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Stream: deprecated: physics

Topic: synthetic formulation of qm

Antoine Van Muylder (Aug 10 2021 at 17:35):

Hello! Does there exist categorical formulations of quantum mechanics?

Spencer Breiner (Aug 10 2021 at 19:24):

Yes, and good textbook:
https://www.cambridge.org/core/books/picturing-quantum-processes/1119568B3101F3A685BE832FEEC53E52

Martti Karvonen (Aug 10 2021 at 20:34):

There's also
https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616
which talks a bit more about categories

Antoine Van Muylder (Aug 10 2021 at 20:58):

great! thanks a lot for the answers

Fawzi Hreiki (Aug 10 2021 at 22:03):

Also, if you’re interested in logic, then you may want to take a look at linear logic (particularly Melliès’ book ‘Categorical Semantics of Linear Logic’) since quantum logic is (in many cases) built on top of linear logic

John Baez (Aug 11 2021 at 19:13):

And if you want something a lot shorter than a book, you could try this paper of mine:

• Quantum quandaries: a category-theoretic perspective.

This is not a substitute for the two books mentioned. It's an explanation of how category theory can help us understand some of the puzzles of quantum mechanics.

John van de Wetering (Sep 16 2021 at 16:33):

Chris Heunen and Andre Kornell just put up a really cool new paper on the arxiv: https://arxiv.org/abs/2109.07418
They find categorical conditions that force a category to be equivalent to the category of (real or complex) Hilbert spaces

Matteo Capucci (he/him) (Sep 16 2021 at 22:50):

Open problem (for categorical physicists): what's the physical interpretation of their mathematical axioms?

James Deikun (Sep 17 2021 at 00:14):

Isn't this sort of an open problem for QM in general? (I'm not a skeptic of QM; but while we have many great explanations of how it's necessarily true, it seems like our explanations of how it's possibly true are kind of lacking.) In fact it's questionable if this is even a totally solved problem for statistical physics...

James Deikun (Sep 17 2021 at 00:18):

I'm not saying this problem is not of interest; in fact one of the best reasons to pursue synthetic QM is that it might come up with some insight in that direction by looking a little farther afield from the formalisms that we know "work". But we won't make progress if we don't give people the same slack in finding a "physical interpretation" of their models that physicists have had (and admittedly had to work for) in the past.

Fabrizio Genovese (Sep 17 2021 at 08:38):

Matteo Capucci (he/him) said:

Open problem (for categorical physicists): what's the physical interpretation of their mathematical axioms?

People like Von Neumann, which came up with the axiomatization of QM in terms of complex Hilbert spaces later said they didn't believe in the Hilbert spaces formalism anymore... So one may as well argue that there is no satisfying interpretation, this is just the stuff that up to now works best.

Fabrizio Genovese (Sep 17 2021 at 08:39):

This said, some axioms such as the presence of a monoidal product seem very natural if interpreted as "taking two systems together"

Matteo Capucci (he/him) (Sep 17 2021 at 10:23):

Fabrizio Genovese said:

Matteo Capucci (he/him) said:

Open problem (for categorical physicists): what's the physical interpretation of their mathematical axioms?

People like Von Neumann, which came up with the axiomatization of QM in terms of complex Hilbert spaces later said they didn't believe in the Hilbert spaces formalism anymore... So one may as well argue that there is no satisfying interpretation, this is just the stuff that up to now works best.

Ha, that's interesting. I find the conceptual framework of von Neumann algebras and Hilbert spaces quite illuminating and more or less natural, up to technicalities.
That is, it makes sense to me to take a system as kind of mystery object which you can only investigate by means of observables. Then you look at the algebraic structure observables ought to have, and you realize you have something like an Hilbert space. Then dynamics is not something that makes your system evolve (you don't have access to it!) but something that makes your observables evolve, i.e. an operator.

Matteo Capucci (he/him) (Sep 17 2021 at 10:24):

Fabrizio Genovese said:

This said, some axioms such as the presence of a monoidal product seem very natural if interpreted as "taking two systems together"

Indeed. There are two monoidal products on Hilb, the biproduct $\oplus$ and the tensor $\otimes$. The first should be putting systems toghether in a non-interacting way (so I'd call this considering two systems together), whereas the second is where interaction happens, in fact mixed tensors model entangled states afaik.

Matteo Capucci (he/him) (Sep 17 2021 at 10:25):

Probably the most mysterious axioms are the one concerning separability of $I$ and its two dagger subobjects (which turn out to be $0$ and $1$)

Matteo Capucci (he/him) (Sep 17 2021 at 10:26):

In fact $I$ is where the magic dust lies, imo. That's what turns out to magically be $\mathbb R$ or $\mathbb C$ and implies everything else is an Hilbert space

John van de Wetering (Sep 17 2021 at 14:40):

Fabrizio Genovese said:

Matteo Capucci (he/him) said:

Open problem (for categorical physicists): what's the physical interpretation of their mathematical axioms?

People like Von Neumann, which came up with the axiomatization of QM in terms of complex Hilbert spaces later said they didn't believe in the Hilbert spaces formalism anymore... So one may as well argue that there is no satisfying interpretation, this is just the stuff that up to now works best.

This statement that von Neumann didn't believe in Hilbert spaces is actually taken wildly out of context by most people. What he meant is that he thought we should be looking at type II von Neumann algebras which have a concept of continuous dimension. But of course no one right now believes that type II von Neumann algebras are the 'correct' notion for quantum mechanics. This was part in the days when von Neumann tried to generalise projective geometry to continuous dimensions.

John van de Wetering (Sep 17 2021 at 14:42):

Matteo Capucci (he/him) said:

Probably the most mysterious axioms are the one concerning separability of $I$ and its two dagger subobjects (which turn out to be $0$ and $1$)

So the axiom that $I$ is a separating object is quite natural, as this is saying that there are 'enough states' to separate the morphisms. In fact, if this isn't the case, you can just quotient the category to make it the case.
The other condition that $I$ is separable is also quite reasonable: Suppose we didn't have an axiom like this, then if we had two categories $C_1$ and $C_2$ satisfying the assumptions, the product category $C_1\times C_2$ would also satisfy the assumptions. So this axiom is saying our category should be 'irreducible' (in the sense that is often used in algebra).

John van de Wetering (Sep 17 2021 at 14:43):

I think for me the axiom that is the hardest to interpret is that every dagger monomorpism is a dagger equaliser. Does anybody know what this means intuitively?

Cole Comfort (Sep 17 2021 at 14:45):

John van de Wetering said:

I think for me the axiom that is the hardest to interpret is that every dagger monomorpism is a dagger equaliser. Does anybody know what this means intuitively?

"Monics are regular if they behave like embeddings"

So I guess $\dag$-monics are $\dag$-regular if they behave like $\dag$-embeddings?

Martti Karvonen (Sep 17 2021 at 14:50):

Actually I believe that any dagger mono $f\colon A\to B$ is always the dagger equalizer of $id, ff^\dag\colon B\rightrightarrows B$, so the actual content of the relevant axiom is that any dagger mono is a dagger kernel (=equalizer of some arrow and the zero morphism), and not just a dagger equalizer.

Martti Karvonen (Sep 17 2021 at 17:34):

So roughly, I think that the axiom will imply that the subspace given by a dagger mono has a complement.

Simon Burton (Sep 18 2021 at 14:29):

There's this issue with the dagger, that it doesn't obey the principle of equivalence ... i'm not sure exactly what this means but i think it has to do with the definition of an abstract dagger being identity on objects, rather than just getting an isomorphism to the original object. So this makes me a bit wary of dagger categories.

James Deikun (Sep 18 2021 at 15:15):

You might be able to fix this by passing to a dagger chirality ...

Martti Karvonen (Sep 18 2021 at 15:55):

Given the answer at https://mathoverflow.net/questions/220032/are-dagger-categories-truly-evil/220111#220111 , I'm tempted to think that there are no issues with "weak dagger categories" - it just happens that the ones we see in practice tend to be strict, and in fact any weak one could be strictified.

John van de Wetering (Sep 18 2021 at 18:50):

I thought the conclusion from that Mathoverflow post was that we shouldn't think of dagger categories as categories with additional structures, rather it is just its own thing entirely. Just like how composition of morphisms in a category is not evil even though it needs the objects to exactly match.
So the title of the paper should really be "Axioms for the dagger-category of Hilbert spaces" :)

John van de Wetering (Sep 18 2021 at 18:51):

Martti Karvonen said:

Actually I believe that any dagger mono $f\colon A\to B$ is always the dagger equalizer of $id, ff^\dag\colon B\rightrightarrows B$, so the actual content of the relevant axiom is that any dagger mono is a dagger kernel (=equalizer of some arrow and the zero morphism), and not just a dagger equalizer.

Ahh, that is nice. Thanks!
So how strong is this axiom generally? Are there many categories wherin all monos are kernels?

Morgan Rogers (he/him) (Sep 18 2021 at 19:01):

It's true in all abelian categories, for example.

John Baez (Sep 18 2021 at 21:13):

Simon wrote:

There's this issue with the dagger, that it doesn't obey the principle of equivalence ... i'm not sure exactly what this means but i think it has to do with the definition of an abstract dagger being identity on objects, rather than just getting an isomorphism to the original object.

What it means is that if a category C is given the structure of a dagger-category and we choose an equivalence between C and some other category D, there is no systematic method to use this equivalence to transfer the dagger-category structure to D.

You sketched not what "violate the principle of equivalence" means, but rather one of the things you'd want to use to show that dagger structures violate this principle.

My statement of what it means is not precise because I didn't say what "systematic method" means. Category theorists have ways of making that idea precise.

But instead of doing that, let me give two examples. Suppose a category C is given the structure of a monoidal category and we choose an equivalence between C and some other category D, say

$F : C \to D$
$G: D \to C$
$\epsilon : FG \stackrel{\sim}{\Rightarrow} 1$
$\iota : 1 \stackrel{\sim}{\Rightarrow} GF$

Then there's a systematic way to make D into a monoidal category. I won't describe it, but you start by defining the tensor product of objects $d,d' \in D$ by

$F (Gd \otimes Gd')$

It gets fun when you define the associator for D and prove it obeys the pentagon identity.

If you try to do the same thing for strict monoidal categories it fails.

John Baez (Sep 18 2021 at 21:16):

John van de Wetering wrote:

I thought the conclusion from that Mathoverflow post was that we shouldn't think of dagger categories as categories with additional structures, rather it is just its own thing entirely.

Yes, that's a widely accepted way out of this problem.

John Baez (Sep 18 2021 at 21:36):

Of course this is a rather dramatic conclusion if we're going to give foundations of quantum mechanics based on dagger categories: it means nature is not content with categories.

Naso (Sep 19 2021 at 03:37):

John Baez said:

And if you want something a lot shorter than a book, you could try this paper of mine:

• Quantum quandaries: a category-theoretic perspective.

This is not a substitute for the two books mentioned. It's an explanation of how category theory can help us understand some of the puzzles of quantum mechanics.

I've just read this and really enjoyed it, thank you! I have two basic questions (as someone without physics training)

1. On p6 you said 'topological quantum field theories are an attempt to do background-free physics, so in this context we drop the background metrics:'.

   Q: Why is the topology not also considered part of the 'background'?

2. On p15 you said 'in special relativity, time evolution corresponds to a change of coordinates $t \to t + c$, which can also be thought of as change of our description of the system.'

   Q: How exactly does that correspond to to time evolution, and why can we even add $c$ to $t$ (when they have different units)?

Simon Burton (Sep 19 2021 at 16:25):

Mike Shulman said:

FWIW, I think usually a definition involving an identity-on-objects functor (or other structure violating the principle of equivalence) should usually be regarded not as structure on a previously given collection of categories but as a definition of a different, sui generis categorical structure. For instance, a dagger-category can be defined as a category with an identity-on-objects contravariant involution, but it's better to think of it as a new kind of categorical structure that happens to have an underlying category rather than as a "dagger-structure" on a previously given category. Same with Freyd-categories, etc.

It sounds like what Mike is saying here is similar to what @John van de Wetering is saying above.

Martti Karvonen (Sep 19 2021 at 16:54):

I like both kinds of answers from the MO thread I linked (1) dagcats are their own thing or (2) if axiomatized in a weaker manner, they aint' so bad -it's just that the strict ones show up in practice

John Baez (Sep 19 2021 at 21:31):

1. On p6 you said 'topological quantum field theories are an attempt to do background-free physics, so in this context we drop the background metrics.'

Q: Why is the topology not also considered part of the 'background'?

'Background' has a specific meaning in physics, namely a field that affects other fields but is not affected by it. A background field is simply posited, rather than obeying some equations that describe how it interacts with other fields. For example the metric in special relativity is a 'background metric', but not the metric in general relativity. People working in quantum gravity sometimes take the attitude that background fields are bad and try to do 'background-free' physics.

2. On p15 you said 'in special relativity, time evolution corresponds to a change of coordinates $t \to t + c$, which can also be thought of as change of our description of the system.'

Q: How exactly does that correspond to to time evolution, and why can we even add $c$ to $t$ (when they have different units)?

In this equation $c$ has nothing to do with the speed of light. It's just an arbitrary constant with units of time. Maybe I should have called it $k$ or $\Delta t$ or something.

John Baez (Sep 19 2021 at 21:31):

The point is that if you wait 10 minutes you're doing $t \mapsto t - 10$. What was 35 minutes into the future is now 25 minutes into the future.

Naso (Sep 20 2021 at 06:34):

John Baez said:

'Background' has a specific meaning in physics, namely a field that affects other fields but is not affected by it. A background field is simply posited, rather than obeying some equations that describe how it interacts with other fields. For example the metric in special relativity is a 'background metric', but not the metric in general relativity. People working in quantum gravity sometimes take the attitude that background fields are bad and try to do 'background-free' physics.

In the TQFT formalism the topology is posited, so should it be considered a sort of background structure? But I guess not a 'field', since it it is a global rather than local structure (unlike the geometry/metric)?

Cole Comfort (Sep 20 2021 at 07:47):

Martti Karvonen said:

Given the answer at https://mathoverflow.net/questions/220032/are-dagger-categories-truly-evil/220111#220111 , I'm tempted to think that there are no issues with "weak dagger categories" - it just happens that the ones we see in practice tend to be strict, and in fact any weak one could be strictified.

I don't think there is any reasonable weakening where one can talk about maps in general being isometries or unitaries. But of course, these things are very important in QM.

John Baez (Sep 20 2021 at 11:14):

Yes, the topology is posited; then people try to get rid of that and "sum over topologies", though these sums tend to diverge. But it's not a "field": for simplicity we can say a field is a section of some bundle over a manifold - or in other words, the sort of thing that can obey a differential equation.

John Baez (Sep 20 2021 at 11:15):

You can try to generalize the concept of "background structure" more and more, to go beyond the definition I laid out and include things like topology, but then it becomes a less and less useful concept in my opinion.

John Baez (Sep 20 2021 at 11:17):

When we have a field theory we have some fields that are "background" and some that are not: the background fields are the ones that are fixed from the start, so classically we don't vary them when we do the variational calculus to get the field equatons, and quantum-mechanically we don't integrate over them in the path integral.

John Baez (Sep 20 2021 at 11:19):

For all non-background fields we get a principle that could be summarized as "for every action there is an equal and opposite reaction": if field A pushes on field B then field B pushes on field A just as much. Background fields are like Aristotle's "unmoved mover": they affect the other fields but are not affected by them. So there's plenty of philosophical/physical reasons to be suspicious of background fields.

John Baez (Sep 20 2021 at 11:20):

But just as importantly, it's rather easy to tell in a theory what are the background fields and what aren't, and we know a lot about how they behave differently.

John Baez (Sep 20 2021 at 11:28):

It's extremely hard to formulate quantum theories without background fields, and topological quantum field theories are perhaps the only examples we understand really well, to the point where they are mathematically well-defined things.

Martti Karvonen (Sep 20 2021 at 12:25):

Cole Comfort said:

Martti Karvonen said:

Given the answer at https://mathoverflow.net/questions/220032/are-dagger-categories-truly-evil/220111#220111 , I'm tempted to think that there are no issues with "weak dagger categories" - it just happens that the ones we see in practice tend to be strict, and in fact any weak one could be strictified.

I don't think there is any reasonable weakening where one can talk about maps in general being isometries or unitaries. But of course, these things are very important in QM.

Well, in the approach Peter Lumsdaine explains in the MO-thread, it seems to me that the groupoid of unitaries is something that makes sense. But I agree, I don't see how to talk about isometries, partial isometries, self-adjoint maps, etc in such a setting. In any case, "all the relevant examples are strict" seems like a good enough reason to not feel bad about working with the strict ones and the concepts that make sense in that setting.

Cole Comfort (Sep 20 2021 at 12:28):

Martti Karvonen said:

Cole Comfort said:

Martti Karvonen said:

Given the answer at https://mathoverflow.net/questions/220032/are-dagger-categories-truly-evil/220111#220111 , I'm tempted to think that there are no issues with "weak dagger categories" - it just happens that the ones we see in practice tend to be strict, and in fact any weak one could be strictified.

I don't think there is any reasonable weakening where one can talk about maps in general being isometries or unitaries. But of course, these things are very important in QM.

Well, in the approach Peter Lumsdaine explains in the MO-thread, it seems to me that the groupoid of unitaries is something that makes sense. But I agree, I don't see how to talk about isometries, partial isometries, self-adjoint maps, etc in such a setting. In any case, "all the relevant examples are strict" seems like a good enough reason to not feel bad about working with the strict ones and the concepts that make sense in that setting.

My experience with this kind of stuff is that you can shift the evilness around, but if you want to be able to compose a map with its adjoint in general, then you can never fully get rid of it. Which I think is compelling evidence that $\dag$-categories should be seen as fundementally different mathematical objects than categories.

Oscar Cunningham (Sep 20 2021 at 12:32):

Is the forgetful functor from dagger categories to categories a right adjoint in the appropriate $2$-categorical sense? If so what are the algebras of the corresponding monad?

Martti Karvonen (Sep 20 2021 at 12:47):

No such biadjunction exists, but if you just think of $Cat$ and $DagCat$ as 1-categories, then the forgetful functor has both adjoints.

Cole Comfort (Sep 20 2021 at 12:49):

Martti Karvonen said:

No such biadjunction exists, but if you just think of $Cat$ and $DagCat$ as 1-categories, then the forgetful functor has both adjoints.

Chris explicity constructs the free and cofree such dagger categories in this way in his thesis, if you need a reference @Oscar Cunningham

Martti Karvonen (Sep 20 2021 at 12:50):

And the algebras of the monad are dagger categories

Oscar Cunningham (Sep 20 2021 at 12:51):

Thanks. I think the fact that it doesn't exist as a biadjunction is a good way to formalise the fact that daggers aren't a structure on categories.

Martti Karvonen (Sep 20 2021 at 12:51):

In a nutshell, the obstruction for a biadjunction is that $DagCat(C,D)$ is always a dagger category and in particular self-dual, but one can choose $C$ or $D$ so that $Cat(C,D)$ is not self-dual, preventing these hom-categories from being equivalent.

Amar Hadzihasanovic (Sep 20 2021 at 13:05):

Something I wrote about in this topic is that if you have the category of Hilbert spaces and isometries, there is a nice construction of Hilbert spaces and short maps as the 1-truncation of the bicategory of Hilbert spaces and cospans of isometries, where the dagger becomes the standard involution of cospans...

I have a “weak conjecture” that maybe the dagger categories that arise in practice are either groupoids, or are hiding some bicategory of spans or cospans in the background.

Amar Hadzihasanovic (Sep 20 2021 at 13:07):

I would personally find that a satisfying solution to the “problem” of dagger categories, but I do not have a strong belief in it.

Amar Hadzihasanovic (Sep 20 2021 at 13:09):

Beyond Hilbert spaces and short maps, it is certainly true of relations, partial injections, and cobordisms.

Martti Karvonen (Sep 20 2021 at 13:10):

How about Hilbert spaces and bounded linear maps?

Amar Hadzihasanovic (Sep 20 2021 at 13:11):

I think it could be possible to get it from something like “Hilbert spaces and rescaled isometries” but I haven't done the work.

Cole Comfort (Sep 20 2021 at 13:12):

Or what about completely positive maps between von Neumann algebras? I guess you could argue that spans are lurking in the background because they can be constructed in terms of adding a generator to hilbert spaces modulo discarding isometries.

Amar Hadzihasanovic (Sep 20 2021 at 13:17):

Anyway I'd be curious to know if the Heunen-Kornell axioms can be modified to give either isometries or “rescaled isometries”.

Amar Hadzihasanovic (Sep 20 2021 at 13:17):

Seems not so obvious since so many of them rely on the dagger!

John van de Wetering (Sep 20 2021 at 13:49):

Cole Comfort said:

Or what about completely positive maps between von Neumann algebras? I guess you could argue that spans are lurking in the background because they can be constructed in terms of adding a generator to hilbert spaces modulo discarding isometries.

von Neumann algebras with completely positive maps are not a dagger category. Resticting to the pure maps (in the effectus theory sense) however does give a dagger category. This is explained in Bas Westerbaan's PhD thesis.

John van de Wetering (Sep 20 2021 at 13:50):

Amar Hadzihasanovic said:

Anyway I'd be curious to know if the Heunen-Kornell axioms can be modified to give either isometries or “rescaled isometries”.

So an immediate problem with this approach would be that your scalars trivialise (although I guess with "rescaled" isometries you mean you do allow arbitrary scalars)?

John van de Wetering (Sep 20 2021 at 13:51):

Btw, not sure if he's active here, but maybe @Chris Heunen wants to take part in this topic as well?

Simon Burton (Sep 20 2021 at 14:33):

This discussion is helping me understand why there is no 2-categorical characterisation of the CPM construction.. (For a while i thought CPM must be related to the Int construction, but then Bob Coecke told me the two are not related. )

John Baez (Sep 20 2021 at 14:41):

By the way, I believe that in physics dagger-structures may ultimately arise from truncating $\infty$-categories with duals. This perspective is lurking in my papers Higher-dimensional algebra and topological quantum field theory, HDAII: 2-Hilbert spaces and Categorification. The idea is that having duals is a property of an $\infty$-category or $n$-category, except at the top level where we can only express it as a structure.

Alonso Perez-Lona (Nov 17 2022 at 02:46):

What is the current status of String Field Theory? Last thing I knew there was some formal formulation of superstring field theory, but I as far as I understand a mathematically rigorous definition isn't available at the moment.

Steve Huntsman (Nov 17 2022 at 13:55):

Alonso Perez-Lona said:

What is the current status of String Field Theory? Last thing I knew there was some formal formulation of superstring field theory, but I as far as I understand a mathematically rigorous definition isn't available at the moment.

From https://arxiv.org/abs/2211.04467:

image.png

Steve Huntsman (Nov 17 2022 at 13:59):

FWIW there is not a mathematically rigorous definition of "vanilla" Yang-Mills: https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap

Jean-Baptiste Vienney (Nov 17 2022 at 16:59):

Thanks for recalling your point of view @Steve Huntsman, very nuanced, as always. You're a professional: giving a little passage of a text, out of context, as answer, and keeping only half of the words in the questions. You could be a very good journalist.

Steve Huntsman (Nov 17 2022 at 17:14):

Jean-Baptiste Vienney said:

Thanks for recalling your point of view Steve Huntsman, very nuanced, as always. You're a professional: giving a little passage of a text, out of context, as answer, and keeping only half of the words in the questions. You could be a very good journalist.

My POV is demonstrably informed if not nuanced. As I said I was very interested in this stuff and took courses in SUSY/strings in 1996/97, then checked back up on it and saw that the area had only regressed in terms of its scientific (vs mathematical) content. The LHC didn't find SUSY and there are no realistic prospects for it. The paper I cite doesn't even mention (let alone dwell on) the utter failure of the landscape/swampland that makes string theory unscientific, and I didn't bring that up this time. So if you think I'm being uncharitable or dishonest then you are plain wrong.

Jean-Baptiste Vienney (Nov 17 2022 at 17:20):

But you didn't answer the question, it was about string field theory.

Steve Huntsman (Nov 17 2022 at 17:21):

Do you think there is a difference between string field theory and string theory?

Steve Huntsman (Nov 17 2022 at 17:21):

It's not like "quantum theory" and "quantum field theory" (which BTW the latter incorporates principles of the former). They are the same thing.

Jean-Baptiste Vienney (Nov 17 2022 at 17:22):

It's written in the paper "One notable attempt to give a fundamental formulation of string theory is the subject of string field theory. A solid foundation was given in [356,357] building on important previous work such as [358]. Unfortunately, the theory, as currently understood, is intractable. Important progress was made, nevertheless, [...]"

Steve Huntsman (Nov 17 2022 at 17:23):

Oh please. Next you're gonna tell me the separate wiki articles mean something.

Daniel Geisler (Nov 17 2022 at 17:24):

I agree with @Steve Huntsman and I think his perspective here is widely held. @Jean-Baptiste Vienney why do you think people should reconsider their positions. Please share your thinking here.

Jean-Baptiste Vienney (Nov 17 2022 at 17:29):

I love the nLab and 80% of the contributions are made by a single researcher in string theory. So I can't believe such people can be completely wrong about their ideas on physics. That's not a good argument but that's my feeling.

Steve Huntsman (Nov 17 2022 at 17:29):

https://www.google.com/books/edition/Lost_in_Math/OvI3DwAAQBAJ?hl=en

Steve Huntsman (Nov 17 2022 at 17:31):

BTW Witten and Atiyah have essentially caused a macro version of your impression on a generation of physicists. The AdS/CFT stuff has just made it worse.

Steve Huntsman (Nov 17 2022 at 17:32):

"Let's take this beautiful correspondence that has no discernable way to connect to the actual physics of the observable universe and turn it into a religion" is true of AdS/CFT and just feeds into the same cult of strings

Steve Huntsman (Nov 17 2022 at 17:35):

And BTW if a mathematically rigorous definition of a realistic QFT is half of a Millenium Problem then defining string (field) theory is even harder/further off.

Steve Huntsman (Nov 17 2022 at 17:35):

Besides being a gigantic waste of time.

Jean-Baptiste Vienney (Nov 17 2022 at 17:37):

Thanks, the book looks definitely interesting. But I would need a proof that string theory makes bad predictions to be convinced that it's not a good theory of physics. The lack of good predictions is not sufficient. And I don't see why we should stop working on it because it's mathematically too difficult. That's not a good argument.

Steve Huntsman (Nov 17 2022 at 17:38):

It makes no predictions! It is incapable of making predictions.

Daniel Geisler (Nov 17 2022 at 17:38):

I think Feynman's position is well considered on the importance of clarity in differentiating what is known with what is popular. I worry that many physicist have lost sight of the boundary between the two.

Xuanrui Qi (Nov 17 2022 at 17:40):

proof that string theory makes bad predictions to be convinced that it's not a good theory of physics

I'm clearly a mathematician and not a scientist, but I'd say that's now how I'd (or anyone would) go about in science... Generally in science you want to apply the principle of Occam's razor (if something is unnecessarily complicated and fails to make good predictions, then it's not a good theory!)

Steve Huntsman (Nov 17 2022 at 17:41):

I could probably overfit a Lagrangian that'll reproduce all known physics with way less than 10^500 or more free parameters. That wouldn't mean that the enterprise wasn't ridiculous.

Daniel Geisler (Nov 17 2022 at 17:44):

I believe even if string theory would not be immediately appropriate in physics it is important in mathematics and does deserve to be studied. To me it seems to lay out a mathematical landscape where everything ends up connected.

Xuanrui Qi (Nov 17 2022 at 17:47):

That's also what physicists I know tell me. The vast majority of them do not believe in string theory being physically valid, although they think it might be good mathematics. Of course a small minority still hold faith in it, but just like some people believe in Santa Claus, more or less :-D

Steve Huntsman (Nov 17 2022 at 17:52):

Daniel Geisler said:

I believe even if string theory would not be immediately appropriate in physics it is important in mathematics and does deserve to be studied. To me it seems to lay out a mathematical landscape where everything ends up connected.

Monstrous moonshine is lovely. So is mirror symmetry. Heck, so is AdS/CFT. But none of this has any relevance to physics.

Steve Huntsman (Nov 17 2022 at 17:54):

I think a sort of decent analogy might be: "this Lagrangian stuff doesn't describe experiments involving motion [let's pretend here to be in a world where this was true]. But optimal control is very mathematically interesting [it's actually useful too]."

Daniel Geisler (Nov 17 2022 at 18:04):

I'm writing paper on extending map to flows; $f^n(x)$ where $n \in \mathbb{C}$. This is a paper that could be published (hopefully) in either a mathematics or physics journal. If I want to established my work in physics I can avoid dealing with issues of convergence. So my work is likely to be more easily publish in a physics journal.

Daniel Geisler (Nov 17 2022 at 18:05):

We talked about what doesn't work. I think it is more fun to talk about what does work. What related subjects are folks working on, if any?

Steve Huntsman (Nov 17 2022 at 18:06):

Daniel Geisler said:

I'm writing paper on extending map to flows; $f^n(x)$ where $n \in \mathbb{C}$. This is a paper that could be published (hopefully) in either a mathematics or physics journal. If I want to established my work in physics I can avoid dealing with issues of convergence. So my work is likely to be more easily publish in a physics journal.

How does it work with (say) the Arnol'd cat map? There is a simple flow one can make from this but it's singular.

Steve Huntsman (Nov 17 2022 at 18:11):

One can make a suspension that's nice--see p. 24-25 of https://arxiv.org/abs/1009.2127 for details. Would be interested in how to make a flow more generally.

Daniel Geisler (Nov 17 2022 at 18:26):

My previous work has been confined to the iterations of complex functions. The work is general and likely works in Fréchet space as it is a recursive Bell polynomial because regular Bell polynomials are. But having an entire cat as a periodic fixed point hadn't occurred to me. :smile: Thanks for the reference, I'll read it and see what could be adopted from my work and report back in.

Morgan Rogers (he/him) (Nov 17 2022 at 19:41):

It feels like this discussion veered off-topic. Does anyone have any further informed perspectives for @Alonso Perez-Lona 's question on String Field Theory?

Xuanrui Qi (Nov 18 2022 at 07:47):

I guess since SFT isn't physics, one should approach it mathematically, which makes the nLab page a good answer to the question. Seems that the latest progress is something related to ∞-Chern-Simons theory.

Xuanrui Qi (Nov 18 2022 at 07:48):

I'm not a string theorist though, so there might be something newer that I don't know.

Daniel Geisler (Nov 18 2022 at 09:24):

Sorry, but I'm confused as to what the topic is supposed to be. The original topic was "synthetic formulation of qm". I assumed QFT was being referred to as QM was axiomatized by Dirac almost one hundred years ago. The subject seems to have changed to string theory, but folks might have more to say about QFT than string theory.

John Baez (Nov 19 2022 at 10:26):

There is a short but useful discussion of string field theory, with references to new work, here:

• Ibrahima Bah, Daniel S. Freed, Gregory W. Moore, Nikita Nekrasov, Shlomo S. Razamat, Sakura Schafer-Nameki, A panorama of physical mathematics c. 2022.

I also recommend this article for a good overview of the latest developments in string-inspired mathematics as a whole!

John Baez (Nov 19 2022 at 10:38):

Unfortunately it seems to omit any discussion of Urs Schreiber's work... but it's a good way to catch up on some other ideas.

Steve Huntsman (Nov 20 2022 at 02:10):

John Baez said:

There is a short but useful discussion of string field theory, with references to new work, here:

• Ibrahima Bah, Daniel S. Freed, Gregory W. Moore, Nikita Nekrasov, Shlomo S. Razamat, Sakura Schafer-Nameki, A panorama of physical mathematics c. 2022.

I also recommend this article for a good overview of the latest developments in string-inspired mathematics as a whole!

Yeah that’s exactly the paper that I managed to irritate folks with earlier in this thread. Go figure