Locality (non-locality) of Algebraic and Topological Objects
http://motls.blogspot.com/2010/09/algebra-and-geometry-secret-siblings.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+LuboMotlsReferenceFrame+%28Lubos+Motl%27s+reference+frame%29
http://matpitka.blogspot.com/2010/09/exact-yangian-symmetry-non-trivial.html
Lubos, a very good presentation perhaps leading up to these issues of geometry and algebra being the same thing (as you say coming closer). Yes, and trying to convey learning to all aspiring levels of people.
I give you the TGD site because it strikes me as emphasizing the geometry where your string approach emphasized the algebra of it all. I mean this in the sense of imaginary numbers, e i pi and all as the unification but the question of other numbers applying- and maybe twistor ideas rather than say Gaussian.
It seems to me that one essential question to answer is for example if these "extra dimensions" are at some point in general space why should they be necessarily discontinuous- and if part of a general and overlapping entangled continuity what is the mathematical generalization behind that. Perhaps we need this answer before we can say M theory for example does or does not lead to a theory of everything, or even the philosopher's God or not because this question of unification may be more fundamental than the topology or twists and knots of string theory. I find the opening as if a ladder or chain of our sense of numbers or worm hole like openings rather vague if we do not have a clear picture or formula with which to treat them.
ThePeSla
Hi,
ReplyDeleteI agree with Lubos that algebraic and geometric (or taken to extreme, topological) views are complementary and that both are necessary. I started from topological and geometric vision and number theoretic vision developed much later.
For instance, the leading idea in geometric approach that the geometry and spinor structure of "world of classical worlds" (WCW) is unique and therefore also physic described in terms of modes of WCW spinor fields. Infinite-dimensional isometry groups leave only imbedding space which are Cartesian products of flat or symmetric spaces and M^4 factor gives maximal conformal symmetries. There are still however many l options between which to choose.
What makes it unique reduces to the question what distinguishes M^4xCP_2 amongst other spaces of form M^4xS (and even more general imbedding spaces) besides the fact that standard model requires it. Here number theoretical vision is needed and classical number fields enter the picture.
Continuity-discontinuity question is very interesting. I see discretization as necessary correlate for finite measurement resolution and also for finite resolution of cognition. Algebraic extensions of rationals appear would the bridge between real physics and p-adic physics of cognition and algebraic physics describes the physics in the intersection between real and p-adic worlds, where life resides.
The discretization at space-time level appears as effective replacement of orbits of partonic 2-surfaces with braids and a connection with topological quantum field theories emerges from number theoretic constraint of p-adicization. What I call modified Dirac equation leads automatically with "periodic boundary conditions" leads to a localization of solutions to braids so that finite measurement resolution would be a feature of dynamics if one accepts that fermions make it possible to get information about geometry.
The symmetric space geometry for building blocks of WCW (all points of space metrically equivalent, generalization of cosmological principle) is also a prerequisite for the p-adicization of integration, one of the basic problems in constructing p-adic physics.
These diametric opposites of mathematics meet each other everywhere even in this infinitesimal glimpse to the richness of mathematics that the work with TGD have allowed me to gain.
Hi,
ReplyDeleteI agree with Lubos that algebraic and geometric (or taken to extreme, topological) views are complementary and that both are necessary. I started from topological and geometric vision and number theoretic vision developed much later.
For instance, the leading idea in geometric approach that the geometry and spinor structure of "world of classical worlds" (WCW) is unique and therefore also physic described in terms of modes of WCW spinor fields. Infinite-dimensional isometry groups leave only imbedding space which are Cartesian products of flat or symmetric spaces and M^4 factor gives maximal conformal symmetries. There are still however many l options between which to choose.
What makes it unique reduces to the question what distinguishes M^4xCP_2 amongst other spaces of form M^4xS (and even more general imbedding spaces) besides the fact that standard model requires it. Here number theoretical vision is needed and classical number fields enter the picture.
Continuity-discontinuity question is very interesting. I see discretization as necessary correlate for finite measurement resolution and also for finite resolution of cognition. Algebraic extensions of rationals appear would the bridge between real physics and p-adic physics of cognition and algebraic physics describes the physics in the intersection between real and p-adic worlds, where life resides. Continued in next posting....
I continue, with the continuity-discontinuity theme. The discretization at space-time level appears as effective replacement of orbits of partonic 2-surfaces with braids and a connection with topological quantum field theories emerges from number theoretic constraint of p-adicization. What I call modified Dirac equation leads automatically with "periodic boundary conditions" leads to a localization of solutions to braids so that finite measurement resolution would be a feature of dynamics if one accepts that fermions make it possible to get information about geometry.
ReplyDeleteThe symmetric space geometry for building blocks of WCW (all points of space metrically equivalent, generalization of cosmological principle) is also a prerequisite for the p-adicization of integration, one of the basic problems in constructing p-adic physics.
These diametric opposites of mathematics meet each other everywhere even in this infinitesimal glimpse to the richness of mathematics that the work with TGD have allowed me to gain.
Thank you for the reply I just saw. Let me think more on it. :-)
ReplyDeleteThePeSla
Wow, I see, and I agree it is an important principle and issue. I also think recent articles on the nature of dimension suggest your approach is right on.
ReplyDeleteThat is in newscientist for example where on the smallest level all things are one dimensional as even two dimensions vanish. But what of all things being points- all is singularities- that would be like a general cosmological principle would it not, metrical in the usual sense or perhaps deeper.
One thing for sure this strikes me not so much as to how we see dimensions as primary or not but how we have to relate hierarchies of the transfinite numbers as if dimensions.
In any case, I agree and think geometry is in a sense more fundamental and primary than our current physics.
Matti
ReplyDeleteI posted earlier but it did not get thru, so I hope I can restate the spirit of the post here:
I had to look up p-adic but I have similar notions in different terms and a few that seem to go beyond the concept.
Energy in the still not unified two physics is defined as action-frequency or mass-velocity of light- but there is a third way, the maximum diameter of the view of an expanding universe. In this sense I imagine the geometry is primary over what is regarded as the actual physical situation.
I think your ideas and sacrifice for them are right on and one day others will catch up to you for the case. I look forward to what more you may find if we are lucky in new enquiry.
The PeSla