, a vector space. And to every cobordism of (n-1)-manifolds, a linear map.

The "symmetric monoidal" condition means it maps disjoint unions to tensors products.

i.e. the image of a disjoint union of two (n-1)-manifolds, is the tensor product of the images of each manifold

>>9377447
woah...okay, i see...Thanks anon! Can you tell me what background do i have to have to get into it? i'm thinking about doing my graduate thesis (a paper you need to right to get a degree) about it

>>9377452
are you a physicist or a mathematician ?

>>9377428
Yes, it is a cobordism from a circle to a disjoint union of 2 circles.

>>9377452
Just standard stuff on manifolds and algebra.

I like this as an introduction: https://arxiv.org/abs/math/0512103

>>9377462
i'm a physicist.. but i mostly study mathematics, thus im interested in such a field
>>9377466
thank you!!
Do you think its too muck for an undergrad-level student (i mean it for my paper before the degree)

>>9377473
ask your professors about it. they know you, and your background.

>>9377473
>Do you think its too muck for an undergrad-level student (i mean it for my paper before the degree)

I don't think so. That math really isn't that bad, and you can blackbox a lot of the physics used to motivate it.

>>9377479
well i'm too embarrassed to ask cause i'm only in my second year of undergrad and i know its too early for even thinking about it. But i like to have things planed beforehand, to have a rough idea of what i would like to study. Thats why i ask your opinion on it.

>>9377490
I'm not the other anon, but if you're thinking about this you need the "standard stuff on manifolds and algebra" he mentioned. This includes, at the very least, courses in groups, rings and fields, in topology, in differential geometry. perhaps some algebraic / differential topology too.

don't be embarrassed. talk to a professor who teachers a class you like and participate in, and be honest. tell them you find this stuff interesting and would like to get there in time for your thesis project. you seem to have a lot of time left to get there.

>>9377502
Well i'm familiar with these subjects at least in an introductory level (currently taking diffgeo)... Well i guess you are right.. but heck i haven't even had my first QM course yet it would be foolish of me to discuss stuff like that, you see what i mean..

>>9377512
It would be a bit weird as a physics student if you haven't even taken QM, but not as a math student.

Maybe add math as a second major?

>>9377516
Thought of that already...i would love to learn all these subjects in a deeper level but i would lose at least 2-3 years in which i can do my masters degree im mathematical physics, thus covering both (and doing only the math i would like to)

>>9377512
dude, you're totally fine, don't overthink it
the worst case scenario: you will write a shitty thesis but guess what, NOBODY gives a fuck about your undergrad thesis

>>9377512
there's nothing weird about being interested in something you know jack shit about. your professors know the feeling well, and will help you out if you're up front about it

>>9377529
well i guess you are right...and i will learn much more in the upcoming two years left for my thesis...although, i've been told it matters if you want to proceed in academia..

>>9377532
Okay, i'll consider asking for help or talking about it at least!

>>9377533
what matters is (proven) drive to study hard and will to succeed. you're in your second year, and it sounds like you're doing quite well. talk to your professors so they can help you get involved with research projects that are right for you, and to get on the path to the things you want to learn.

>>9377466
I never thought I'd see someone actually use penrose notation

>>9377540
okay i hear you..Thanks anons, this really helped me!

>>9377543
Not quite the same thing.

The nlab entry talks about manifolds being cobordant or cobordism classes. But aren't any two manifolds of the same dimension going to be cobordant?

>>9377813
No because cobordisms are required to be compact.

>>9377916
ok. But if the two manifolds are compact then they'll be cobordant?

>>9377813
https://math.stackexchange.com/questions/1385708/examples-of-manifolds-that-are-not-boundaries

>>9377919
cool, thanks

>>9377917
I mean the n-manifold, determining whether 2 closed (n-1)-manifolds are cobordant, is compact.

cocks

>>9378307
10/10 comment

>can't describe reality because reality has geometric structure

TQFT BTFO

>>9377447
>im a fat faggot
kys retard

>>9379077
RIP Voevodsky, 2016-2017. We shall all unite under your HoTT-Coq (also known as UniMath).

>>9377447
It also needs to be modular and maps the empty cobordism to [math]\mathbb{C}[/math].
In general a TQFT is a tuple [math](\mathscr{T},\tau)[/math] where [math]\mathscr{T}[/math] is the functor you've mentioned and [math]\tau[/math] is the quantum invariant. [math]\mathscr{T}[/math] encodes all the categorical information (i.e. naturality and functoriality of glueing patterns) while [math]\tau[/math] makes sure the TQFT is a topological field theory (i.e. computes topological invariants with values in the R-module).
The category of 2-cobordisms can be finitely generated by cusps, pants and cylinders which satisfy certain algebraic relations, which is why 2D TQFTs are isomorphic to commutative Frobenius algebras. The quantum invariants is just the Euler characteristic.
In the case of 3D TQFTs more work is needed since the Euler characteristic is not a cobordism invariant. Knot theory and Dehn surgery theory has been used to embed ribbon graphs into [math]S^3[/math] and surgering 3-manifolds from it, which keeps track of 3-cobordisms in an invariant way. The quantum invariant in this case can then be defined via knot invariants.
4D TQFTs are yet harder, because there exists exotic 4-manifolds that are h-cobordant but does not admit diffeomorphic differential structures. This is a problem for studying 4D Yang-Mills from the TQFT/CFT point of view, and I'm my opinion presents one of the biggest roadblocks to the Yang-Mills gap problem. In fact it has been proven a few years back that 4D unitary TQFTs induce a non-positive definite inner product on [math]\mathbb{C}[/maths]-linear spaces, which is a fatal problem if you want your UTQFT to describe any physical theory. This is precisely due to the fact that the knot-based quantum invariants generalized from 3D TQFT is not a h-cobordism invariant on exotic Mazur 4-manifolds.

>>9379848
lol i only understand the words manifold and invariant

>>9379927
>doesn't know what pants are

>>9377382
dat thigh gap tho

hhhhhhhhhhhhhhhhhnnnnnnnnnnnnnnnnnnnnnnnnnggggggggggggggggggggh

>>9379848
What's a cusp and why does it have to map the empty cobordism to C?

>>9381690
Sorry I meant that the empty set maps to C. The cusp is a cobordism between a loop and the empty set so it's assigned a C-linear morphism from a C-space V to C.
The choice that [math]\mathscr{T}[/math] maps the empty set to the ground ring R (C in the case of unitary TQFTs) is just a normalization condition.

>>9381713
>just a normalization condition

meaning?

also, nice pic.

>>9381744
Sorry for the slow response, my phone battery died as I was at the gym.
To see why the functor maps the empty set to the ground ring R, note that [math]\mathscr{T}[/math] maps the manifold [math]\overline{M}[/math] with the reverse orientation of [math]{M}[/math] to the dual [math]\mathscr{T}(M)^*[/math] of the R-module [math]\mathscr{T}(M)[/math], so since [math]\overline{\emptyset} = \emptyset[/math], we have [math]\mathscr{T}(\emptyset)^* = \mathscr{T}(\emptyset)[/math] and hence by the non-degeneracy (prove this!) of the pairing [math]\mathscr{T}(\emptyset) \times \mathscr{T}(\emptyset)^* = \mathscr{T}(\emptyset) \times \mathscr{T}(\emptyset) \rightarrow R[/math] we see that there is an isomorphism [math] \mathscr{T}(\emptyset) \cong R[/math], since the pairing would just be a multiplication by an element in R. Hence for any cobordism [math]W[/math] that ends/starts with the empty set, we can pre/post-compose the R-linear homomorphism [math]\mathscr{T}(W)[/math] with the isomorphism [math] \mathscr{T}(\emptyset) \cong R[/math] such that the new [math]\mathscr{T}(W)[/math] maps whatever space to the empty set (or vice versa). This isomorphism is unique up to isotopy so this doesn't change the categorical structure of R-modules.
This "shifting" of morphisms can be considered as a "normalization".

Hey T(ouhou)QFT poster, what do you make of this

http://news.rice.edu/2017/12/18/rice-u-physicists-discover-new-type-of-quantum-material/
Forgot link

>>9382430
is the gym worth it?

>>9382523
Really interesting. I was actually studying topological superconductors before I switched to mathematical physics.
One of the most powerful tools for studying quantum cirticality is conformal field theory, since a condensed matter system has infinite correlation length at the critical point and hence achieves conformal symmetry. In 2D the symmetry group is the modular (Mobius) group of [math]\mathbb{C}[/math], which is so large that all observables in the theory can be described by an affine Lie algebra. This means that the conformal blocks satisfy a finite set of differential KZ equations and so the [math]n[/math]-point correlation functions of the CFT can be solved exactly at the critical point.
General mathematical structures of CFT have been outlined by Seiberg and Moore, and connections to category theory made by Nayak and others. The latter is manifestly important since this gives a way to frame CFTs in terms of certain types of TQFTs, and quantum criticality described by the CFT can be computed via topological methods. An example would be the fact that the non-trivial particle statistics arising from the fusion relations in a CFT can be computed via Vafa's theorem in the corresponding TQFT.
>>9382736
It was good. Did some oly lifts while flirting with my bf.

>>9382806
Do you think mathematics is unreasonably effective at describing reality?

>>9382960
I wouldn't say it's unreasonable. Math was created to serve the needs of physics so it's only natural that it finds application in it, and sometimes even the other way around. No matter how far the mathematicians abstract away from the original, physically motivated first principles physics will always progress to a point where that level of abstraction is needed (or can be used) to describe reality. Even Diophantine equations can be used to describe the duality of quantum Hall states.

>>9382993
anon what is pic related?

>>9383968
Turaev.

>>9377382
Two holes

>>9382993
if I would take the disjoint union of 5 circles, which combinations remain after the operation?

>>9384652
Uhh please repeat the question.

The red pill on topological QFT is that Topological and QFT are both meme words

>>9385414
fuck off

>>9385414
Based role-player.

>>9377432
Based Lurie. Pls give him an Abel prize.
>>9377473
>do you think it's too much
Yes, and frankly it'd probably be too much for your supervisor too unless he's doing string theory, and even then he'd only be familiar with specific examples of TQFT like Chern-Simons, WZW or topological theta models in the best scenario. In any other cases he'll either direct you to other profs or (implicitly) tell you to stop worrying about things (specifically the categorical definition of a TQFT a la Atiyah) you won't ever use in physics.
Mathematical physics is actually part of mathematics, so you'd be better off asking math faculties about TQFT than physics faculties. The examples above are already rich enough for you to spend years studying if you want to continue to be a physicist.
This is not to say that you shouldn't pursue your interests, but I'm just laying out what I've went through to bare so that you won't waste 2 years for your masters like me doing something you're not as interested in.

>>9385408
hi hank pls halp

Let M be an even dimensional smooth manifold.
I want to find an example M such that "Kahler cone ≠ symplectic cone" with non-empty Kahler cone, satisfying the following conditions:
M admits a Kahler structure.
ω is a symplectic form on M.
There is no Kahler structure (M,ω,J) such that [ω]=[ω]∈H^2 (M;R).

>>9382806

>In 2D the symmetry group is the modular (Mobius) group of C, which is so large that all observables in the theory can be described by an affine Lie algebra.

Can you explain this a little more? Are you saying that at the critical point you can start describing things with affine Lie algebras? I don't really understand what this signifies.

>>9386573
oh i see... my prof is studying string but yet i dont know if he would know all that stuff... you say that cause you ended up not liking the subject?

>>9388176
In the presence of scaling symmetry near the critical point you can write the energy-momentum tensor as a holomorphic part and an antiholomorphic part, which can be evaluated using contour integrals. Then you can build up the generators of the conformal symmetry with this using Cauchy integral formula
[math]L_n(z) = \int_\gamma \frac{dw}{(z-w)^{n+1}}T(w)[/math] as well as its dual part [math]\overline{L}_n[/math], and from the commutation relations of [math]T[/math] you can deduce the commutation relations of the generators [math]L_n[/math], which turns out to be the Virasoro algebra [math][L_n,L_m] =
L_{n+m} + \delta_{n+m}\frac{c}{12}[/math]. Given a simple Lie algebra [math]\mathfrak{g}[/math], the generators form the basis for the affine Lie algebra [math]\mathfrak{g}_\mathbb{C} = (\mathfrak{g}^+ \otimes \mathfrak{g}^-) \oplus c \mathbb{C}[/math] of [math]\mathfrak{g}[/math], where [math]\mathfrak{g}^+[/math] is generated by [math](L_n,f), f \in \mathbb{C}((z))[/math] and [math]\mathfrak{g}^-[/math] by [math](\overline L_n,f), f \in \mathbb{C}((z))[/math], where [math]\mathbb{C}((z))[/math] is the algebra of Laurent series in [math]z[/math] (you can generalized this to sheaves of Laurent series [math]\mathcal{O}_R[/math] on a Riemann surface [math]R[/math]). The Verma module that describes the physical states are then defined to be the set of vectors [math]v \in V[/math] such that [math]U(\mathfrak{g}^+)v =
0[/math], and the "primary field operators" of the CFT lie within the loop subgroup [math]LG^+[/math] of the loop group [math]LG[/math] of the central extension [math]\hat{\mathfrak{g}}_\mathbb{C}[/math] of the affine Lie algebra [math]\mathfrak{g}_\mathbb{C}[/math] of the Lie algebra [math]\mathfrak{g}[/math].
It is then possible to define a Hitchin's connection on [math]LG^+[/math], the flatness condition of which gives rise to the KZ equations, and these are the equations that your conformal blocks satisfy.

>>9388103
If I recall correctly Kahler manifolds are symplectic manifolds with a Kahler potential 0-form [math]K[/math] such that locally [math]\omega = \frac{\partial^2 K}{\partial z\partial \overline{z}}dzd\overline{z}[/math]. You need to elaborate on what a "Kahler cone" is since as far as I know these cones of a 2-form [math]\omega[/math] are defined to be vector fields [math]\xi[/math] such that [math]\omega_\xi(\eta) = 0[/math] for all vector fields [math]\eta[/math].
>>9388424
It never hurts to ask, and it's always good to let your prof know where your interests lies. They don't want a student who's begrudgingly going through the motions on something they don't have interest in either.
>you say that cause you ended up not liking the subject?
I studied condensed matter theory focusing on topological defects, but it wasn't mathematical enough. I was mainly interested in the mathematics that came out of the physics instead of the physics itself, while theoretical physics is still very much tied to experimental results, meaning that what I say needs to be at least experimentally feasible and isn't just some interesting mathematical result. This fact pushed me over the edge while pursuing a PhD.
Again, this is just anecdotal and you should decide on what you want to do based on what your prof suggests, not what some random internet stranger says.

>>9388780
>I was mainly interested in the mathematics that came out of the physics instead of the physics itself
are you me? hahaha
i hear you anon..thank for the advice, much appreciated. I will see if i can do my ungergrad thesis with a math prof. although i think those subjects tend more to the interest of physicists than mathematicians (at least i haven't found one math prof in our department interested at QFTs...the closest i got is applications of diff geo which is awesome bit not something i would want for my undergrad thesis)

>>9388780
t-thank you p-professor y-yukari

>>9388764

Thanks anon. I don't know about some of the math you're using, and I had to look up stuff to try to follow, so let me just recap in simpleton terms and you tell me if this is correct:

We have an energy-momentum tensor, which can be described as a polynomial including all negative powers (which is what I think holomorphic and antiholomorphic is saying).

From this we get generators (of what, and how do you get generators from the energy-momentum tensor? I know how to do integrals in the complex plane, but is there some sort of procedural way to obtain generators I'm not aware of?).

The generators are that of a scale transformation, so that [math]mathcal{g}^+[/math] and [math]mathcal{g}^-[/math] are the spaces of the transformations for zooming in and out of your system.

The second half is still too confusing for me. I'd need some concrete examples of physical systems using these concepts and definitions before my brain accepts them as meaningful. Or I need more math classes.

>>9388764
Are you by any chance the Russian anon from Lomonosov Moscow State University? If so, how rigorous was your gauge fields and fibred spaces course? I'm disappointed by my university's take on it and am thinking about transfering to either your uni or some uni affiliated with Max Planck Institute after this semester (currently third semester undergrad in physics).

>>9389340
>including all negative powers
All powers. That's what Laurent series mean.
>(which is what I think holomorphic and antiholomorphic is saying)
Not quite. A function [math]f[/math] is holomorphic if [math]\overline \partial f = 0[/math] and antiholomorphic if [math]\partial f =
0[/math].
>of what
The Virasoro algebra.
>and how do you get generators from the energy-momentum tensor?
With the formula up there.
>The generators are that of a scale transformation
No, the generators are the generators of the Virasoro algebra, which describes the algebra of the field operators, not the symmetry.
>>9389390
No, though I'm affiliated with Perimeter. Here the courses are taught from physical motivations but can become quite rigorous depending on what stream you're following. Since I'm following the mathematical physics stream (along Ed Witten), all courses are extremely rigorous.

>>9389760
>though I'm affiliated with Perimeter
You lucky son of a bitch. I'm applying there now (Waterloo too since the guy I work under knows Achim Kempf and Brian Forrest). From what a friend of mine said it's tough as all hell to get in but looking it over if definitely seems like it's worth it. I figure I might have a somewhat remote chance of getting in since I got a package from them with a letter saying to apply. You doing the PSI program or the PhD?

Thigh gap goals

>>9389778
PhD. Another option for you is to apply for other universities around Perimeter and transfer into it later through doing PSI and some research. It's pretty bothersome but, as you've said, it's worth it.
Good luck anon.

>>9382736
You should see his thighs :hearteyes:

i was wondering if any anon would be willing to answer a question i have regarding the nature of 4D topology

>>9390088
>nature of 4D topology
>4D
The most topologically non-trivial dimension there is. Shoot, maybe someone will know

>>9389823
>his

>>9390091

Alright, I am attempting to extrapolate a 3D argument.

Suppose you have a 2+1 space with a stationary particle tracing out a world line. Now imagine that over a period of time you take a line segment and form it into a circle around that particle. You then break the circle and remove the line segment. When one looks at the world volumes this will result in a 'threading' action where the the particle world-line is threaded through a hole in the line segment world-sheet.

Now my question is, when we upgrade to 4D. i.e. when we temporarily surround a stationary particle in 3D with a 2D surface shaped into a sphere, and then break that sphere and remove the 2D surface, do we obtain the same kind of 'threading' action as we saw in the previous case.

>>9389760

> No, the generators are the generators of the Virasoro algebra, which describes the algebra of the field operators, not the symmetry.

I was under the impression a Lie algebra describes *a* symmetry in the system. Is there a short answer for how the Virasoro algebra connects to field operators and their commutation relations?

Also, is there a generator for scale transformations that fit in all this?

Is there a generator for scale transformation?

>>9389760
What would you say is a priority in terms of fields of math for undergrad looking to get into mathematical physics?
I've been through most graduate courses, but again my university is teaching them in a pretty bad way. I've been taking as many courses as i could on math department, but now i'm hitting barriers of our legal system. For example i'm not allowed to take non-stable K theory because it has mandatory prerequisites which are only available to math students. I still go to lectures and do all asignments, i even took the exam and passed it with full score, but from legal standpoint it is as if i never took the class. This applies to most of the courses on math department i went through.
At this point i think i have pretty solid understanding of MSc-level math, my focus being on higher gauge theories using methods of supergeometry, simplical homotopy and homotopy algebras. This is a topic i'm very interested in, but unfortunately none of the proffessors here are. For this reason i'm thinking of switching into environment that is more suited to me. Upon reading it, i was thinking of extending arxiv.org/abs/1604.01639 which unfortunately is out of interest of all the proffessors here. The only people who were interested in this were from Perimeter Institute, Max Planck Institute and MIPT. The most realistic choice for me is either Max Planck-affiliated uni or MIPT-affiliated uni (pretty much just Lomonosov Moscow State is worth mentioning) as i can't, due to my criminal record, leave EU (i can get student visa to Russia, but certainly not to US or Canada).
Do you know of any worthwhile institute in EU or Russia (apart from MIPT to which i can't apply) that would allow me to focus on this topic while giving me a good foundation for mathematical physics? I'm mainly interested in specific people that share my interest, rather than a curriculum.

>>9390511
>I was under the impression a Lie algebra describes *a* symmetry in the system.
This isn't always true, and even if it were it wouldn't mean Lie algebras has to describe symmetries anyway. In fact the special thing about CFTs is that the entirety of its operator algebra can be described by an affine Lie algebra [math]given[/math] its symmetries.
>Is there a short answer for how the Virasoro algebra connects to field operators and their commutation relations?
The Virasoro algebra defines a Verma module on which the algebra relations gives rise to KZ equations, and the solutions are the correlation functions (i.e. conformal blocks) of the primary fields.
>Also, is there a generator for scale transformations that fit in all this?
No, it's something else entirely. The translation, rotation and scaling symmetries are symmetries of the Hamiltonian which gives you the sufficient (and necessary) conditions to decompose it into anti-/holomorphic parts. That's the extent to which these symmetries are significant.
>Is there a generator for scale transformation?
Yes, and it's easily derived by putting [math]x^\mu \rightarrow x^\mu + \partial^\mu \epsilon(x)[/math] into the variation of the Hamiltonian.
There are several good books on the basics of CFT like DiFrancesco, Henkel, Kohno or Ueno that you can look into.
>>9390695
>What would you say is a priority in terms of fields of math for undergrad looking to get into mathematical physics?
Diff. top./geo., alg. top., functional analysis, group/representation theory, cohomology theory, Seiberg-Witten/Donaldson theory, Deligne-Mumford compactification, etc.
I don't know much about the institutes in the EU so I can't help you there.

>>9390154
>do we obtain the same kind of 'threading' action as we saw in the previous case.
It would seem so, at least thinking about standard R^4, you should actually ask this on math or physics stack exchange.

>>9390978
Sorry for bothering, but i was more interested in the etc. part, could you please expand on that? I have nobody else to ask and as i am, i have trouble using my math knowledge in physics. As in i can comfortably read mathematical papers, but i have trouble reading hep-th papers that are more focused on physics. I think it's mostly because these assume some basic level of knowledge in physics which i lack as i've been focusing on the math. All i've done in physics is pretty much just reading Landau and an introductory string book by Polchinski.
To give you a better idea of the bad state i'm in, i read papers from mathematical journals and understand them just fine, but i can't get into string theory papers (unless it's by mathematician), it just seems arcane and obfuscated by layers upon layers of notation, convention and assumptions. Is it just a barrier that i'll get through with more energy, or do i lack some fundamental component to be useful in physics?

>>9391202
At some point the distinction vanishes. If you're looking for physics background I suggest you read Landau-Lifshitz.
>obfuscated by layers upon layers of notation, convention and assumptions
Example? I've never seen this happen.
>Is it just a barrier that i'll get through with more energy, or do i lack some fundamental component to be useful in physics?
Hard to say without an example.

>>9390978
>There are several good books on the basics of CFT like DiFrancesco, Henkel, Kohno or Ueno that you can look into.

Ok, cool. I have a lot on my reading list right now though. I've been trying to teach myself QFT this past semester after switching research from plasma-physics, but progress is slow.

I've been trying to read the "intro" books like Mark Srednicki, Pier Ramond, and Peskin and Schroeder, which is good for basic knowledge but seem much less rigorous or lacking in giving an intuition when explaining the basic mechanics and how you obtain any particular Hamiltonian, or why a certain way to calculate a probability amplitude works.

The one colleague of mine 'teaching' me QFT just has me draw Feynman diagrams and write down the integrals, (and not even normalized so that I can calculate a probability of a scattering process).

The most insight I've gotten into this subject has been from condensed matter concepts and thermo. It seems to motivate the subject more as you can have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. I mean, this is essentially what's happening in QFT, right? We just don't know the toy model at the plank-scale. Otherwise it just seems to me like Hamiltonians and Lagrangians fall out of the sky.

>>9391781
Read Baez's Guage Fields, Knots and Gravity.
>have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. I mean, this is essentially what's happening in QFT, right?
Not really.

If the physical information contained in a ket is not affected by multiplication by a non-zero complex number why isn't quantum theory built on rays and not vectors?

>>9392325
It is.
https://en.wikipedia.org/wiki/Wigner%27s_theorem
https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction

>>9392336
Oh thanks.

>Wigner early work laid the ground for what many physicists came to call the group theory disease[1] in quantum mechanics – or as Hermann Weyl (co-responsible) puts it in his The Theory of Groups and Quantum Mechanics (preface to 2nd ed.), "It has been rumored that the group pest is gradually being cut out from quantum mechanics. This is certainly not true..."
Bastards.

>>9392315
I'll check this one out.

Do you have any good overarching intuitions for what's happening then? I thought it was a reasonable base intuition with analogies I perceived between QFT and condensed matter.

>>9392561
>Do you have any good overarching intuitions
Not really, I just know that integrating out momentum shells to get renorm flows isn't a proper group operation since the Hubbard-Stratonovich transform doesn't have unique inverses, so this whole thing isn't very well understood mathematically (by myself at least) and I try not to have any impressions about what renormalization actually is before I do.

>>9382806
shut the fuck up hank

>>9392948
Well the way I've been thinking about renormalization as integrating/ averaging over smaller scale phenomena, I wouldn't expect there to be a unique inverse. The process of smoothing out/ integrating over variables throws out information.

post thighs

>>9393226
That's precisely what happens -- the renormalisation group is actually a semigroup!

Wew lads, I'm hete struggling with some QM, what the fucknis this wizzardy?

>>9377447
Damn, I know next to nothing about differential topology, but I can actually see now why that is very useful. Category theory is no meme after all it seems.

>>9393807

>>9394517
lol same here

>>9394517
>>9396493

At it's core QM is just probability theory but with the additional rule that you can let probabilities 'interfere' with each-other in a special subspace that has phases. (Trying to ask questions like what's the probability of A and B happening can also give you nonsensical results like negative probability.)

In this sense, QFT is nothing new but now you throw in probabilities of certain interactions happening like particles being created/ destroyed, and you make everything Lorentz invariant.

I just had QFT last semester and it was the most fun and toughest subject I have ever learned. But reading this thread I have just a very superficial knowledge and this makes me sad.

Still a very interesting thread. Thanks guys.

>>9398325
yup, OP here. thanks anons, never thought this thread would be so interesting

>>9396654
This entire post is wrong.
>At it's core QM is just probability theory but with the additional rule that you can let probabilities 'interfere' with each-other in a special subspace that has phases.
Are you referring to superposition? That's a very convoluted way of saying it.
>Trying to ask questions like what's the probability of A and B happening can also give you nonsensical results like negative probability.
No. Negative probabilities are never accepted in QM. One of the heavy-lifting needed in geometric quantization was the construction of "square-root" forms such that the sections of which induce a polarized Hermitian line bundle needed for the integrality conditions, so that its fibre integration corresponds to a bona fide probability measure.
Are you perhaps referring to fuzzy probability and the Kochen-Specker theorem? Because that [math]is[/math] a meaningful quantum theory.
https://arxiv.org/abs/1207.1744
>In this sense, QFT is nothing new but now you throw in probabilities of certain interactions happening like particles being created/ destroyed,
QM has this too. In fact IMO proper QM should be taught from [math]\mathbb{C}^*[/math]-algebras so that misunderstandings like these wouldn't be so prevalent.
>and you make everything Lorentz invariant.
Relativistic QM with full Lorentz invariance can be formulated without field theory. The problem is that these theories cannot describe quantum mechanical potentials due to Ekstein's no-go theorem.

Does anybody know if category theory is necessary? In fact, when is the right time (in which level) to start learning category theory?

>>9401101
Necessary for what? It's not necessary to do anything that engineers and physicists do (which is not to say that it couldn't be used there, just that those type don't know about it - except for some rare field theoriest.)

Raw category theory only requires cheap logic and known basic set theory helps. For the examples where it was first applied - and for which the theory was initially cooked up for, you'd want to know some abstract algebra. It was first used for some algebra tied to topology. But it turned out that it's really a way to look at mathematics in general, a bit form the outside. It can be argued that it's metamathematics, like logics can be used that way as well.
It's to a large extent a study of reasoning about "equalness". Some like it and some don't. It's probably good to learn it even if you're not in STEM, but once you're capbably of understanding it, you're able to do math already, so that point is moot.

>>9401117
well since we are in this thread i am referring to QFTs. I get what you say and from what i have read on Category theory (very little) i look at it as something more general and compact that covers most of maths. (correct me if i'm wrong, haven't read actual cat.theory yet..)

That being said, if i'm interested in it, what is the appropriate level for me to start learning it? undergrad/grad/postgrad? Thanks.

>>9401127
the only prerequisite is a stock of example. I'd start with your first encounter with general topology and/or manifolds. if you know how a product topology is defined, you're good to go.

>>9401101
Not really to define a TQFT, you could write a list of axioms. It is just easier, and more insightful, to say it is a symmetric monoidal functor.

However there are times where category theory could be essential. Various moduli spaces often appear in QFT.

Moduli spaces are not always nice spaces, usually stacky, and you need category theory to study stacks.

>>9401215
thanks anon

>>9401199
only topology and manifold theory? i was under the impression that you need to have a solid grasp of rings, groups etc.

>>9401359
>i was under the impression that you need to have a solid grasp of rings, groups etc.

You do. Lie groups, lie algberas, and their representations are essential to QFT.

>>9401390
oh okay..and for those of course i should have a background on manifold theory. I see, thanks anon

>>9401215
>Various moduli spaces often appear in QFT.
To expand on this, instantons solutions to Hermitian Yang-Mills theories can be characterized by curves defined by selfduality equations (such as Seiberg-Witten type equations) in the moduli space of connections.
There are two main ingredients for constructing local solutions to a Yang-Mills theory: self-duality and guage invariance. The gauge invariance under an endomorphism [math]\operatorname{End}(V)[/math] of the [math]G[/math]-bundle in question forces us to mod out the space of self-dual connections by its gauge transforms. This quotienting is analogous to the definition of the cohomology space of an exact sequence
[eqn]0 \rightarrow \Gamma(M,\mathcal{U})\rightarrow \Omega^2(M)\otimes \mathfrak{g} \rightarrow \Omega^{2,+}(M)\otimes \mathfrak{g} \rightarrow 0[/eqn]
called the instanton complex. This is an example of a Cartan equivariant cohomology, which involves sheaves in its full generality.
A crucial ingredient in the theory of equivariant cohomology is the fibre integration, which corresponds to Feynman path integrals in the context of Yang-Mills theory. Fibre integration can also reduce global operations (integration) to local operations (summation) via the Atiyah-Bott integration formula, which gets you properties of global solutions for free sometimes. However to get global solutions in general you need to compactify the moduli space, which involves Deligne-Mumford constructions, Teichmuller spaces, stacks, etc. which I'm not very familiar with tbqhwy.

>>9401359
I was talking about category theory. sadly TQFT is beyond my scope at the moment

>>9399782

>This entire post is wrong.

I oversimplified, with a bit of my own working heuristic interpretation to compare it more with classical probability theory. Let me explain myself a little more and let me know what you think. I don't want to carry around and perpetuate misconceptions.

>No. Negative probabilities are never accepted in QM. One of the heavy-lifting needed in geometric quantization was the construction of "square-root" forms such that the sections of which induce a polarized Hermitian line bundle needed for the integrality conditions, so that its fibre integration corresponds to a bona fide probability measure.
Are you perhaps referring to fuzzy probability and the Kochen-Specker theorem? Because that is a meaningful quantum theory.
https://arxiv.org/abs/1207.1744

With two bits of information in classical probability theory you always have [math]P(A \vee (\wedge) B) = P(A) + P(B) - P(A \wedge (\vee) B)[/math]. Playing around with density matrices and pure states with two bits, you can get analogous expressions, expect the cross terms where [math]P(A \wedge B)[/math] or [math]P(A \vee B)[/math] can be negative when added to other probabilities.

True, that within the formalism of QM, a well posed question will never let you obtain a negative probability on it's own, so everything is ok there. But just trying to interpret the cross terms by analogy to classical probability theory you can get an apparent negative probability. Here's an example with double-slit:

https://en.wikipedia.org/wiki/Negative_probability#An_example:_the_double_slit_experiment

>QM has this too. In fact IMO proper QM should be taught from C∗-algebras so that misunderstandings like these wouldn't be so prevalent.

Repill me on C*-algebra. I thought it was a generalization of the Fourier transform, how you can represent L^2 integrable functions with a bunch of different basis? And how does that have to do with particle conservation? (P2)

>>9399782
>QM has this too. In fact IMO proper QM should be taught from C∗-algebras so that misunderstandings like these wouldn't be so prevalent.

Continued... The norm squared is conserved for a fourier transform, how does the relate to particle conservation?

Relativistic QM with full Lorentz invariance can be formulated without field theory. The problem is that these theories cannot describe quantum mechanical potentials due to Ekstein's no-go theorem.

Interesting. What's the formalism for a Lorentz invariant probability density function btw, much less an invariant wavefunction? Would you just apply the additional constraint that it must integrate to 1 in every reference frame? Could you instead make event probability density functions where the normalization is over all of spacetime?

>>9402035
>But just trying to interpret the cross terms by analogy to classical probability theory
Similar to the confusion caused by the Aharonov-Bohm effect, this is just an artifact of thinking about quantum mechanical system wrongly. If you interpret probabilities of mixed states (photons at the 2 slits) as that of pure states (photons at [math]each[/math] of the 2 slits) of course you're going to get the wrong results.
>Repill me on C*-algebra. I thought it was a generalization of the Fourier transform, how you can represent L^2 integrable functions with a bunch of different basis?
Von Neumann algebras, which is a special kind of unital associative [math]\mathbb{C}^*[/math]-algebra, underlies the foundations of quantum mechanics. The ladder operators (from SHO, say) are generators of the Heisenberg algebra, which is a specific kind of Von Neumann algebra on which a representation on [math]L^2[/math] exists, [math]then[/math] Fourier transforms can be used along with other results from harmonic analysis. [math]\mathbb}C^*[/math]-algebras aren't so much a generalization of Fourier transform, but rather a generalization of the Heisenberg algebra.
>What's the formalism for a Lorentz invariant probability density function btw, much less an invariant wavefunction?
It just require the existence of a representation of the spin group compatible with the Heisenberg algebra. The only problem is that there's only the trivial representation so only vacuua can be defined as a consequence of Ekstein's theorem.

>>9402195
Is this paper good
https://arxiv.org/abs/quant-ph/0101012

>>9402280
I'm not familiar with quantum computation so I can't pass judgement on that specific paper, but I generally dislike how they proceed from Hilbert spaces and build up representations of the quantum algebras instead of [math]\mathbb{C}^*[/math]-algebras themselves, but that's just me. This looks like a fine survey for understanding the difference between quantum and classical probability but not the actual foundation of quantum mechanics. His axiom 5 seems much stronger than the cyclicity of the vacuum, for example.

>>9402195

>Similar to the confusion caused by the Aharonov-Bohm effect, this is just an artifact of thinking about quantum mechanical system wrongly.

That was kind of my point though. I was trying to explain quantum results (true quantum probabilities from pure states) in terms that someone only familiar with classical probability theory (Bayesian probabilities due to lack of information) in order to highlight the differences. IF you try to interpret everything in terms of classical probability theory (treating everything like a mixed state), you get apparently nonsensical results. When of-course what's really happening is that you're missing all the magic of interference happening in Hilbert space, which is the main departure from classical logic.

>Von Neumann algebras, which is a special kind of unital associative C∗-algebra, underlies the foundations of quantum mechanics.

I've actually been interested for awhile in more formal approaches to QM than what my courses have offered. I'll check this out.

>It just require the existence of a representation of the spin group compatible with the Heisenberg algebra. The only problem is that there's only the trivial representation so only vacuua can be defined as a consequence of Ekstein's theorem.

So what you're saying is... we need field theory for Lorentz invariance.

>>9402459
>IF you try to interpret everything in terms of classical probability theory (treating everything like a mixed state)
Did you switch "pure" and "mixed"? Treating things as mixed states are in no way a departure from "classical" probability. Do we conclude that gauge invariance is a useless condition due to the fact that an observable effect (i.e. the AB effect) breaks it, or do we invoke more subtle arguments (i.e. topology) to reconcile this? Same thing here, just replace gauge non-invariance with negative probability and topology with mixed states.
>So what you're saying is... we need field theory for Lorentz invariance.
That's not what I was saying at all.

>>9402476

>Did you switch "pure" and "mixed"?

No, I meant mixed there.

>Treating things as mixed states are in no way a departure from "classical" probability.

Yes, I know. But pure states are a departure, and can't be interpreted in terms of mixed states, or else you get negative probabilities.

Just to be clear, I'm NOT advocating for negative probabilities. I'm literally saying they are nonsensical and are a reason why classical probability theory fails to explain quantum phenomena.

>>9402556
>No, I meant mixed there.
No I'm pretty sure you switched them around everywhere. By assuming [math]I(x) = |f_1(x)+f_2(x)|^2[/math] is the probability density it's already implying that [math](f_1(x),f_2(x))[/math] is in a pure state (with a diagonal density matrix).
>I'm literally saying they are nonsensical and are a reason why classical probability theory fails to explain quantum phenomena.
And this is wrong because classical probability is still the foundation of QM. As I've said, a result that violates a well-established theory/statement must be due to our lack of proper understanding of it, not because of the fault of the theory. I (or anyone I know) would not say probability theory, classical or otherwise, fails to describe quantum mechanics in the same way that I wouldn't say the AB effect undermines gauge invariance.
Anyways this is now leaning towards philosophy so I'll stop replying if you're gonna continue down this road.

I thought this was about the wigner quasiprobability distribution, am I incorrect?

t. dumb undergrad

>>9402573

>No I'm pretty sure you switched them around everywhere. By assuming I(x)=|f1(x)+f2(x)|2 is the probability density it's already implying that (f1(x),f2(x)) is in a pure state (with a diagonal density matrix).

Diagonal density matrix is mixed state though? Pure means that [math] \hat{\rho} [/math] is written as [math]\hat{\rho} = \left| \psi \right> \left< \psi \right| [/math], so that for two bits you can only have stuff like: [math] \hat{\rho}= |\alpha|^2 |A><A|+\alpha^*\beta |B><A| + \alpha \beta^* |A><B| + |\beta|^2|B><B|[/math].

Where mixed states you can have just diagonal:
[math]\hat{\rho} = |\alpha|^2|A><A|+|\beta|^2|B><B|[/math].

>Anyways this is now leaning towards philosophy so I'll stop replying if you're gonna continue down this road.

Yes, I'm getting tired anyway, it's late here. I think a lot of this is about semantics and non-consequential interpretations of the theory. G'nite.

>>9402593
Yeah that's completely flipped from how I understand it: a state is mixed if it can be expressed as a convex sum of state operators. Haven't done any concrete calculations in years.