Re #99

In the KU-example above, for the particle on the boundary of the 2d PCS theory, the role of $A$ is played by tensoring with the prequantum line bundle of the boundary theory. Traditional geometric quantization is recovered here in the form that “the Hilbert space of quantum states of the particle is the 2-expectation value of its prequantum line bundle regarded as a boundary operator of the cobounding 2d PCS theory”.

In the tmf-example above, for the string on the boundary of the 3d CS theory, the role of $A$ is similarly played by tensoring with the Kalb-Ramond B-field 2-bundle.

Presumably this continues with something to do with the C-field. And what then if

]]>the C-field is simply a cocycle in J- twisted Cohomotopy ?

Ok, I did that.

]]>And we wouldn’t then have to change the currently misleading link to ’polynomial (infinity,1)-functor’ at polynomial functor.

]]>Sounds good to me!

]]>Ugh. I’m not familiar with the functor-homology version, does anyone know what that is?

There’s not much content at polynomial (∞,1)-functor currently, and the definition suggests that anything one might want to say about that kind of polynomial (∞,1)-functor could equally well be placed at n-excisive (∞,1)-functor. So could we take over polynomial (∞,1)-functor for this, aligning terminologically with our page polynomial functor, and include a hatnote pointing the reader to n-excisive (∞,1)-functor or Goodwillie calculus if that’s what they’re after (like we already have at polynomial functor)?

]]>I’ve added it to (infinity,1)-operad. I would also add to a suitable page on polynomial functors, but we’ve already reserved polynomial (∞,1)-functor for the Goodwillie version.

This seems a recipe for confusion:

The notion of “polynomial functor” we consider here should not be confused with the notion of “polynomial functor” introduced by Eilenberg and Mac Lane and subsequently used in the study of functor homology, nor with the notion occurring in Goodwillie’s calculus of functors.

Their version is a homotopified version of what we have as polynomial functor. They note:

The fundamental nature of these three operations is witnessed by the fact that they correspond precisely to substitution, dependent sums, and dependent products, the most basic building blocks of type theory [HoTT].

Not sure I’m up to cleaning out the Augean stables of mathematicians’ naming conventions.

]]>I hadn’t seen this reference. Best to add a pointer to it to a relevant entry!

]]>Is

- David Gepner, Rune Haugseng, Joachim Kock,
*∞-Operads as Analytic Monads*, (arXiv:1712.06469)

what became of that forthcoming paper, Gepner, D. and J. Kock, Polynomial functors over inﬁnity categories, mentioned in #76?

I guess at the time I was wondering if it might be useful for Quantization via Linear homotopy types. Was def 3.20 the first appearance of the $\infty$-version of polynomial functor?

]]>So we could still say something of Mellies’ structural/logical split persists - weakening and contraction available in the classical world, logical rules as linear operators in the quantum world?

Hmm, or does the shift to dependent type theory muddle things up? I guess there’s type formation and term intro/elim/computation going on in the base and in the linear fibres.

]]>First of all it was Mike who emphasized that once we talk about *dependent* linear type theory then this adjunction arises canonically. In dependent linear type theory there are already two “universes of discourse”: the intuitionistic logic of the base types and the linear logic of the linear types parameterized over them. The adjunction that gives the exponential modality is just one small aspect of this more general “interplay of two universes”.

Then of course in the application to cohomological quantization we find very naturally familar names for these “universes of discourse”: that of the cartesian base category is the discourse of pre-quantum (“classical”) geometry/physics, whereas the other one is that of quantum geometry/physics. The fact that one is parameterized over the other is really the content of geometric quantization.

]]>Mellies makes a big deal about the factorization of ! in linear logic in his Categorical Semantics of Linear Logic, e.g.,

One lesson of categorical semantics is that the exponential modality of linear logic should be described as an adjunction, rather than as a comonad. The observation is not simply technical: it has also a deep eﬀect upon the way we understand logic in the wider sense…This reﬁned decomposition requires to think diﬀerently about proofs, and to accept the idea that

logic is polychrome, not monochromethis meaning that several universes of discourse (in this case, the categories L and M) generally coexist in logic, and that the purpose of a proof is precisely to intertwine these various universes by applying back and forth modalities (in this case, the functors L and M). In this account of logic, each universe of discourse implements its own body of internal laws. Typically, in the case of linear logic, the category M is cartesian in order to interpret the structural rules (weakening and contraction) while the category L is symmetric monoidal closed, or ∗-autonomous, in order to interpret the logical rules.

In view of #56

$! Z = \Sigma_+^\infty \Omega^\infty Z : \mathbf{H} \to Mod(\ast),$could we have a similarly florid description of what’s going on? Does something like that distinction between structural and logical rules persist?

]]>Right, elsewhere (at *order-theoretic structure in quantum mechanics*) I had emphasized the observation that the foundational theorems in quantum mechanics taken all together show that the idea that quantum observables form a Bohr topos – the glorified incarnation of the Jordan algebra of observables – is, essentially not a speculation but a derivable fact.

Now in the discussion of “quantization via linear homotopy-type theory” I have been adopting alltogether the “dual” perspective (in the sense that Bohr toposophy is Heisenberg picture where quantum logic as linear type theory is Schrödinger picture).

I don’t know yet how these two “pcictures” eventually want to merge to a single whole.

We had talked about this issue once before, when we were discussing “coordination” (I am providing links just for completeness for eventual bystanders). There I had speculated that the issue is the following:

suppose you manage to form the full extended quantization of 3d CS in $tmf Mod_4$. Then it ends up producing for you statements such as “the space of states on this manifold here is this tmf-$k$-module”. This raises a clear coordination issue: what does that *mean physically*?

We know by inspection how to do this coordination in some cases (e.g. an element of a $tmf$-1-module may be regarded as a Witten genus, hence as a partition function of a string). But we (well, me) don’t know how to “coordinate” systematically and generally.

Maybe Bohr toposophy is telling us that we should figure out to assign some kind of topos to any $E$-$k$-module such that the internal logic of this topos is the logic of statements about the universe of quantum states that make up this $E$-$k$-module.

But this is just a vague hunch. I have no idea at the moment about how to make this more concrete.

]]>I wonder if there something Jordan algebra-like about collections of such higher operators.

]]>And the KU and tmf cases have an equivalent of the A that you had in the Hilbert space case as an operator?

Yes, sorry I had been a bit telegraphic. That operator is the “integral kernel”, the “$\Xi$” in section 4.5.

In the KU-example above, for the particle on the boundary of the 2d PCS theory, the role of $A$ is played by tensoring with the prequantum line bundle of the boundary theory. Traditional geometric quantizatin is recovered here in the form that “the Hilbert space of quantum states of the particle is the 2-expectation value of its prequantum line bundle regarded as a boundary operator of the cobounding 2d PCS theory”.

In the tmf-example above, for the string on the boundary of the 3d CS theory, the role of $A$ is similarly played by tensoring with the Kalb-Ramond B-field 2-bundle.

Presumably at some point there must be scattering and absorption probabilities to calculate.

Yes, and that’s what I meant by transgression to codimension 0, which is where these scattering amplitudes appear (as discussed at *S-matrix – formalization*).

So what is currently described in the notes is, as highlighted in the second summary section, QFT extended down to the point but described only in three possibly high codimensions. This needs to be completed, eventually, clearly.

On the other hand there is already quite a bit of correlation function data appearing even in this higher codimension. Not the least for instance by AdS3-CFT2 we know that the spaces of states of the 3d CS theory compute “pre-correlators” (conformal blocks) of the WZW 2d CFT.

And for instance the value on the boundary of the Spin-Chern-Simons theory in the codimensions that we describe being the String orientation in tmf and hence on homotopy groups the Witten genus captures a correlation function of the string: the one on the torus without insertions, the partition function.

But this needs to be better understood in generality. Does every QFT have an extended quantization over a coefficient $\infty$-ring whose elements itself encode the partition function of the theory?

I don’t know yet.

]]>And the KU and tmf cases have an equivalent of the $A$ that you had in the Hilbert space case as an operator?

What you are wondering is probably how these “categoriefied” or “homotopyfied” statements transgress back to just expresssions in complex numbers.

Presumably at some point there must be scattering and absorption probabilities to calculate.

]]>So the relation between the dagger-structure and the inner product on Hilbert spaces is the following:

if a vector $| \psi\rangle\in H$ in a (finite dimensional, for simplicity) Hilbert space $H$ is thought of as a morphism

$| \psi \rangle \colon \mathbb{C} \longrightarrow H$then $(-)^\dagger$ of that is

$\langle \psi | \colon H \longrightarrow \mathbb{C}$and the pairing $(-)^\dagger \circ A\circ \circ (-)$ (which is a linear endomorphism of the ground rign, hence identified with an element of that ring) produces the probability of any observable operator $A$.

All these ingredients lift to the field theoretic quantization in linear homotopy-type theory now.

For instance for the quantization of a compact symplectic manifold as the boundary theory of the 2d Poisson-Chern-Simons theory then the state is a $KU$-linear map

$KU \longleftarrow KU_\bullet(X)$given by the K-theory class of the prequantum line bundle on $X$, and then the dagger structure allows to pair to a $KU$-linear map

$KU \longleftarrow KU_\bullet(X) \longleftarrow KU$which represents a single K-theory class. The virtual vector space representing that is the Hilbert space of quantum states on the system described by the phase space $X$ .

Or for the string, then the ground field is tmf and the tmf class

$tmf \longleftarrow tmf_\bullet(X) \longleftarrow tmf$is the refined Witten genus, hence the partition function of the string.

What you are wondering is probably how these “categoriefied” or “homotopyfied” statements transgress back to just expresssions in complex numbers. This I don’t fully know yet. I understand the transgrssion to lower codimension completely for the pre-quantum data, but not yet properly after quantization. Thomas Nikolaus had presented a related result in Vienna: where topological T-duality traditionally gives an equivalence of D-brane charges in $KU$, one really expects this to be the transgression of an equivalence of classes in $tmf$. He shows that this this is the case under some assumptions and conjectures that it is true generally. But this needs further investigation.

]]>And how do you pass from KU and tmf to measurable quantities? Is there something like $z \mapsto z^{\ast} z$, for complex numbers? Don’t we have to pass from codomains of genera to the reals at some point?

]]>Michael Wright. I’d mentioned your work in a couple of talks he has attended.

Thanks. We had agreed to stay in contact, but somehow we didn’t. I have created now an nLab entry *Michael Wright* and also one for his *Archive for Mathematical Sciences and Philosophy*. I was told fascinating stories about this archive. Too bad that it leads an offline analog life for the time being.

You relate the probabilistic nature of quantum physics to the dagger-compact structure, as laid out in section 4.5. Is that the place to look?

So the fact that the probabilistic nature of QM is well captured by the dagger-structure on a suitable linear category (say of finite-dimensional Hilbert spaces) is something very much emphasized by the “categorical quantum mechanics” crowd in the wake of Abramsky and Coecke. Maybe I should have added more of a comment on that in the file.

The observation that I meant to add in that section 4.5 is that in linear homotopy-type theory this becomes a bit deeper still. Here the dagger-structure becomes the existence of Umkehr maps/fiber integration maps via Poincaré duality in generalized cohomology and gives rise to a rigorous and useful path integral. Or put more simply: we have probability amplitudes not just with values in the complex numbers, but with values in rings like KU and tmf. I suppose it remains to be seen what to make of this relation…

Regarding Regensburg, yes, there will be plenty of interesting connections…

]]>The gentleman who runs that archive

Michael Wright. I’d mentioned your work in a couple of talks he has attended.

I’d like to understand the following better:

So we have a refined kind of logic – linear homotopy-type theory – which describes quantum field theoretic processes in their probabilistic nature and is at the same time such that making propositions about these QFT systems means “wave function collapse”.

You relate the probabilistic nature of quantum physics to the dagger-compact structure, as laid out in section 4.5. Is that the place to look?

A Regensburg trip sounds interesting. It would be great to see some of these ideas penetrate further into number theory, such as arXiv:1209.6451.

]]>My lightning appearance and disappearance at the meeting in Paris limited the amount of possible discussion.

Over joint lunch after the talk one point that proved to resonate with some participants was the observation that the gauge principle in physics is the same as what makes types be homotopy types.

One participant said he enjoyed the appearance of motivic structures.

Another participant wondered if I could see in the formalization that I was talking about the point that he was advertizing in his talk, namely that the quantum phase should be thought of as a remnant of gauge equivalence. (My reply: maybe sort of, vaguely, but maybe not quite the way he is envisioning).

The observation that string-theoretic structures appear by themselves from the fromalization (not being explicitly asked for) led some to wonder.

The gentleman who runs that archive with plenty of historical records of talks by Lawvere videotaped my talk and expressed a fair bit of general interest.

Before my talk I had the chance to chat with Gabriel Catren and his group a bit and I expressed and we agreed that the existence alone of a group and grant like his (pairing physics with philosophy and mathematics) is quite remarkable. Indeed, he recounted a story of how a fellow professor, not recognizing him, vividly expressed his disdain for the European Union giving money to anything with both “philosophy” and “physics” in the title.

Myself, I feel that there is a bit of a technology distance between what I am advertizing and what most other participants who were at the workshop in Paris are working on.

That was of course different at the ESI meeting afterwards. There I didn’t dwell on any foundational questions but concentrated on the mathematical technology. Among the participants it seems that Uli Bunke and Thomas Nikolaus were those who appreciated the stuff genuinely. In fact there was some nice connection to Thomas’s talk (which was an impressive list of unpublished results on the generalized cohomology of T-duality). If all goes well then Joost and maybe myself will go to Regensburg for a little by summer or end of the year to talk more.

]]>So, any interesting reaction to the talks, especially from the philosophers there at the first one?

]]>Re #88,

So we have a refined kind of logic – linear homotopy-type theory – which describes quantum field theoretic processes in their probabilistic nature and is at the same time such that making propositions about these QFT systems means “wave function collapse”. All this from the logical substrate, nothing put in “by hand”. Seems to be rather beautiful to me. (But I guess this point deserves to be further elaborated on eventually…)

This would be just the kind of account that we should take to philosophers of physics.

Re # 90, is there any room in this picture for the ’critical’ aspect in the Kantian sense? His ’Copernican revolution’ was to take our knowledge of the world not to be of the thing-in-itself, but the world as taken up within our faculty of reason, a combination of the categories of the understanding (objecthood, causality, necessity, etc.) and the contents of our intuition. This wasn’t meant just to apply to a person’s everyday judgements. Even the theoretical resources of the leading science of the day, Newtonianism, arose out of the categories.

Looking at what you just wrote, Kant would find enormously problematic your “external perspective on this universe described by $Z$”. There can be no such thing for him. But then maybe that’s why you’re a Hegelian ;) – the categories aren’t just to be read off from our judgements, but derived in Logic, the same Logic that drove the creation of the universe.

]]>I have come to the heretic opinion that the problem of observers in “quantum cosmology” is not any different than that in “classical cosmology”. Here is what I mean:

Assume for the moment, as lots of people around us do, that once it is done then quantum gravity is a TQFT

$Z \;:\; Bord_n^\sqcup \longrightarrow Mod_n^\otimes \,.$(of course that’s a bit of an assumption here, but let’s go with this to have something a bit concrete to talk about).

Imagine that this arises from a quantization process the way it is discussed in the notes that this thread is about

$Z \;:\; Bord_n^\sqcup \stackrel{\exp(\tfrac{i}{\hbar}S)}{\longrightarrow} Corr_n(\mathbf{H})^\otimes \stackrel{\int(-)d\mu}{\longrightarrow} Mod_n^\otimes \,.$Here the first functor is the “classical” (prequantum) theory, the second is the quantization step.

So feeding closed $n$-manifolds into this $Z$ it tells us about the spaces of states of “cosmologies” of shapes these manifolds, and how they propagate.

The intermediate functor gives classical states, the total functor quantum states. In either case, since, we imagine, all these functors are constructed (as done in the notes in high codimension) using homotopy-type theory and linear homotopy-type theory, we may formally analyze them within this logical framework. There are propositions about the states, and in the quantum case they come with projection/collapse operators and so forth. Nothing left mysterious, all formal logic.

Except one thing: now you ask “Fine, so this is the external perspective on this universe described by $Z$. But what do “observers inside that universe” see and feel? “

Yes, that’s a super-difficult question. In fact, how do we even see if “inside $Z$” there is anything like an observer at all?

But this problem is not one of quantum, but one of cosmology: because the same problem we have for the functor $Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H})^\otimes$. If I hand you such a functor, hence a “classical cosmology”, how do you know what “observers inside this cosmology” are and feel.

This is a deep riddle. It’s related to that Penrose triangle (not to mention Mr. H. yet again) which we discussed another time in another thread. But whatever it is, I come to think, for the above reason, that it’s not an issue related to “quantum”.

]]>I can’t seem to find it now, but we did discuss once the problems with quantum gravity. Something like QFT/QM yields predictions for events in regions of spacetime in which an experiment takes place, and how could this be extended to the whole universe.

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