Sunday, September 06, 2009

My defence statement

Following von Neumann known from his factors of type I, II, and III, one could classify mathematicians to those of type I and type II. I hope that mathematicians do not feel insulted by this kind of classifications;-).

Mathematicians of type I believe in the uniqueness of those mathematical structures, which are really God given. Classical number fields and groups would be the basic example about God-given-ness. Also physics would be fixed uniquely by the condition of the mathematical existence of the theory and by its maximal richness.

Mathematicians of type II prefer genericity. Everything that can be imagined is possible. In physics this would mean that space-time dimension four just happens to be 4 in this particular corner of the multiverse and symmetries of physics - if real rather than figments of our imagination - just happen to be just those of standard model in this little pocket of multiverse.

String theorist began as mathematicians of type I and were fascinated by the idea that they had the theory of everything allowing to calculate proton mass to arbitrary number digits within decade or two. As they suffered a phase transition to M-theorists they woke up as mathematicians of type II and realized how wonderful it after all is that M-theory predicts nothing. Lubos is one of few exceptions. Personally I am also still of type I and I can only represent excuses for my conservatism. I hope that jury bothers to listen for humanitarian reasons if not for anything else.

  1. The Kähler geometry of the worlds of classical worlds is unique already for loop spaces. In 3-D context there are excellent hopes that even the imbedding and therefore symmetries of physics can be fixed from the existence condition. The basic outcome is that infinite-dimensional symmetries are not figments of our imagination: they are absolutely essential for the infinite-dimensional geometric existence. Extended super-conformal invariance indeed fixes space-time dimension correctly and implies M4×S decomposition for imbedding space.

  2. The isometries of this particular infinite-D space which has the special property that it exists and also of corresponding finite-dimensional imbedding space must have very special meaning. Number theoretic one is the natural guess. Classical number fields and their complexifixations are natural here.

  3. The choice of S remains to be fixed from number theoretic considerations. The existence of M8-M4×CP2 duality ("number theoretical compactification") with M8 is interpreted as hyper-octonionic plane of complexified octonions with Minkowskian metric, means the possibility to map space-time surfaces identified as hyper-quaternionic surfaces of M8 to M4xCP2. This map preservers induced metric and Kahler form. Standard model symmetries are the outcome.

M8-M4× CP2 duality implies also the core element of gauge invariance and stringy picture.

  1. Only if the hyper-quaternionic plane assignable to a point of space-time surface contains preferred M2 (hypercomplex plane) of M8 having interpretation as plane of non-physical polarizations, it can be labeled by a point of CP2 so that the map of X4 in M8 to X4 in M4×CP2 exists naturally and maps M4 point to M4 point and hyper-quaternionic plane to CP2 point.This is the physical essence of gauge conditions. It seems to be possible to assume that the choices of the preferred polarization plane is local.

  2. The duality also implies decomposition of space-time surface to string world sheets parameterized by partonic 2-surface and in finite measurement resolution implying the replacement of partonic 2-surface by a discrete set of points, the replacement of space-time surface with string world sheets.

Therefore the existence of the geometry of the world of classical worlds plus classical number theory would imply both gauge invariance and stringy description in finite measurement resolution besides M4×CP2 and 4-dimensionality of space-time.

That was my defence. Jury can decide;-).


At 7:48 AM, Anonymous Mathematician type II said...

As long as you keep your universalist orthodox dogmas to yourself and don't force your belief systems upon the rest of Being and Creation, you're OK, pal. :)


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