Dirac's Hidden Geometries

I find this interesting because I like to visualizze as much as possible, and I sometimes think the basis of the leading ideas in science would had to follow a progression. Klein's Ordering of Geometries was one such road that seem to make sense. The basis of relativity lead through in geometrical principals?

Such an issue with string theory had to have such a basis with it as well, although how do you assign any views to the very begininngs of the universe below planck length? Well there are images to contend with what are these and how are they derived? Rotations held in context of te progression of this universe and all thoughts held to the very nature of particle creation and degrees fo freedom?

[PAUL DIRAC]

When one is doing mathematical work, there are essentially two different ways of
thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way.

While I am very far from being the mathematician, I understand that this basis is very important. Such summations in mathmatical design, leave a flavour, for conceptiual ideas to form in images, so I understand this as well. It is a progression of sorts I think, as I read, and learn. Geometry lies at the very basis of all such progressions in science?

So Feynmans toy models arose from the ideas of Dirac?

Professor Shiing-Shen Chern

HOUSTON JOURNAL OF MATHEMATICS
Editorial, Electronic Edition Vol. 28, No. 2, 2002

Shiing-Shen Chern -- famed mathematicianCarrie Sturrock, Chronicle Staff Writer
Thursday, December 9, 2004

He was a great mathematician partly because of the quality of his research as well as his ability to convey it to other people, UC Berkeley mathematics Professor Robin Hartshorne said. Professor Chern turned the once- dormant field of differential geometry, which deals with the mathematical description of geometric figures, into a lively field of study. He had the greatest impact on global differential geometry and complex algebraic geometry, which are fundamental to many areas of mathematics and theoretical physics.

Big Ideas.....To String Theory

Plato said:

yes to Gauss and gaussian coordinates, not forgetting, Saccheri, Bolyai and Lobschevasky along this lineage of geometers

On the Hypotheses which lie at the Bases of Geometry

So A continuation from this, and reference to important papers for consideration.

I was actually looking for papers on S.S.Chern and I have been having difficulty tracking down one of his papers entitled,"Relativity and Post Reimannian Differential Geometry," published in 1980. As I look, I usually come across interesting sites for consideration. They do indeed lead from one spot to another willy-nilly.

So I thought I would show the transition to topics that I compiled for reference in relation to string theory.

So having gone through a list here as follows, I came upon the article from a site called "Big Ideas". It was nice then, that I link to the site in question and the article for consideration. Talk about getting off the beaten path.:)

My intentions was to see how Gauss's and S.S. Chern's work correlated together and developed in line with Reimann. Hence the paper in question I was looking for. If anything had change my perspective, Gauss and Reimann were instrumental here and the understading of the metric. Gaussian coordinates help united much for me into the picture General Relativity had taken me too in see the dynamcial nature of the graviational field.

Big Ideas



It is of course from 2003, but always interesting nonetheless.




I understand Clifford's hesistancy on articles that have come out and some trepidation also seen by P.P. Cook on the issue Horizon of Hawkings in his article here. My focused is well set to this horizon as well, as th e question of blackhole types etc, and how such theoretical positions arise fromthis horizon. This was important to me that I move to the understanding of conformal ideas from tha horizon.

But articles, as best they can, hopefully can bring the lay person up to speed on what these ladies and gentlemen are doing with string theory and such. They help me in the generalized direction, so I hope all things are not to lost for Clifford and Paul in their entrancement of observation. "Disgust" to something fine in the media of consideration.

I noticed Paul's link to Jan Troost's site and have seen that site develope from inception, so it was interesting to continue to see the summation of string theory on his site as well. There are really good sites out there that I have kept track of, to help orientate my thinking in regards to to theoretical thinking at it's finest.

The Total Field

For me this title above strikes a cord somehow in the struggle and regard, leading in our comprehensions to the extension of the standard model. By bringing gravity into the picture and descibing the graviton teaming in the bulk of expression.

The general theory of relativity is as yet incomplete insofar as it has been able to apply the general principle of relativity satisfactorily only to grvaitational fields, but not to the total field. We do not yet know with certainty by what mathematical mechanism the total field in space is to be described and what the general invariant laws are to which this total field is subject. One thing, however, seems certain: namely, that the general principal of relativity will prove a necessary and effective tool for the solution of the problem for the toal field.

Out of My Later Years, Pg 48, Albert Einstein

Well now the reason why this paragraph strikes such a chord with me, has everything to do with the information that I have progressed through, in order to reach this vision Lisa Randall does not think one asa layman is capable of? Now I should be fair here, and I am not judging personalities, but the essence of the statements about "observation" and "vision".

Lisa Randall:

Most people think of "seeing" and "observing" directly with their senses. But for physicists, these words refer to much more indirect measurements involving a train of theoretical logic by which we can interpret what is "seen."

Now in my quest for comprehension, such building has gone on in my conceptual foundations, are ones that we are carefully lead through in theoretical developement. Ah so we see where such extensions have gone beyond th elayan's view then? To have such things of expression, in the computer world, as numerical relativity, is a nice way in which to round out the data and experience. But as she points out, we are talking about Physicists.

Lisa Randall:

Remarkably, we can potentially "see" or "observe" evidence of extra dimensions.

Those Russion Dolls

Well now. I have this strange picture in my head about "time variable images" we seen of the earth in measure, and such a statement above, by Einstein. It is information on the "total field" that struck immediately in my mind about all those things that lead one through to the comprehension of general relativity. It is indeed, about "gravity" and it indeed seen in the larger aspects of the cosmological scale. But then, how would such a thing take us down to scale in our look at quantum mechanical views. Other components of earth that efect time avraiableness and we are indeed driving this image of scale down to the component parts of our earth?



So I have this picture of earth here. I know its not so pretty, but it describes in greater context the world as you have not seen it before. This advancement in observation, is much more inherent in our culture now, that the grade with which we assign physicists and the lay persons, are really never that far apart. What was accomplished, was that leading infomration and theorectical developement paved the way for an "illustrous view" as to those I impart now. They were already there but never seen in context of each other and as a total field.



So now as I think about Lisa's words, I recognize more deeply the sigificance of how far our vision has been taken, not just in terms of the physicists view, but of how far we had been taken in layman terms as well. What then else retains this view about the total field that I had not show and in it writing, other images come to mind as follows.

So developing this sense in terms of relativity and views of Einstein in regards to the total filed had consequences in my mind about how we view things in new ways.

If conceived as a series of ever-wider experiential contexts, nested one within the other like a set of Chinese boxes, consciousness can be thought of as wrapping back around on itself in such a way that the outermost 'context' is indistinguishable from the innermost 'content' - a structure for which we coined the term 'liminocentric'.

Now it has to be understood, that the total field is one which has inclusiveness such as these boxes indicate, that such views of our blue marble earth, do not consider as we lay "one" over the top of another. Such extensions to our views of earth, lead me to understand the complexity of these views in ways that we had not considered before, and with such a synoptical view, what indeed shall this total field say about earth? So that's where I am at. Much like, Glast, in it's own synoptical view about the range of our vision.

So we have this frame of reference now to consider. Our apprehensions about earth(some who share the climatic valuation) that we can now say, that Inverse square law contains information in relation to "these boxes". That if taken to "new heights" our climatic valuations about this new view of earth, how shall we judge now, that such Kaluza Klein modes held in relation to the expanding nature of this point(circle) can have energy valuations assigned right from the supersymmetrical vision ofa beginning, to have phases (symmetry breaking)with which our views have been generated, in what we see of earth now?



While indeed then, "light had been joined to gravity" how shall we wrap again the views of this earth, in what is now a teaming in this new place, where differences exist in our views. Strengths and weaknesses, are measures in this new abstracted view?

So we have this total view in mind, about the "total field" and I have taken us to a a abstracted space within the idealization of what exists here now as earth arose from some beginning point. To what the earth encapsulates.

How we view then such comsological events has a greater story as we look deep into space, and see the valution of those same cosmological events streaming past all things in existance, that such a gravitational view has arrows pointing in a certain direction. To ideas about comsological expansion and such. This has gone to far I think about our place in this new abstracted view of the universe:)

Point--> Line-->Plane <---> Point<-- String<-- Brane

Under the heading of Klein`s Ordering of the Geometries :

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

Now the usual thinking here has been placed under intense thinking by the introduction of a new way in which to look at "geometry" that has gone through a "revision" in thinking.

New trigonometry is a sign of the times

Lubos Motl introduces this topic and link in his blog entry and from this this has caused great consternation in how I am seeing. I see Lisa Randall might counter this in terms of what the brain is capable of, in line with this revisionary seeing, and comparative examples of this geometry Lubos links.

Dangling Particles,By LISA RANDALL
Published: September 18, 2005 New York Yimes

Lisa Randall:

Most people think of "seeing" and "observing" directly with their senses. But for physicists, these words refer to much more indirect measurements involving a train of theoretical logic by which we can interpret what is "seen." I do theoretical research on string theory and particle physics and try to focus on aspects of those theories we might experimentally test. My most recent research is about extra dimensions of space. Remarkably, we can potentially "see" or "observe" evidence of extra dimensions. But we won't reach out and touch those dimensions with our fingertips or see them with our eyes. The evidence will consist of heavy particles known as Kaluza-Klein modes that travel in extra-dimensional space. If our theories correctly describe the world, there will be a precise enough link between such particles (which will be experimentally observed) and extra dimensions to establish the existence of extra dimensions.

But first before I get to the essence of the title of my blog entry, I like to prep the mind for what is seemingly a consistent move towards geometry that has it's basis in applicabilty to physics, and move through GR to a vast new comprehsnsion in non-euclidean geometries. Must we now move backwards that we had gained in insight, or was it recognition of the "length scales" that we now say, how could such a dynamcial view ever be assigned to the eucildean discription under the guise of brane world recognitions?

Moving Backwards?

What exactly do I mean here?

Well the idea is that if you move to fifth dimensional views, and there are ways to wrap this within our "Brains":) We then see the dynamcial nature of our neurons have found acceptable ways in which to see this brane feature. As well as, approaches in use of new processes in geometerical considerations as those linked by Lubos.

Dealing with 5D world



Thomas Banchoff is instrumental here is showing us that fifth dimensional views can be utilized in our computer screens, and such comparisons, reduce to a two dimensional frame, makes it very easy to accept this new way in which to attack the dynamcial nature of reality.

How indeed now could our computer screen act a liason with the reality of our world, when see from screen imagery effects, that all the rules of order have been safely applied for inspection and consistancy in physics approaches.

The Earth in Gamma Rays

This is a false-color image of the Earth in three gamma-ray energy bands, analogous to the colors red (lower energy), green (mid energy) and blue (higher energy) in the visible spectrum. The lower right shows a composit image.(Image credit: NASA/CGRO/EGRET/ Dirk Petry)

But Petry created this gamma-ray Earth image to better understand the impact of "local" cosmic-ray and gamma-ray interactions on an upcoming NASA mission called GLAST, the Gamma-ray Large Area Space Telescope. GLAST is planned for launch in 2007. Its main instrument, the Large Area Telescope, is essentially EGRET's successor.

What's in a House

Plato:

The ‘meeting hall” is a interesting version of native design

Alejandro Rivero:

(Plato, could you ellaborate, here or in your weblog, about Netherlands “meeting hall” concept?

I was thinking about the educational community in regards to the building, and how such communities are really like these families, and the "long houses" that provided for such large native families to live side by side.

The program shows how Native Americans used materials available to create a comfortable and safe living environment. It also looks at the sophisticated social environments in which such things as the Iroquois Longhouse and the West Coast Cedar Houses, both multiple family dwellings, operated. A look at art forms, both fine and utilitarian, in relationship to the architecture is also included in the program.

Using natural logs inside house of educational center, brings a nice natural feel and environment for such lectures. That's all.

Time to be Grounded:) Not!

In Garden and Writing by Clifford of Cosmic Variance, this piece reminded me of the "realities" and the approaches of "circumstance". These are our lives?

Things that we do to help us, and remind us of the way to approach this streaming dialogue deeper held, to bring outward, to the pen felt drawn word or picture? Death's door, loosing our figures of time, books and words, that help change who we are?

Probabilistic valuation set to condensate or soliton approach mattered defined in conceptual logic frame?

A "poetry of emotive charge" for the wants of a better time( a fresher air), lost to the reality all around us to engines, planes trains and auotmobiles.:) A home forgotten? A earth fresher begotten?

IN a soft moment, Rivero reminds of the way things use to be, and in these memories things retrospect bring us closer to all things that make us who we are. No where are their maps to progress this character of ours, that we see this master plan, but relegated and caught in our microcosm classroom world held, only the memory of the world.

It is always much larger, when we physically revisit "ole domains":)

So yes to getting the hands dirty and feeling earth touched, that such grounding, makes time for new moments appear. Better words, from a ethered view of information that exists all around.

From a mystic drawl of a cowboy slang, to a refined English gentleman? It doesn't matter when you change the overcoat, because the seasons hold their own regard for how we shall address the circumstance.

Back to work:)

CFT and the Tomato Soup Can

As always, the layman trying to develope the mathematical views?:)

Greg Kuperberg on Sep 15th, 2005 at 12:11 pm

Conformal maps of the Earth are a great introduction to complex analysis. If you identify the Earth with the Riemann sphere, then the Mercator map is exp(i*z), while the quincuncial projection is a Weierstrass elliptic function. Or you could view it as a 2-to-1 conformal projection from a torus to a sphere with four ramified points. I imagine that it is relevant to one-loop calculations in string theory in that guise.

At what level has this map then progressed if we held such views to the "horizon and boundary conditions." That is now replaces what we talk about of earth, and now relay the mass consideration to events in the gravitational field? Has the mathematic hypothesized now, gone through a revision, and needed support of mathematical views?


Campbell's Soup Can A. Warhol

What mathematics would move our perception to the gravitational views seen there? Gary Horowitz relays the outside label of a can of a soup as the conformal surface, while the soup, the spacetime fabric?

On planet Earth, we tend to think of the gravitational effect as being the same no matter where we are on the planet. We certainly don't see variations anywhere near as dramatic as those between the Earth and the Moon. But the truth is, the Earth's topography is highly variable with mountains, valleys, plains, and deep ocean trenches. As a consequence of this variable topography, the density of Earth's surface varies. These fluctuations in density cause slight variations in the gravity field, which, remarkably, GRACE can detect from space.

So one would look at topography as something much different then what is laid out on this globe as "hills and valleys"?

So now this map, has this extra feature to it.

Holography encodes the information in a region of space onto a surface one dimension lower. It sees to be the property of gravity, as is shown by the fact that the area of th event horizon measures the number of internal states of a blackhole, holography would be a one-to-one correspondance between states in our four dimensional world and states in higher dimensions. From a positivist viewpoint, one cannot distinquish which discription is more fundamental.

Pg 198, The Universe in Nutshell, by Stephen Hawking

While on this topic it behooves me to think of the "horizon" and the mathematical construct that has taken us there. While we see to explain the nature of the effect in a fifth dimensional view, it had been reduced to "temperature" as a relation of this conformal view?

"D-branes provide the fundamental quantum microstates of a black hole that underlie black hole thermodynamics"

As much as one would try and ignore this position, you cannot get away from the mathematics or the approach and what this has culminated too.

I like Peter and his no nonsense views, but he has gone to far in rejecting the basis of "mathematical dialogue" in face of what D brane issue had been taken too?

Why would he reject mathematics on the one hand demonstrative of a particular point of view to which it has developed, then, ignore what position it had taken both string theory and Lee Smolins attempts at the disciption of the blackhole dynamics, from the views of that horizon?

With regards to the conformal field theory approach. While I am in my infancy, I recognize the views of Bekenstein Bound, and the hologrpahical approach. One must first learn to crawl, then walk I know, but how indeed does one get to the vision held, when he himself(who ever you like) cannot explain how such a mathematics like string theory, arose to help with our views of reality?

In 1919, Kaluza sent Albert Einstein a preprint --- later published in 1921 --- that considered the extension of general relativity to five dimensions. He assumed that the 5-dimensional field equations were simply the higher-dimensional version of the vacuum Einstein equation, and that all the metric components were independent of the fifth coordinate. The later assumption came to be known as the cylinder condition. This resulted in something remarkable: the fifteen higher-dimension field equations naturally broke into a set of ten formulae governing a tensor field representing gravity, four describing a vector field representing electromagnetism, and one wave equation for a scalar field. Furthermore, if the scalar field was constant, the vector field equations were just Maxwell's equations in vacuo, and the tensor field equations were the 4-dimensional Einstein field equations sourced by an EM field. In one fell swoop, Kaluza had written down a single covariant field theory in five dimensions that yielded the four dimensional theories of general relativity and electromagnetism. Naturally, Einstein was very interested in this preprint .

While one may use sites to give indicative values ot the information, can we ignore these assumptions mathematically driven. It paved the way for how we view things that we did not see before. Go ahead reject it then:)

Are we not looking for the Trigger?:)

Atlas and Proton-proton Collisions

Depth of Perception

I am responding to the link here on Cosmic Variance and the related article, Cosmic Violence. I do not want to tie up their space, so my "further response" is being given here.

I speak of Glast in the context of that "Window on the Universe" view. This helps to orientate our deeping recognition of those events, but does not include the realization of where high energy considerations are taking us as well.:)

What is happening at the beginning of our Universe? High energy implications and lower energy determinations reveal prospective views about that same universe? How is it such a view created by such particle collisions could not be drawn to a certain time in our universe?

By getting to the "high energy times", we are also getting to the circle (think the planck epoch to now) valuation of that early universe? There are always results of energy dissipaton of these early cosmological events, so it needed a consistant way in which to look at this?

The machine, dubbed ATLAS (A Toroidal LHC ApparatuS), is one of four facilities to be located at a powerful accelerator, the Large Hadron Collider (LHC), now under construction near Geneva, in Switzerland.

If we were to accept the circle and strong curvature as evident from our early universe considerations, (think of the circle and the planck epoch diagram as a blackhole?), then what happens when our views have been taken to suspersymmetrical points of view and the whole picture becomes locked within the model computation that Andrey Kravtsov does for us. The relaization is that this circle when taken down to planck has extremely strng gravitational considerations, and when and how do we reach this level of consideration on the time and birth of this universe?

IN Regards to Mathematical Constructs

Such an article presented by Peter Woit (How Much Mathematics Does A Theoretical Physicist Need To Know?), had me thinking in terms of what the quoted italicized statement below might mean in terms of the consistancy of mathematics developed?

http://www.math.columbia.edu/~woit/wordpress/?p=256#comment-4918(my comment below)Click on post and you now see the numbered posts alignment. What's the point?

Plato said:

If the Horizon exists as a mathematical construct, would we dissallow any mathematical counterpart that would lead from this, to incorporate other perspectives?

"D-branes provide the fundamental quantum microstates of a black hole that underlie black hole thermodynamics"

Developement of the mathematics would have been consistent then in how strng theory had developed?

So we know getting to the depth of perception necessary, had to include physics views here in order to develope the framework. High energy consideration could not have done it on it's own, so the topic is masked in theoretical definitions that we are not to accustomed too?:)

Yet it deals with a specific time frame in the developement of the early universe that is below Planck length. Below the "Planck epoch" (this holds a measureable time frame just after the beginning of the universe?)is the realization and "time valuation" that we assign this new perspective view, when we take physics in hand and abstract mathematics to it's fruitation?

While the link has been maintained to Peter Woits Blog, the post has not. It had been supplemented by Dickt's post.

This won't deter the documents and valuation of what string theory had to offer, and refused acknowledgement by Peter Woit to the progress, such developements might have taken string theory too?:)Tricky post like I wrote, acknowledges not only string theories position but Lee Smolins pursuate as well:)