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Stream: deprecated: physics

Topic: discrete quantum theory

Ben Sprott (Apr 29 2021 at 17:07):

I like to think about the problem of Quantum Gravity. After doing my MSc as a member of the Perimeter Institute, I started wondering about quantum theory itself. I also read a lot about Category Theory and advanced ideas and analysis of modern physics. I always loved the casual approach to quantum gravity and how it dovetails with Category Theory.

One of the problems that I felt would get in the way of the standard approach to Quantum Gravity, was that it seemed to call on a quantum theory with a continuum of pure states. How could that be an aspect to the theory in a finite, early universe with nothing around to test it? I felt this would break down.

I am working on a derivation of a truly finite quantum theory. I have written background papers which describe a new way of doing Science.. These, I believe, form a foundation for a physics with a truly finite quantum theory. In fact, this quantum theory, derived from casual ideas, might itself form the theory of quantum gravity, though I have never tested that out.

I have started a question over at Math Overflow where I am asking for a computation of a monad composition.. I am going to offer a 2-3 hundred point bounty for it in a few days.

Ben Sprott (Apr 29 2021 at 18:03):

(deleted)

Ben Sprott (Apr 29 2021 at 23:21):

According to my earlier papers, the theory that one derives from the data of quantum experiments is the Eilenberg-Moore category of the monad which models data.

My experiences with simple quantum experiments, with no entanglement and large numbers of single qubits, came from polarization experiments. In those experiments, it is quite common to rotate a wave plate and observe some statistical data. Since the Distribution monad is related to the multiset monad via a natural transformation, we can say that, beneath the intensities output by power meters, are Multisets of detector clicks. These Multisets of, say left/right clicks, have a Symmetry property that you discover by rotating the wave plate.

We normally understand this Symmetry to be an O(1) group of operators on a Hilbert space. I am suggest that we replace this continuous symmetry with a different one via cyclic lists. These lists are exactly what comes out of these types of experiments in terms of the data.

Jade Master (Apr 30 2021 at 23:59):

You said in one of the questions that a cyclic list monad is a list where each element points to the next. I don't see how it's different from the ordinary list monad...what I mean by that is it seems like you can turn every list into a cyclic list in a unique way.

Spencer Breiner (May 01 2021 at 02:01):

But not vice versa. You can't turn a cyclic list into a list without knowing where to start.

Jade Master (May 01 2021 at 02:46):

Oh I see thanks.

John Baez (May 01 2021 at 04:32):

In the world of species we have the species of cyclically ordered sets and the species of totally ordered sets, and there's a morphism from the latter to the former, but it has no inverse.

John Baez (May 01 2021 at 04:33):

There are $n!$ total orderings of an n-element set but only $(n-1)!$ cyclic orderings.

Jade Master (May 01 2021 at 04:53):

Interesting

Joachim Kock (May 01 2021 at 09:17):

The cyclic-order endofunctor is not a monad, for the same reason as mentioned by Spencer: given a cyclic order of cyclic orders, there is no canonical way to flatten it to a single cyclic order, because to do that it would be necessary to cut open the innermost cyclic orders, and there is no canonical place to cut and start.

But the cyclic-orders endofunctor does fit into the decomposition-space framework: if one builds a simplicial set of nested cyclic orders, that simplicial set is a decomposition space. (If one does that for a monad, the simplicial set is the two-sided bar construction, and the monad structure translates into this simplicial set being Segal. So the interesting bit about cyclic orders is that in a sense it admits a two-sided bar construction without being a monad, and that this is not Segal but 2-Segal (2-Segal is synonymous with decomposition space).

Joachim Kock (May 01 2021 at 09:30):

Now that we have commutative diagrams, maybe it is time to write out the decomposition space axioms! (I have already polluted a couple of threads speaking about decomposition spaces without giving the definition :-( But here it is.)

Here is first the Segal condition: this is a pullback:

$$\begin{CD} X_2 @>{d_2}>> X_1\\ @V{d_0}VV @VV{d_0}V \\ X_1 @>>{d_1}> X_0 \end{CD}$$

It says that the $2$-simplies are just 'composable $1$-simplices'.

And here is the decomposition-space condition: these two are pullbacks:

$$\begin{CD} X_3 @>{d_3}>> X_2\\ @V{d_1}VV @VV{d_1}V \\ X_2 @>>{d_2}> X_1 \end{CD} \qquad \begin{CD} X_3 @>{d_0}>> X_2\\ @V{d_2}VV @VV{d_1}V \\ X_2 @>>{d_0}> X_1 \end{CD}$$

It looks very mysterious until you start working with the conditions. Here is a combinatorial interpretation: let's take the first square. To say that it is a pullback is to say that the fibres of the vertical maps are isomorphic. So pick a $2$-simplex $\sigma$ in the lower left-hand $X_2$. Map it over to $X_1$ via the horizontal map $d_2$ which deletes the last part of $\sigma$, leaving only the first part. Now compare the two fibres. The fibre of the left-hand vertical map $d_1$ is the set of all ways to subdivide the first part of $\sigma$. The fibre of the right-hand vertical map is the set of all ways to subdivide the first part of $\sigma$. So the whole axioms says that subdividing a part of a simplex does not depend on what is outside that simplex. It's a locality condition.

Joachim Kock (May 01 2021 at 09:39):

Of course, to read these conditions, some familiarity with simplicial language is required. At first it can look confusing with all those face maps and all those indices. But it is a fantastic toolbox (useful in algebra, topology, geometry, combinatorics, etc.) The main secret to the language is that indices always refer to what is deleted! In this way, if $X$ is the nerve of a category, $d_1:X_1 \to X_0$ is the source of an arrow, and $d_0: X_1 \to X_0$ is the target of an arrow, which can look backwards at first sight. But it has to be like this.

The point of the Segal condition is that the maps I wrote horizontally are top face maps, and the vertical ones are bottom face maps. The general Segal condition says that top and bottom face maps form pullbacks against each other.

The point of the decomposition-space condition is that the horizontally-written maps are top face maps (in the left-hand diagram) and bottom face maps (in the right-hand diagram), whereas the vertical maps are inner face maps. The general decomposition-space condition says that outer and inner face maps form pullbacks against each other.

Spencer Breiner (May 01 2021 at 15:51):

John Baez said:

In the world of species we have the species of cyclically ordered sets and the species of totally ordered sets, and there's a morphism from the latter to the former, but it has no inverse.

Is it a quotient?

John Baez (May 01 2021 at 16:04):

Yes. There's an automorphism $\alpha$ of the species of linearly ordered sets that says, like Jesus said, "the last shall be first": it takes the linearly ordered finite set

$x_1 \lt x_2 \lt \cdots \lt x_n$

and maps it to the linearly ordered set

$x_n \lt x_1 \lt \cdots \lt x_{n-1}$

Taking the coequalizer of $\alpha$ and the identity, we get the species of cyclically ordered sets. So there's a regular epimorphism from the species of of linearly ordered sets to the species of cyclically ordered sets.

(But species form a Grothendieck topos, so any epimorphism is regular.)