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TGD: Physics as Infinite-Dimensional Geometry
Note: Newest contributions are at the top!
In the earlier blog posting I told that the boundary condition for the Kähler-Dirac equation is massless Dirac equation with the analog of Higgs term added. The interpretation in terms of Higgs mechanism however fails since the term can be also tachyonic. Intriguingly, p-adic mass calculations require the ground state conformal weight to be negative half odd integer. This raises the question whether the the boundary condition for Kähler-Dirac equation could be equivalent for the mass formulation given by the condition that the scaling generator L0 annihilates the physical states for Super Virasoro representations. This equivalence is suggested by quantum classical correspondence.
If this is the case, the two mass shell conditions would be equivalent. This possibility is discussed more precisely below.
Well-definedness of the em charge is the fundamental on spinor modes. Physical intuition suggests that also classical Z0 field should vanish - at least in scales longer than weak scale. Above the condition guaranteeing vanishing of em charge has been discussed at very general level. It has however turned out that one can understand situation by simply posing the simplest condition that one can imagine: the vanishing of classical W and possibly also Z0 fields inducing mixing of different charge states.
R01= e0 ∧ e1-e2∧ e3 , R23= e0∧ e1- e2∧ e3 ,
R02=e0∧ e2-e3 ∧ e1 , R31 = -e0∧ e2+e3∧ e1 ,
R03 = 4e0∧ e3+2e1∧ e2 , R12 = 2e0∧ e3+4e1∧ e2 .
R01=R23 and R03= R31 combine to form purely left handed classical W boson fields and Z0 field corresponds to Z0=2R03.
Kähler form is given by
J= 2(e0∧e3+e1∧ e2) .
e0∧ e1-e2∧e3 =0 ,
e0∧ e2-e3 ∧e1 ,
4e0∧ e3+2e1∧e2 .
Also classical Z0 vanishes if a2= 2 holds true. This guarantees that the couplings of induced gauge potential are purely vectorial. One can consider other alternaties. For instance, one could require that only classical Z0 field or induced Kähler form is non-vanishing and deduce similar condition.
See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article.
The intriguing general result of class field theory) -something extremely abstract for physicist's brain - is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this).
Does this mean that it is not absolutely necessary to introduce p-adic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also p-adic integers and numbers are. What is intriguing that p-dic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2-adic units consisting of bit sequences which can have literally infinite non-vanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and p-adic norm approaching to zero expresses the reduction of the insignificance.
1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions?
Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.
2. Adelic variant of space-time dynamics and spinorial dynamics?
As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the space-time dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).
The basic idea is that appropriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.
Octonions, quaternions, quaternionic space-time surfaces, octonionic spinors and twistors and twistor spaces are highly relevant for quantum TGD. In the following some general observations distilled during years are summarized.
There is a beautiful pattern present suggesting that H=M4× CP2 is completely unique on number theoretical grounds. Consider only the following facts. M4 and CP2 are the unique 4-D spaces allowing twistor space with Kähler structure. M8-H duality allows to deduce M4× CP2 via number theoretical compactification. Octonionic projective space OP2 appears as octonionic twistor space (there are no higher-dimensional octonionic projective spaces). Octotwistors generalise the twistorial construction from M4 to M8 and octonionic gamma matrices make sense also for H with quaternionicity condition reducing OP2 to to the twistor space of H.
A further fascinating structure related to octo-twistors is the non-associated analog of Lie group defined by automorphisms by octonionic imaginary units: this group is topologically six-sphere. Also the analogy of quaternionicity of preferred extremals in TGD with the Majorana condition central in super string models is very thought provoking. All this suggests that associativity indeed could define basic dynamical principle of TGD.
The construction of Kähler geometry of WCW ("world of classical worlds") is fundamental to TGD program. I ended up with the idea about physics as WCW geometry around 1985 and made a breakthrough around 1990, when I realized that Kähler function for WCW could correspond to Kähler action for its preferred extremals defining the analogs of Bohr orbits so that classical theory with Bohr rules would become an exact part of quantum theory and path integral would be replaced with genuine integral over WCW. The motivating construction was that for loop spaces leading to a unique Kähler geometry. The geometry for the space of 3-D objects is even more complex than that for loops and the vision still is that the geometry of WCW is unique from the mere existence of Riemann connection.
The basic idea is that WCW is union of symmetric spaces G/H labelled by zero modes which do not contribute to the WCW metric. There have been many open questions and it seems the details of the earlier approach must be modified at the level of detailed identifications and interpretations. What is satisfying that the overall coherence of the picture has increased dramatically and connections with string model and applications of TGD as WCW geometry to particle physics are now very concrete.