Quantum Gravity and the Standard Model (Sundance + PI)

~~Mike2~~ · Mar16-06, 08:02 PM

Originally Posted by Kea

Dear me, f-h, this is just what category theoretic M-theory is all about. Yes, I mean String theory. I'm afraid a lot of people would consider that very off-topic.

Would you recommend I start off with :

Elementary Categories, Elementary Toposes (Oxford Logic Guides) (Paperback)
by Colin McLarty

at:
http://www.amazon.com/gp/product/019...lance&n=283155

Or do you know an easier introduction? Thanks.

**Kea** · Mar16-06, 08:24 PM

Originally Posted by Mike2

Elementary Categories, Elementary Toposes (Oxford Logic Guides) (Paperback)
by Colin McLarty

Excellent choice. More a logician's viewpoint than a category theorist's, but probably the best book out there for working through.

**Kea** · Mar16-06, 09:37 PM

Originally Posted by selfAdjoint

Very interesting! The modular group, the Pythagorean spinors, and the connection to the Lorentz group! As he says, there has got to be something neat in there somehow.

John also mentioned the trefoil knot. This has a nice simple Jones polynomial, namely

$LaTeX Code: J(t) = t + t^{3} - t^{4}$

Now it so happens that a very nice HEP guy here mentioned today the logarithmic relation between $LaTeX Code: m_{e}, m_{\\mu}, m_{\\tau}$ which I will write in the form

$LaTeX Code: \\sqrt{m_{e}} = r e^{\\pi a} \\hspace{1cm} \\sqrt{m_{\\mu}} = r e^{2 \\pi a} \\hspace{1cm} \\sqrt{m_{\\tau}} = r e^{3 \\pi a}$

This means we can plug it into the Koide formula

$LaTeX Code: m_{e} + m_{\\mu} + m_{\\tau} = 4(\\sqrt{m_{e}} \\sqrt{m_{\\mu}} + \\sqrt{m_{e}} \\sqrt{m_{\\tau}} + \\sqrt{m_{\\mu}} \\sqrt{m_{\\tau}})$

to get

$LaTeX Code: 1 + e^{2 \\pi a} + e^{4 \\pi a} = 4(e^{2 \\pi a} + e^{3 \\pi a} + e^{\\pi a})$

or rather,

$LaTeX Code: 1 = 4e^{\\pi a} + 3e^{2 \\pi a} + 4e^{3 \\pi a} - e^{4 \\pi a} $

which for $LaTeX Code: t = e^{\\pi a}$ reads as the very simple

$LaTeX Code: r(1 - J(t)) = 3(\\sqrt{m_{e}} + \\sqrt{m_{\\mu}} + \\sqrt{m_{\\tau}})$

Hmmm....

~~Mike2~~ · Mar16-06, 09:37 PM

Originally Posted by Mike2

Does catagory theory and topos theory try to define sets independent of any underlying point set topology? Is this why it is useful in background independent efforts? Is topos theory the underlying mathematics of Algebraic QFT which define an algebra of operators (and not states) in order to get away from the background dependence of states?

Originally Posted by Kea

Hi Mike

The answer to all your questions is yes.

I've noticed some similarities amoung various efforts that I'd like to consider. This effort by Sundance, et al, looks a lot like Torstens effort to connect geometry to the operator algebra of the SM. They state that it is common to use knot theory to develop an operator algebra used in the SM. I wonder if these efforts are connected? Are the ribbons in the work at hand just knot theory in disguise? Perhaps it is the same as knot theory with an added dimension. (Sorry, I've only read the abstract)

Also, the web of graphs in LQG also looks like piecewise linear knot theory. Perhaps just a subset of all the links and nodes can be interpreted as intertwining knots use to develope the operator algebra of the SM. Has anyone considered that?

And if we intertwine the loops in string theory, perhaps that also is knot theory in disguise.

And perhaps a subset of the lattice of CDT might be interpreted as piecewise linear knot theory and develop the operator algebra of the SM from that.

As you can see. I've not achieved a synthesis yet. I'd like your opinion as to how likely it is that this algebraic QFT developed with the use of knot theory underlies all these different efforts. Thanks.

**Kea** · Mar16-06, 09:59 PM

Originally Posted by Mike2

...how likely it is that this algebraic QFT developed with the use of knot theory underlies all these different efforts.

Firstly, the name algebraic QFT is the subject of Schroer et al (talked about recently here and on NotEvenWrong). Although they do admire knots and CFT and such things, there is still a vast gulf between this way of thinking and the way of thinking of which I am thinking. It really is a matter of there being an awful lot of things that need sorting out before different approaches can be linked (excuse the pun).

But this is all OT. Sorry, Mike.

**Kea** · Mar16-06, 11:08 PM

Originally Posted by Kea

$LaTeX Code: r(1 - J(t)) = 3(\\sqrt{m_{e}} + \\sqrt{m_{\\mu}} + \\sqrt{m_{\\tau}})$

The trefoil knot is the simplest example of a torus knot. These are created by winding a string

times about one axis and

times about the other. Let

denote the Jones polynomial for a torus knot. The trefoil is the

knot. Now, considering

, all torus knots are naturally normalised to the value of

. So it seems that this normalisation is somehow associated with a choice of mass scale.

Hmmm...

**Kea** · Mar17-06, 12:03 AM

Goodness, this is so distracting. I must go for a walk. It feels like being trapped inside an artist's impression of a Bohr atom, like one of those nauseating joy rides.

**arivero** · Mar17-06, 11:15 AM

Originally Posted by Kea

This means we can plug it into the Koide formula

Hmmm....

How happens a thread on Sundance gets Koide inserted along

?

**marcus** · Mar17-06, 11:31 AM

Originally Posted by arivero

How happens a thread on Sundance gets Koide inserted along ?

Is this off topic? You know far more than I do, so I ask the question sincerely. To my limited understanding, it looks actually ON topic in a sense-----because of braids -> knots (like the trefoil) -> Jones polynomial -> pretty algebra like Koide

the Kiwi wild parrot called a kea is a notorious mischief-making bird members of this species have been known to drop a hiker's boot off a cliff---when he left his boots outside his tent while he was taking a nap.
they are strong intelligent and destructive birds----problematical during the day much as raccoons are problems during the night, if you know raccoons

part of Kea's persona is this new zealand kea bird. and she in fact DOES sometimes drag thread irretreivably off topic and sometimes will even drop them off a cliff. As with the wild parrot, this happens inevitably and although it may cause dismay, is not a proper subject for complaint.

but IN THIS CASE it seemed to me that the chain of mental associations was exceptionally elegant and fascinating

or were you giving a back-handed compliment? Not seriously objecting?
Sometimes I can't tell.

**arivero** · Mar17-06, 11:40 AM

I'd not say it is off topic, indeed I would prefer most of the rest of physics discussions online to be considered offtopic! I was amazed because I was looking Sundance restricted to one generation of particles and wondering about how the triplication was to be got.

I was also amazed about relating the formula to knots. Amateurs should notice that we physicists are educated against knotting during the introductory years, because of the bad legend of a previous attempt (XIXth century!) to explain the periodic system via knot classification. Still, Jones polinomial is always a subject of advanced seminars from time to time.

Ah, I missed the remark on logarithmic relationships between masses. It is something I have not classified yet?

**Kea** · Mar17-06, 04:57 PM

Originally Posted by marcus

As with the wild parrot, this happens inevitably and although it may cause dismay, is not a proper subject for complaint.

Hee, hee.

**Kea** · Mar17-06, 09:08 PM

Of course, in the real world the mass ratios are not equal. But, arivero, you have discussed quantum-group like mass logarithms yourself over on the thread

http://www.physicsforums.com/showthr...t=46055&page=8

Remember CarlB's appearance with the mass matrix? To quote Carl:

I should mention what all this has to do with Higgs-free lepton masses.

Consider the Feynman diagrams (in the momentum representation) where each vertex has only two propagators, a massless electron propagator coming in, and a massless electron propagator coming out, and a vertex value of $LaTeX Code: m_{e}$ is being generated. When you add up this set of diagrams, the result is just the usual propagator for the electron with mass. Feynman's comment on this, (a footnote in his book, "QED: The strange theory of matter and light"), is that "nobody knows what this means". Well the reason that no one knows what it means is because these vertices can't be derived from a Lorentz symmetric Lagrangian.

But what the above comment does show is that it is possible to remove the Higgs from the standard model (along with all those parameters that go with it), if you are willing to assume Feynman diagrams that don't come from energy conservation principles.

~~Mike2~~ · Mar17-06, 09:23 PM

Originally Posted by Kea

Firstly, the name algebraic QFT is the subject of Schroer et al (talked about recently here and on NotEvenWrong). Although they do admire knots and CFT and such things, there is still a vast gulf between this way of thinking and the way of thinking of which I am thinking. It really is a matter of there being an awful lot of things that need sorting out before different approaches can be linked (excuse the pun).

But this is all OT. Sorry, Mike.

As I understand it, you think more in terms of catagory and topos theory. I wonder if there is a connection between this and knot theory in that these knots are defined on 3D support manifolds whose union and intersection relate to operator algebra with the use of knot theory.

**Kea** · Mar17-06, 09:53 PM

Originally Posted by Mike2

As I understand it, you think more in terms of category and topos theory. I wonder if there is a connection between this and knot theory in that these knots are defined on 3D support manifolds whose union and intersection relate to operator algebra with the use of knot theory.

Yes, Mike. This is well known.

~~Mike2~~ · Mar18-06, 09:16 AM

Originally Posted by Kea

Originally Posted by Mike2

As I understand it, you think more in terms of catagory and topos theory. I wonder if there is a connection between this and knot theory in that these knots are defined on 3D support manifolds whose union and intersection relate to operator algebra with the use of knot theory.

Yes, Mike. This is well known.

I'm guessing here, but it sounds like the intertwining of these "knots" is not a differential feature, but is a description of something global. So it sounds like they are defining the union and intersection of sets in terms of something global and not in terms of elements of the underlying sets. This sounds like background independent set theory, independent of the background of underlying elements. Does this sound correct? If this backgound independent set theory is the basis of catagory/topos theory, then how can you deny the relevance of knot theory to all these different efforts when you do acknowledge the use of catagory theory for the same goal? Thanks.

**arivero** · Mar19-06, 01:32 PM

Originally Posted by Kea

Of course, in the real world the mass ratios are not equal. But, arivero, you have discussed quantum-group like mass logarithms yourself over on the thread
http://www.physicsforums.com/showthr...t=46055&page=8

Yeah I have (and you

). It was only the "today" remark that drove me to think I was missing something. BTW, in http://www.physicsforums.com/showpos...&postcount=171 I remarked that the only published note I am aware about mass logarithms is

Andreas Blumhofer, Marcus Hutter Nucl.Phys. B484 (1997) 80-96 http://arxiv.org/abs/hep-ph/9605393

I got to locate to M Hutter but not Blumhofer. Both of them have left physics; it seems that the system generates a very high rejection (abandon) rate even within people following the standard study path, such a thing enerves me.