Stroop Theory (lives in category land)

[marcus]

if there is any place where string ideas and loop quantum gravity might join hands and live happily ever after it is in the Baez paper
quant-ph/0404040
and a few other papers to which this one is the gentlest introduction I have found so far.

this is where Baez introduces the idea of a "star" category
and more specifically a TENSOR star category-----(a star category with something like a tensor product defined------that is not the official math term for it but for brevity sake that is what I will call this type of category).

the category of SPACETIMES is a tensor star category

the category of HILBERTSPACES is a tensor star category

This could be a giveaway hint that quantum gravity might eventually INVOLVE THIS sort of CATEGORY---because QG is about connecting spacetime dynamics to a hilbertspace of quantum states. This type of category is rather special and it is a remarkable coincidence that hilbertspaces (which is where quantum states are vectors in)
and spacetimes (which is what Gen Rel is the dynamics of) should both be this same type of dingus.
=================

now I have already made several faux pas of math language, so I have to back up. But the message i want to get across, before doing that, is that we should find out what this kind of category is.

It will turn out that string WORLDSHEETS and non-string QG SPINFOAMS come up rather naturally in this context

 

[marcus]

The hard part is the categorical definition of tensor product. The STAR part is easy.
If you look into quant-ph/0404040
you will quickly see what a "star" category is. that is no problem.
It is just a category where every map f:X->Y has a DUAL map f*:Y->X
which is its PARTNER
(CATEGORISTS CALL MAPS "MORPHISMS")
and a simple example to consider is Hilbertspaces with the morphisms being continuuous linear maps----think matrices----and the dual of a map being the familiar ADJOINT map----think about flip-conjugating the matrix. We do this as freshmen. All the categorists are doing is ABSTRACTING the idea of adjoint map.

and it is a nice reciprocal thing: f**=f, and (fg)* = g*f* and the IDENTITY map is its own adjoint. Just as you expect doing it with matrices in the example
=========================

"star" category is Baez made-up terminology which he says might disturb some serious Categorists. Some of them like to always use the correct terminology and they might say "A Category with Dual Morphisms". they are stubborn sum*****es and always want to force you to use their nomenclature.

But in this thread we follow Baez 0404040 and call it "star" because the star reminds us of the adjoint symbol f*
=========================

Categorists would probably call a TENSOR star category by the longer correct term
"symmetric braided monoidal category with dual"

but for our purposes, a "symmetric braided monoidal" is too long to say and that kind of category is simply one that has something like a TENSOR PRODUCT defined on it.
=========================

the absolute minimum you need to understand what is going on is to check out an undergrad algebra text and be sure you understand what the tensor product of two hilbertspaces is.

there are a couple of thin books by Paul Halmos that used to be good for this. but by now there must be a ton of other. it takes a couple of pages.
=========================

One always suspects, after first being confronted with a generous helping of indigestible terminology, that mathematicians (especially Categorists) are crazy.

this is a natural and i believe basically reasonable reaction.

However it is uncalled for in this case because you have not seen the PICTURES yet.

So best you go to Baez 0404040 and look at the pictures and you will see pictures of string worldsheets and QG spinfoams and you will see a PICTORIAL ANALOGY TO FEYNMAN DIAGRAMS and you will see that these things are recognizable as spacetimes, sort of hoses that connect a "before" space to an "after" space. These spacetime hoses that connect before to after are a concrete manifestation of a dynamic PROCESS and the traditional mathematician's name for them is COBORDISMS.

And these cobordisms (which are spacetiime dynamic process in a very pictorial form) ALSO FORM A TENSOR STAR CATEGORY.

The pictures are very simple, almost self explanatory. Putting two cobordisms (call them worldsheets or call them hose diagrams, whatever) side by side make their TENSOR PRODUCT and putting one on top of another and welding makes the composition of mappings. And the DUAL is just flipping upside down so you get the process running in reverse----exchange "before" with "after".

this seems so familiar as to be deja vu. We may have had a PF thread about this in April 2004 when the paper came out. It was the run-up to Rovellis Loop/Spinfoam conference at Marseille.
===================

so all I can say is probably some of us should review quant-ph/0404040 and a couple of related papers like the Baez Chronology of Physics Since Maxwell (which is physics from a categories viewpoint)

If you know something about categories you know that if you have two categories and you find a way to ANALOGIZE between them-----like a way to connect relativityspaces to hilbertspaces----the analogy is called a FUNCTOR.

ultimately a Quantum Gravity, or a General Relativistic quantum physics might appear to some frazzled unshaven guy on a mountain as a FUNCTOR from something like Cob to something like Hilb-----from the spacetime and matter processes that connect before to after-----to the hilbertspace of quantum states that represents a persons knowledge about the system. So he will come running down from the mountain yelling "I HAVE SEEN THE GREAT FUNCTOR"

there is a real danger of this happening, so we probably all ought to know how to say Wie geht's and Merci beaucoup in Category language.

 

[marcus]

this is just to correct terminology

the category of SPACETIMES is a tensor star category

the category of HILBERTSPACES is a tensor star category

Baez called the first one of these 4Cob (Cob is short for cobordism)
and it is the category where the objects are 3D spaces
and the morphisms are 4D spacetimes that connect a "before" 3D space to an "after" one----IOW "4-cobordisms"

Baez called the second one Hilb
and it is the category where the objects are hilbertspaces and the morphisms that connect them are bounded linear operators, or continuous linear maps, whatever you want to call them: matrices basically.

because Hilb is a STAR category, and every morphism has a dual morphism, there is the idea of a SELF ADJOINT map, where f*=f
and those are the OBSERVABLES in a quantum theory.
=====================

BTW I should post a warning (which I think Baez makes several places, if not this paper some other closely related paper).
the warning is that THIS MIGHT ALL NOT WORK. it might be useless.

a lot of beautiful elegant mathematics turns out not to solve the problem---not do what you want it to, in the end.

For us at PF, these tensor star categories have been around since spring of 2004, and for some people here, like Kea, a lot longer.

I am only just now beginning to think I have to buckle down and learn more about it. Of course it might not work out. But recent stuff connected with Baez 31 May Colloquium at Perimeter is making me think it is too risky not to know the tensor category basics.
Ordinary category basics I sort of know, tensor ("symmetric braided monoidal") category basics I don't know. Maybe some others are in the same boat.

 

[marcus]

I accidentally spoilt this post by trying to edit it. It was something I wrote past bedtime yesterday.

Anyway here is some useful information in case anyone wants links.

the April 7 2004 version of Baez 0404040 is here
http://math.ucr.edu/home/baez/quantum/
the May 31 perimeter colloquium page is here
http://math.ucr.edu/home/baez/quantum_spacetime/
the actual slides/lecturenotes for the colloquium talk are here
http://math.ucr.edu/home/baez/quantum_spacetime/qs.pdf

 

[arivero]

So it's off to bed. Maybe Kea or Arivero will explain tomorrow.
.

Lets try Kea, I am unable to help here.

In fact I am amazed about how category theory enters play. A lot of people expected it to have arole in physics via morphism between representations in Hilbert space; Doplicher did some work on it with the so-called intertwinning operators, and this is related to Morita equivalence. In some sense it was the low-profile role of categories. A "high-profile role" was expected to come via topoi theory, or perhaps via sheaves. It is amusing that Baez approach does not fit with these a-priori expectations, but follows an original path.

 

[Hurkyl]

Hrm. How about a T* category? We already have B* and C* algebras, and H* categories!

A lot is understandable but I came to where he said "zig zag identities" and got stuck.
I could only see pictures and couldn't translate them into write-down equations.

But you're not supposed to! Part of the whole point is that this is higher-dimensional algebra, and requires higher-dimensional formulas! So instead of our boring one-dimensional equations, we're supposed to get used to two-dimensional equations. (No, I'm not used to two-dimensional equations, but I'm getting there) And there's this whole conjecture that higher category theory will just turn out to be topology in disguise, which is good, because category theory is algebra!



So we have this funky diagram:

               |
               |
    /-\        |
   /   \       |
  /     \      |
  |      \     /
  |       \   /
  |        \-/
  |
  |

What does it mean?? Well, for simplicity, let's assume this curve is oriented downwards, and that we have a canonical element f of our T*-category. So, everywhere this curve is heading downwards denotes f, and upwards denotes f*.

But wait! There are two secret lines in this diagram... it really should look like:

     :         |
     :         |
     ^         |
    / \        |
   /   \       |
  /     \      |
  |      \     /
  |       \   /
  |        \ /
  |         v
  |         :
  |         :

where the dotted lines denote the multiplicative identity.

But wait, there are more secrets! The diagram is hiding some important dots:

     :         |
     :         |
     O         O
    / \        |
   /   \       |
  /     \      |
  |      \     /
  |       \   /
  |        \ /
  O         O
  |         :
  |         :

So what are these dots? They represent the morphisms in our T*-category. The upper-left dot is our counit map, and the lower-right dot is our unit map. The other two dots are identity maps.


So now how do we read it? Well, we have to break it into pieces:

     :         |
     :         |
     :         |
     :         |


     :         |
     O         O
    / \        |


   /   \       |
  /     \      |
  |      \     /
  |       \   /


  |        \ /
  O         O
  |         :


  |         :
  |         :
  |         :
  |         :

The first, third, and fifth parts of this picture denote objects in our T*-category. The second and fourth parts denote morphisms in our T*-category. This whole picture is simply a diagram denoting the composition of two morphisms! We simply drew our objects as big pictures, and our morphisms as filaments between the pictures!

     :         |
     :         |
     :         |
     :         |

I've already mentioned that the dotted line denotes the identity element 1, and the dashed line denotes our canonical element f. Well, placing things side-by-side denotes tensor products. This is nothing more than the object $1 \otimes f$.

     :         |
     O         O
    / \        |

The dot on the left denotes the counit map i, and the dot on the right denotes the identity map 1. (Sorry I use 1 so much) So, this is simply the morphism $i \otimes 1$. This morphism points from the object above to the object below.

   /   \       |
  /     \      |
  |      \     /
  |       \   /

This is the object $f \otimes f^* \otimes f$

  |        \ /
  O         O
  |         :

The one on the right, remember, is the unit map e. So, this is simply the morphism $1 \otimes e$.

  |         :
  |         :
  |         :
  |         :

And this is simply the object $f \otimes 1$.



If we were being traditional, this diagram would be drawn as:

         i o 1                 1 o e
1 o f  -------->  f o f* o f --------> f o 1

So, this denotes nothing more than the product:

$$(i \otimes 1) \cdot (1 \otimes e)$$


Since we have a canonical way to replace $f \otimes 1$ and $1 \otimes f$ with f, we can view the above as a map f ---> f, and the zig-zag identity simply asserts that this is equal to the identity!


I guess if we wanted to be complete, we could explicitly add in the isomorphisms that map $f \rightarrow 1 \otimes f$ and $f \otimes 1 \rightarrow f$:

          |f
          |
          |
          o
         : \
        :   \f
     1 :     \
      :       \
     :         |
     :         |
     :         |
     O         |
    / \  *     |
   /   \f      |
f /     \      |f
  |      \     /
  |       \   /
  |        \ /
  |         O
  |         :
  |         :
  |         :
  \         :1
   \       :
    \     :
     \   :
      \ :
       o
       |
       |
       |f

 

[selfAdjoint]

I was interested to see that the "star" operation is a pre-sheaf. In the most general definition (see Spanier's Algebraic Topology) a pre-sheaf is just a contravariant functor, say F, that takes objects and morphisms A -> B into F(B) -> F(A).

Marcus, you say we need to get our minds around tensor product of Hilbert spaces, and that's true, but I also think that monoidal is a block to understanding. Baez gives a formal definition but we need to noodle around with it to see how it - and particularly that commutative diagram - really work. Also the non-cartesian property. Only when we are very clear iin our minds what these "stroop" categories do and are can we move forward.

 

[marcus]

Thanks Alejandro, selfAdjoint, and Hurkyl.
Hurkyl I hope writing this helpful post didn't make you seriously late for work!

 

[Mike2]

if there is any place where string ideas and loop quantum gravity might join hands and live happily ever after it is in the Baez paper
quant-ph/0404040
and a few other papers that this one is the gentlest introduction to, that I have found so far.

I've also been thinking about how strings might unite with LQG, but it does not have directly to do with category theory. Is this thread open to discussion without category theory?

 

[marcus]

Mike regretfully I think not
I would like this thread to help as a CRUTCH to understanding these papers which I mentioned before:

the April 7 2004 version of Baez 0404040 is here
http://math.ucr.edu/home/baez/quantum/
the May 31 perimeter colloquium page is here
http://math.ucr.edu/home/baez/quantum_spacetime/
the actual slides/lecturenotes for the colloquium talk are here
http://math.ucr.edu/home/baez/quantum_spacetime/qs.pdf

this is not just categority in GENERAL but actually a particular AREA of applied category theory
which is the inscrutable and unexpected application of category theory to topological quantum field theory and ultimately quantum gravity.If we want to make headway on the main thing, then I don't think we can efficiently go outside category theory here, and should try to gnaw away at what Baez calles the "HIGHER ALGEBRA" approach to Gen Rel and QM.

 

[Hurkyl]

I've rewritten my earlier post. It's better now! (Though I don't mention the connection between this and 2-categories) I've saved off the old text if you still want to see that

 

[marcus]

Hear, O Israel: the LORD our God is one:

5 and thou shalt love the LORD thy God with all thine heart, and with all thy soul, and with all thy might.

6 And these words, which I command thee this day, shall be upon thine heart:

7 and thou shalt teach them diligently unto thy children, and shalt talk of them when thou sittest in thine house, and when thou walkest by the way, and when thou liest down, and when thou risest up.

8 And thou shalt bind them for a sign upon thine hand, and they shall be for frontlets between thine eyes.

9 And thou shalt write them upon the door posts of thy house, and upon thy gates.

this is the Schema
http://www.bible-researcher.com/shema.html
also known as Deuteronomy 6 verses 4-9

 

[Hurkyl]

That's not quite the old text I meant!

 

[marcus]

ay Hurkyl,

now let's talk about the "creation and annihilation" morphisms that you mentioned, i for initiate and e for erase.

in category Cob (the category made of spacetimes) the UNIT is just the null manifold, the nada.

because the tensor mult is simply to put one space BESIDE another.

so the i_H morphism just CREATES TWO UNIVERSES WITH OPPOSITE ORIENTATION, side by side, out of the null so it is a morphism like this:

i_H : nothing -> H x H*

but to be kosher the x should be a tensorproduct and it is sometimes just not worth going into TEX so I will use the "cool" for tensor and I will say UNIT for "nothing"

i_H : UNIT -> H H*

and the PICTURE for i_H is simply a fat inverted letter U, like made of tube or spaghetti, a fat intersection sign (if you know set theory)

========
OK that is in the case of category Cob. It is an operation or morphism which creates two equal universes of opposite orientation out of the nada---the unit universe. and of course there is a SPACETIME COBORDISM DRAPED BETWEEN THEM which is the fat inverted U.
=========

Now what about HILBERSPACES. what about the category Hilb. There the UNIT is just the complex numbers C, which is the simplest hilbertspace you can have.

Now here, notice that for a given hilber H, the tensorproduct H H* is simply some maps from H to ITSELF. You can figure out what those maps are by thinking about multiplying a column vector times a row vector-----i.e. multiplying two n-dim vectors in the "wrong" order so you get an nxn matrix instead of just a number. so H H* is just some linear maps from H to itself. Now how do we interpret the "creation morphism"?

i_H : C -> H H*

well what this does is it takes an complex number z in C, and it sends it to the MAP from H to itself that is simply multiplying a vector by the scalar z! And that is a kind of map that is in H H* so everything is cool.
==============

Both these categories Cob and Hilb are important so we have to do examples in both cases. now we have done the i_H in both cases and still have to do the e_H morphism in both cases.

this is the destructo morphism that annihilates the tensor product H* H, of the dual of any object H in the category, tensor with itself. This is the ERASURE morphism e_H

In the Cob case it has to give NOTHING. it has to start with two equal universes (with opposite orientation) side by side and it has to conver that to the unit of the category which is the Null manifold where there isn't anything there.

I think we have enough momentum to see how that goes so I will stop here for a while.

 

[arivero]

By the way, it is interesting to do not become obsessed by the arrow in the cathegorial notation. Axiomatically there is a function "source(a)" and a function "target(a)" ranging for all arrows "a", and that is all. Most cathegories can be dualised by using the functions in a reverse way ("reversing the arrows").

 

[Hurkyl]

The matrix "algebra" of all matrices over a field provides a very nice example of the category... and one that people actually use as a category (though they usually don't realize it!). That's a good example of something you usually don't picture with arrows.



Maybe I'm getting too far ahead, but one of the things that intrigued me is that you can start adding more levels -- instead of considering 3-spaces, and 4-spacetimes between them... you can consider 2-dimensional stuff, 3-space between them, and 4-spacetimes between the 3-spaces. This gets mapped not into Hilb, but 2-Hilb. I spent a bit of time trying to fathom what that all means, but didn't get far enough to make me happy.

 

[marcus]

...too far ahead... one of the things that intrigued me is that you can start adding more levels ...I spent a bit of time trying to fathom what that all means,...

there is no schedule that you can rush or get out of synch with. this thread is a good place to help us try to fathom the multilevel stuff.

the big motive (for me) is I see people like Freidel making progress with 3D gravity
and casting about for ways to extend to 4D

and one gets the suspicion that they, in desperation, may have to "jack the algebra up" in order to get the nice Freidel things to happen in one higher Dee.

wilson loops might have to become "wilson homotopies" that are paths between loops------and those might transmute into the faces of a spinfoam---or some such pocus hocus.

in james thurber's book THE WHITE DEER there is a moment at which "the King gave a great haarooof! like a lion tormented by magical mice" and I sympathize with him, these suspicions that higher algebra may hold keys to extending what already works so well up one Dee, these suspicions itch in my fur and are a bother.

so please pursue it on any path and schedule you know how!

 

[Hurkyl]

I was trying to imagine the next level as sort of like Loop Quantum Gravity. Remember that in a spin network, the nodes represent chunks of space-time, and edges represent the faces between them. The edges are labelled with representations of SU(2), and the vertices with "intertwining operators" between the representations.


As a warning, this is very much speculation.


So I was trying to picture the next level of a TQFT in this way. When we're up a level 4Cob consists of:

0-cells that are 2-dimensional manifolds.
1-cells that are 3-dimensional cobordisms between 0-cells.
2-cells that are 4-dimensional cobordism-like things between the 1-cells

So, I imagined that maybe the 0-cells are our primitive area elements of space -- they are what the edges represent in a spin network. The 1-cells are our primitive volume elements of space -- they are what the nodes of a spin network represent.


And then on the algebraic side, some objects of 2Hilb look like categories of group representations! Let's say we limit ourself to these, so that our TQFT is a (2-)functor from 4Cob into the 2-category of categories group representations.

So, to each of our area elements, our TQFT assigns a "state space" that is none other than a category of group representations. In other words, our TQFT assigns a group to each area element. Then, to select a state, we merely pick a representation of our group.


Then, we have our volume elements which tell us how to get from one product area elements to another product of area elements. In the algebra world, this is a functor that tells us how to transport a group representation on the source side over to become a group representation on the target side... and then when we take the 2-Hilb "inner product", the result is the Hilbert space of intertwining operators.

So, to construct a state, we pick one of these intertwining operators.


That's as far as I got -- I couldn't picture how the 4-dimensional stuff works! But that's probably because I never got that far in the LQG picture either.

 

[Mike2]

wilson loops might have to become "wilson homotopies" that are paths between loops------and those might transmute into the faces of a spinfoam---or some such pocus hocus.

As I understand the situation, recently a gauge theory has be shown to be a representation of a string theoy (what paper is that). And also gauge theory is connected to knots as talked about in J.C. Baez' book "Gauge Fields, Knots and Gravity". So I have to wonder if string theory might be an alternative description of links and knots? Perhaps the link invariants might equate to the string states, etc.

 

[marcus]

I think you both probably realize where I am is page 19 of the "Quantum Spacetime" lecture notes
http://math.ucr.edu/home/baez/quantum_spacetime/qs.pdf

the April 7 2004 version of Baez 0404040 is here
http://math.ucr.edu/home/baez/quantum/
the May 31 perimeter colloquium page is here
http://math.ucr.edu/home/baez/quantum_spacetime/
the actual slides/lecturenotes for the colloquium talk are here
http://math.ucr.edu/home/baez/quantum_spacetime/qs.pdf

From what Hurkyl says, it sounds like he is also on that page. Maybe Mike as well. But just to be extra explict about where I am hung up and what I am looking at I will try to paste it here as well.
Nope, it doesn't copy. So I will have to play the scribe and trans-type it

===quote page 19 and 20===
In 3D quantum gravity---more generally in any extended topological quantum field theory---we have
a 2-category where:
* objects describe kinds of matter
* morphisms describe choices of space
* 2-morphisms describe choices of spacetime

In 3D quantum gravity this matter consists of point particles---see the work of Freidel et al...

In 4D topological gravity this matter consists of strings---see my papers with Crans, Wise, and Perez.

3. In higher gauge theory we have fields describing parallel transport not just for point particles moving
along paths...
but also for strings tracing out surfaces...
I've developed this in papers with Bartels, Crans, Lauda, Schreiber and Stevenson...
===end of sample===
[I am leaving out a lot of pictures as i transcribe]

 

[Mike2]

As I understand the situation, recently a gauge theory has be shown to be a representation of a string theoy (what paper is that). And also gauge theory is connected to knots as talked about in J.C. Baez' book "Gauge Fields, Knots and Gravity". So I have to wonder if string theory might be an alternative description of links and knots? Perhaps the link invariants might equate to the string states, etc.

Sorry, I forgot to finish my though,... damn TV anyway...

I forgot to mention that links and knots have been related to loop quantum gravity through gauge theories that have also been related to strings theory. So perhaps in this way LQG is related to ST through the knots of gauge theory. Perhaps the link invariants might impose another constraint on string theory to get rid of the landscape. Or perhaps I'm just dreaming.

 

[Hurkyl]

* objects describe kinds of matter

This is the part that really bothers me. Especially

In 3D quantum gravity this matter consists of point particles

On the topological side, our 2-morphisms represent the 3D space-times... our 1-morphisms represent 2D spatial slices... so our objects are supposed to be somehow 1-dimensional... so I'm completely confused by the claim that they're point particles!

Now, if they were strings, I'd be happy. Or if we built a 3-category whose objects represented our point particles, the 1- and 2- morphisms denote 1- and 2-dimensional space between stuff, and 3-morphisms were space-times.

But what they actually present seems plain wrong!

 

[marcus]

Hurkyl I think what he means about 3D gravity is that matter is point particles stretched out in time. In Freidel's papers esp with Livine he has matter be points in 2D space and so there is a worldline in 3D.
this worldline is a conical singularity with a deficit angle corresp to the mass of the particle. So in a sense Freidel 'removes' this worldline from the space and gets matter that way. It worked rather well and got a lot of people's attention

I think what he is saying is that-----well this may all be obvious to you and I may only kidding myself that i am responding---by saying that matter has to be RINGS in 3D and therefore TUBES in 4D and that it won't work anymore to merely do just what Freidel did, and remove worldlines and have a conical singularity.

So you get these strange tube and pants almost surgical anatomical pictures in the Baez Crans Wise paper.

For me I like the intuition in 3D about the conical singularity along the worldline when you loop around it and get a deficit and that becomrd the mass----that was very nice, and I am trying to find how that story continues analogously in 4D.

maybe you can 'go around' a pants diagram in 4D? maybe curvature 'lives' on 2D things there, instead of being concentrated at LINES the way it is in 3D. these D-2 things sometimes called the "hinges" or the "bones"-----in triangulations gravity they are where the curvature is concentrated. so far it eludes me but I don't think it will forever.

Oh yes, the membrane tension corresponding to mass. Maybe I am just having a bad brain day.

the other thing is that EVERYTHING IN SIGHT IS BEING MOTIVATED BY CATEGORIC THINKING, so we are proceeding on autopilot here. I may sound a bit nervous but I'm not scared at all.
Have to go up the hill for exercise, back later.

 

[Hurkyl]

Hurkyl I think what he means about 3D gravity is that matter is point particles stretched out in time

But that doesn't mesh with everything else --- we have morphisms between the particles... they are supposed to be space! But if our particles are stretched out in time, then the morphisms have to be space-time.

For me I like the intuition in 3D about the conical singularity along the worldline when you loop around it and get a deficit and that becomrd the mass----that was very nice, and I am trying to find how that story continues analogously in 4D.

But then you remind me of this, which solves everything.

Our "matter" is not point particles stretched out in time -- our "matter" is the loops around the conical singularity! Or, at least that's how we capture it in the category.



The 3D case was nice, because there is only one closed connected 1-manifold: the circle. So, all of our objects have to be built out of circles. And what lies inside of a circle? Why, a point particle, of course!

But in 4D, we have a whole zoo of fundamental shapes: spheres, projective planes, torii, klein bottles, 2-holed torii, et cetera. What might lie inside them?

Well, maybe spheres engulf point particles, and torii contain closed strings. Maybe, a 2-holed torus contains some silly string state that's in a figure-8 shape! But Klein bottles are nonorientable... they can't hold anything!

Oh wait, we're saved -- we're looking at the category of orientable manifolds and cobordisms... so our fundamental objects are simply the torii with n holes.


But let's go back to the 3D case. Okay, so we picture our point particles by drawing little loops around them. So... what is the meaning of our morphisms? Maybe we have stringy things connecting our point particles, and our morphisms are simply the tubes around them? Oh wait... if we do some mental gymnastics...

Imagine we have two circles sitting on our table, and a tube connecting them in the shape of an arch. I -think- that we can flatten it out, and "stretch" it so that our tube has become a flat surface (going all the way off to infinity). So, our cobordism really looks like the exterior of the two circles. So, maybe our cobordisms really can represent space! Or, maybe we have a really big loop (the "universe") on the outside, and we only have to stretch our cobordism out to that. (it's now a flattened pants diagram!)

Yes, when I flatten the picture out like that, it looks like other things I've seen, so I think it makes sense. And now it's pretty easy to see that our 2-morphisms really do give how things evolve over time!

 

[arivero]

The matrix "algebra" of all matrices over a field provides a very nice example of the category... and one that people actually use as a category (though they usually don't realize it!). That's a good example of something you usually don't picture with arrows.

A good one. The points are vector spaces on the field F, and a matrix NxM has F^N as targen and F^M as source, if acting on a "column" vector, and the inverse asignment if acting on a "row" vector, this double possibility exemplifying the concept of duality between categories.

 

[selfAdjoint]

Marcus and Hurkyl, have you really got into understanding of what 2-categories are and do? It seems to me that undestanding this behavior will enable you to envision the geometric/topological situation better. Just what exactly is a 2-morphism and what are its axioms (note use of zig zag axioms in physics, cited by Baez in the talk).

 

[selfAdjoint]

A good one. The points are vector spaces on the field F, and a matrix NxM has F^N as targen and F^M as source, if acting on a "column" vector, and the inverse asignment if acting on a "row" vector, this double possibility exemplifying the concept of duality between categories.

Well a practical "duality" in mathematics is the algebraic versus topological thinking. Algebras and their behavior are good for some developments and commutative diagrams are good for others. Note that the whole subject of this thread can be described as a riff on Feynman diagrams!

 

[marcus]

Marcus and Hurkyl, have you really got into understanding of what 2-categories are and do? ...

Hurkyl is the point man on this one. I was thinking this morning of posting simply "hurkyl I owe you one" or some such message, but decided to leave it unsaid

thanks arivero also!

so far AFAICS Baez has NOT given a very complete set of hints, or it is hard to see how they fit together. any attempt to guess how it fits involves risk of making mistakes. my feeling is that it is better for learning to take that risk and go ahead and guess. Probably Baez will eventually explain it better in some TWF. For now, thanks to H. for conjecturing how it might go.

 

[Hurkyl]

I have a good idea about what they are. The timing of this thread is interesting because I encountered "Grothendieck's dream" last week, and have been talking to an algebraic topologist friend, so I can understand how n-groupoids relate to homotopy n-types. I can't really go beyond n=2 yet... but then again the experts can't go all that much further, so I don't feel bad.


I sort of feel that, on the topological side, the 2-category is entirely transparent. I can picture loops, cobordisms of loops, and "cobordisms" of cobordisms. (And now I can picture them even better after marcus's #23) We have this clean algebraic structure, and realizing it's a 2-category just let's you do bookkeeping. I can't picture n-holed torii, cobordisms of n-holed torii, and cobordisms of cobordisms, but knowing that it's a 2-category doesn't help.

I had been making some headway into figuring out 2-Hilb, but I've forgotten all of it. Though I did mention earlier that I realized that there's a subcategory whose objects are simply groups... or more accurately, categories of unitary group representations.

So some eTQFTs are something like a choice of group to represent (exponentiated) momenta!




Thinking some more about the 3-D case, our 1-morphisms are cobordisms of collections circles... so they're 2-manifolds that have boundaries. We can always sew disks onto those circles to turn it into a compact, oriented 2-manifold. The compact, oriented 2-manifolds are, of course, the n-holed torii. (Including the sphere as a 0-holed torus)

And once we've done that, I think we can homeomorphically move those disks anywhere on our 2-manifold that we please... so, up to homeomorphism (!), a 1-morphism is nothing more than a choice of the global (spatial) topology for our universe.

If we always stick to the same genus, I can picture the 2-morphisms. If it's a sphere, I can puncture it then flatten it out to the plane... otherwise, I can take one of those plane models where we identify edges and stuff, but at the moment I'm having trouble imagining things where the number of holes change.

 

[arivero]

Well a practical "duality" in mathematics is the algebraic versus topological thinking.

This is not just reversing arrows, but also different structures, related then with "functors". A functor is covariant or contravariant depending of when it reverses arrows.

The main example here is of course topological spaces versus boolean algebras.

 

[Dcase]

Observations with image examples:

In some ways planetary orbital loops are like quantum loops as R is to 1/R.
http://www.grazian-archive.com/quantavolut...figs/sb_f14.jpg

The Earth rotates and displaces as the sun moves somewhat like a cannon ball fired through a rifled barrel.
http://modelingnts.la.asu.edu/html/UGC.html

David Hestenes discusses the importance of the helix in 'The Kinematic Origin of Complex Wave Functions'.
http://modelingnts.la.asu.edu/html/UGC.html

Thus in some ways planetary helical displacements are like quantum helical strings as R is to 1/R.

Speculative questions:

1 - Is it possible that various loop theories can be integrated into helical string theories?

2 - Conversely, can one find loop theories that are derivatives of string theories?

3 - Could loop and / helical string theories be applied to both GR gauges and the QM gauges?

4 - Did Smolin give up too soon in his attempt to relate quantum loops to strings (via the helix)?

5 - Is twistor string theory a subset of the Monstrous Moonshine proved by Borcherds?

6 - Should the string-D and time-D of the Monstrous Moonshine be complex as are the other 24-D due to the j-function (via e^ir2Pi)?

Note - #6 would accommodate helical strings as complex harmonic oscillators and complex time (consistent with Hawking imaginary time and perhaps 2T physics of Bars.

 

[marcus]

Hello Dcase,

I see you posted something about your helix ideas at Motl blog and mentioned this thread in passing.

Your ideas (about helixes) are distinctive, but, as I assume you realize, are not closely related to the topic of this thread.

In this thread I mainly wanted to talk about Baez paper
Quantum Quandaries; a Category-Theoretic Perspective
http://arxiv.org/quant-ph/0404040

I am afraid the thread title "stroop theory" is rather fanciful

However category theory does give a loose flexible framework where one can see analogies between, not only the cobordisms of Gen Rel and the Feynman diagrams of quantum physics, but also perhaps in a very general way between string worldsheets and spinfoams.

As you surely recognize, these constitute no more than ANALOGIES and not any firm mathematical connection

my advice would be:

1. make a separate thread for your helix theories, because they don't belong in this thread discussing quant-ph/0404040

2. read quant-ph/0404040 ("Quantum Quandaries") and see what you can make of it

3. discuss that John Baez paper with us in this thread, if you would care to. further comment "Quantum Quandaries" by you or anyone would be very welcome (as long as focused on that) because it would help to continue discussion of the paper

This is re the Doug post of 29 June 2006 8:52 PM at Motl blog on the "Barton Zwiebach letter" thread

 

[Dcase]

An alternative perspective may be helpful

Hi Marcus:

1 - Please tell me if this is acceptable for continuing discussion in this thread.

Baez does have 34 references for this paper. I definitely agree with his abstract statement that there is a QM / GR “analogy“! I can visualize planetary orbital loops as analogous to quantum loops and twistor [helical] strings as analogous to planetary orbital trajectories.

Although Penrose, Witten, Hestenes and Freedman emphasize the helix, I suspect that even they may underestimate the true significance of this entity.
http://library.ictp.it/FP-DB/docs/1993/Dirac-1993-Lecture-Freedman.pdf

I suspect that the loop and helix are harmonic oscillators capable of telling time and transmitting information in music, electricity, QM, probably GR and in nucleic acids as in information science. They are likely complex because of usually being correlated with a charge, often ionic. They are important in solar magnetic reconnection [GD Holman], geodynamics [GA Glatzmaier] and observed at the galactic core magnetic fields [M Morris].

My goal is for someone smarter than I to take up where Smolin left off and be able to unify loops with [helical] strings!.

2 - I was somewhat suspicious that your thread was meant to be ‘fanciful’, but this term ’stroop’ that you coined, expresses my interest in how QM may relate to GR and perhaps all intervening gauges. I truly suspect that loops [circles, ellipses and possibly hyperbolas] may have such a relationship to helical strings.

I think that Euler with his identity circle proved this for that specific case [circle].
Electrical engineers have used phasors for 25-30 years more than physicists have used Schrodinger’s wave equation - see Figure 3 of Complex representation of Fourier series – e^jwt plotted in three dimensions is a helix.
http://www.complextoreal.com/tfft2.htm

This can be composed from 2D representations:
Let circle [o] +{or x?] sinusoid [~] => helix [o~] when interpreted as a 2D architectural diagram.
In Figure 18.1 of Zwiebach's "A First Course in String Theory", the left most sinusoidal curve is helix from my perspective may be a representation of a complex-D24, string-D, time-D like Borcherds Monstrous Moonshine. [I remain unclear why the string and time D are also not complex.]

Gabriel Kron, an electrical engineer for General Electric, had an interesting paper: 'Electric Circuit Models of the Schrödinger Equation', Phys. Rev. 67, 39-43 (1945)
http://www.quantum-chemistry-history.com/K...ronGabriel1.htm

I am uncertain if this is true for Riemannian or Gaussian curvature.

I have a great deal of respect for the astronomer Fed Hoyle who coined the term ‘Big-bang’ in a somewhat ‘fanciful’ manner. This term became significant in the literature.
I am hoping that ‘stroop’ may similarly be applied one day since I suspect that loops are related to helical strings.

3 - I quite agree that I am being more analogous [remember the Baez abstract] than rigorous, but must disagree that that there is no firm mathematical connection. I have been trying to make this clear by using words such as speculate rather than conjecture or from my perspective. As much as I respect Hilbert, even his 10th problem was disproved.

I have been absent from pure mathematics for a long time.
I do commonly use biometrics.
Most often I attempt to integrate from this polymorphisms set [history symptoms, physical signs, laboratory data] with the goal to establish a differential diagnosis and formulate a treatment plan.
This process is somewhat analogous to QM, but the probabilities in medicine are very ill-defined for decision analysis. There are attempts underway to improve this process. But the current state of affairs makes for a more analogous than truly rigorous process.
[The Mayo School of Graduate Medical Education (MSGME) HSR 5850 Medical Decision Making.]
http://www.mayo.edu/msgme/crtp-curriculum.html

I have been trying to become more rigorous by attempting to update my mathematical abilities utilizing the internet and various texts. I may be misinterpreting some of what I read.
The web:
a - David Hestenes has an excellent site on geometric calculus and algebras. I particularly found this useful in attempting to understand Grassmann, Clifford and Lie Algebras and the kinematics of complex wave functions.
http://modelingnts.la.asu.edu/
b - John Baez has an excellent site on mathematics ‘This Week's Finds in Mathematical Physics‘. I am particularly interested in weeks 233 and 234, but there is some nugget I think I can learn from nearly each week - such as the addendum to week 73 on biological chiralty.
http://math.ucr.edu/home/baez/TWF.html
c - MathWorld is a great reference site.
http://mathworld.wolfram.com/
d - I have used many other sites but not as frequently as these three.
Texts:
I am reading or have read some texts of the Scientific American Book Club by Nahin, Moar, Livio, Slatner and Seife.
Recently I completed ‘The Limits of Mathematics’ by GJ Chaitin with his interesting ideas on the need for experimental mathematics and his work on incompleteness and definable but not computable probability.

Carlo Rovelli sparked my helical interest in ‘Loop Quantum Gravity’.
http://relativity.livingreviews.org/Articles/lrr-1998-1/index.html

Specifically consider the section 6.10 ‘Unfreezing the frozen time formalism: the covariant form of loop quantum gravity’. On this page - Figure 3: The elementary vertex - reminds me of vertex algebra - while Figure 4: A term of second order - reminds me of a twisted cylinder.
http://relativity.livingreviews.org/Articles/lrr-1998-1/index.html

Once there is a cylinder, consider Generalized Helix: “The geodesics on a general cylinder generated by lines parallel to a line l with which the tangent makes a constant angle.” Squirrels use such a geodesic when climbing the trunk of a tree. Ballistics and celestial mechanics also appear to use a helix in their trajectories.
http://mathworld.wolfram.com/GeneralizedHelix.html

4 - I should add that Lubos did think my perspective a “joke”, but was kind enough to provide two arXiv papers from 2001 to demonstrate that others have had a similar perception.
a - ‘D(NA)-Branes’ by Simeon Hellerman, John McGreevy, Stanford
http://arxiv.org/abs/hep-th/0104010
b - ‘Super D-Helix’ by : Jin-Ho Cho, Phillial Oh, SU-Korea
http://arxiv.org/abs/hep-th/0105095

Awaiting your critique.

 

[marcus]

Hi Marcus:

1 - Please tell me if this is acceptable for continuing discussion in this thread.

Thanks for asking. No it is not acceptable AFAICS. Everything you have posted in this thread appears quite off topic.

If you will kindly read the first 30 posts on the thread, you will see that we have been discussing some category theory ideas that Baez introduced us to.

the most concise outline of the topic is in the lecture
Higher-Dimensional Algebra: a Langauge for Quantum Spacetime
I guess I could abbreviate it as HDA:QS

an easy partial introduction was given in quant-ph/0404040

Awaiting your critique.

Your ideas, which you lay out in your posts, may have some interest, but I CANNOT COMMENT ON THEM IN THIS CONTEXT because it would just crowd this thread with off-topic stuff. If you want people to critique your ideas, why not start a thread?

 

[marcus]

Unless there is some objection, I'd like to continue with the topics we were discussing in the first 30 posts of the thread which are the categorical ideas in
HDA:QS (with a partial and gentle introduction in quant-ph/0404040).
http://math.ucr.edu/home/baez/quantum_spacetime/

Here is a thumbnail summary of what HDA:QS is talking about.

there are 23 slides, with an important break between 14 and 15, which is where he starts talking about TWOCATEGORIES.

Slides 1-14 are about MONOIDAL (one)CATEGORIES with DUALS for objects.

this is the type of category I was earlier calling "tensor star" as a kind of nickname----in the spirit of the introductory essay 0404040 "Quantum Quandaries".

It seems too imposing to call these things "monoidal categories with duals for objects."

The point that John Baez makes in slides 1-14 is that
both Hilb and nCob are this kind of category, and Set isn't, and this is a HINT

And then in slides 15-23 he follows out what he thinks are the consequences of taking this hint seriously----and he talks about TWOCATEGORIES.

Notice that this is a persuasive and suggestive (not strictly logical) development of ideas. he is taking us into his intuition. So there is an intuitive step at slide #15.

To me, that means we should try to deeply assimilate the intuitive content of what he says in 1-14-----what it suggests to him that Hilb and nCob are both a special kind of category unlike the category of Sets (which is conventionally the basis of mathematics).

So I will devote a post to that.