** 1. Exact solutions of the TGD counterpart of the Chern-Simons action**

As I wrote the previous posting I had not yet realized that classical field equations can be solved exactly: the only solutions are those for which CP_{2} projection of 3-dimensional light-like 3-surface is at most 2-dimensional. Thus the construction of theory at this level reduces to that of constructing light-like 3-surfaces of imbedding space as representation of partons.

** 2. Solutions of the modified Dirac equation and eigen modes of modified Dirac operator**

Also the modified Dirac equation can be solved exactly and solutions can be added to the generalized eigen modes of the modified Dirac operator: this defines the super-conformal symmetries in TGD sense. The hypothesis is that the Dirac determinant defined by the product of the eigenvalues gives rise to the exponent of Kähler action. If true, it would allow to construct the theory solely from the data provided by the light-like partonic 3-surfaces.

The allowance of only the eigenvalues belonging to a particular algebraic extension of rationals defining the extension of p-adic number field could guarantee the finiteness by leaving only finite number of eigenvalues to the spectrum. One would obtain number theoretic hierarchy of physics serving as a correlate for a cognitive hierarchy if one accepts p-adic physics as a physics of cognition and intentionality.

The earlier hypothesis that the super-canonical conformal weights relate in a simple manner to the zeros of Riemann Zeta finds additional support from the more detailed structure of the spectrum of eigen values if the super-canonical conformal weights are identified as eigenvalues of the modified Dirac operator. According to earlier idea, the existence of p-adicization might quantize the eigenvalues rather than boundary conditions or finiteness requirement.

** 3. Super-conformal symmetries**

The basic super-conformal symmetries follow trivially from the almost-topological character of the theory and actually generalize. The light-likeness condition brings in the metric of the imbedding space and manifests itself in eigenvalue spectrum of the modified Dirac operator inducing breaking of the super-conformal invariance and the loss of topological field theory property. Therefore TGD would be as near as possible to a topological quantum field theory and somewhat ironically, the breaking of the super-conformal invariance and the emergence of gauge and gravitational interactions as well as classical Kähler dynamics would follow from light-likeness condition.

** 4. TGD counterpart of super-space formalism**

The precise relationship between super-conformal invariance of TGD and that of super-string models has been quite a stressor since the super-space formalism in the standard form breaks down in TGD. The reason is that Majorana spinors are replaced by Weyl spinors (quarks and leptons correspond to different H-chiralities). That super-conformal invariance makes sense has been clear for a long time but one cannot never exlude the possibility that some subtle error combined with overall optimism could have led to a self deception.

It is now however clear that the super-space formalism exists also in TGD framework but involves a different definition of the Grassmann integral measure. The formalism means the replacement of the radial lightlike coordinate of partonic 2-surface with its super-counter part involving second quantized spinor field consisting of eigenmodes of the modified Dirac operator behaving like Weyl spinors. Only the fermionic fields appear besides imbedding space coordinates as dynamical variables in this case. Super-space formalism gives the modified Dirac action as one integrates over the Grassmann parameters using a modified integration measure

dθ^{*} Γ^{r}dθ,

where Γ^{r} is the modified gamma matrix associated with the lightlike direction and ^{*} denotes Dirac conjugation (it would be nice to have also bar in html symbol repertoire). The modified gamma matrices associated with the other coordinates vanish identically so that super-symmetrization occurs only in the radial direction. Also super-symmetrized canonical and Kac-Moody symmetries are formal symmetries of the Chern-Simons action.

The generalization of the super-space formalism to a higher- dimensional case is obtained by the replacement of the product

∏_{i}dθ _{i}dθ^{*}_{i}

with the wedge product

dθ^{*} γ^{1}dθ ×_{w} dθ^{*}γ^{2}dθ ×_{w}...

completely analogous to the bosonic volume element

dx^{1} ×_{w} dx^{2}×_{w}...

Here ×_{w} symbolizes wedge product. It would be interesting to whether this Grassmann integral allows to construct a physically acceptable super-symmetric field theories based on Weyl- rather than Majorana condition. The requirement that the number of Grassmann parameters given by 2D is the number of spinor components of definite chirality (counting also conjugates) given by 2×2^{D/2-1} gives critical dimension D=8, which suggest that this kind of quantum field theory might exist and define the quantum field theory limit of TGD.

The last section of chapter Construction of Quantum Theory represents the detailed form of the argument above.

## 15 comments:

Hi, Matti.

By "partons", you mean "quark-lepton pairs" or "quark-gluon pairs"?

About Grassmann integrals :

I briefly checked in "Statistical field theory I", ch. 2, by Itzykson and Drouffe. The thing is, the construction of the Grasmann integration followed the bosonic case. They write, p. 51 of the Cambridge Univ Press version : "As Berezin showed, this structure [the bosonic one] has its exact parallel in the fermionic case". Further (p.52, § 2.1.2 "Integrals"), one can read : "The parallel with the Fock-Bargmann construction can be extended by defining anticommuting integrals as linear operations on the functions f(eta), with the seemingly paradoxical property that they can be identified with the (left) derivatives !".

It's clear it's some kind of a "paradox"... But it's far from being the first one in theoretical physics, is it ? :-)) Maybe your construction, explicitely involving the gamma matrices, will make things more mathematically correct and justify both left and right derivatives as well as integration rules.

Dear Philippe,

thank you for a comment.

I mean by parton a lightlike 3-surface carrying any state constructive using fermionic creation operators of quark or lepton type. Quarks, leptons, and gauge bosons are the simplest examples of partons. Partonic 3-surface results when I glue CP_2 type extremal representing elementary particle to space-time sheet with Minkowskian signature of metric: the signature changes at lightlike elementary particle horizon. In this case M^4 projection (and quite generally for non-vacuum partonic 2-surfaces) is a random lightlike

curve and this gives rise to classical Virasoro conditions. Gravitational four-momentum is obviously not conserved and inertial momentum is identified as time average of it: the TGD counterpart of Higgs mechanism results in this manner without Higgs and lightlike randomness justifies p-adic thermodynamics.

TGD counterpart of Higgs boson is also there and is a slightly more complex structure resulting in topological condensation of space-time sheets to space-time sheets. Higgs involves two partonic 3-surfaces corresponding to the lightlike 3-surfaces separating the wormhole contact of CP_2 size

and with Euclidian signature of induced metric from space-time sheets with Minkowskian metric. Bi-parton would be the appropriate term here.

I see that Grassmann integral and Grassmann variables as a technical trick, which might be helpful in calculations also in TGD. It is also possible to extend the notion of surface to that of super-surface by positing linear constraints between thetas. One can however do without the notion of super-space and in this manner one can really understand the real meaning of super-conformal symmetry.

The gamma matrices of the configuration space, the world of classical worlds, define what I call super-canonical algebra: there is not need to introduce super-space. Also the super-symmetries of the modified Dirac action have clear meaning as symmetries of the generalized eigenvalue equation for Dirac operator.

On my side, I confirm the change of metric on light-like surfaces. :-)

It's even possible on space-time itself, without the need for a fibration. There's a trick, of course ;-) : how could you keep the SO(3,1) pseudo-riemannian dynamics while keeping the ability of compactifying into SO(4) in a general way, i.e. for all velocities ?

There's not even some "paradox" here, there's a total apparent contradiction...

Anyway, all this confirms the direct observability of super-luminic phenomenas and enables to reconciliate the Einstein's principle of local interactions with the non-local experiments of Aspect, Zweig and Zeilinger on correlated pairs of protons. :-)))

Keep on exploring that transition,I feel like you should get more surprises !

It is the induced metric the signature of which changes. One can consider some extreme situations to get grasp of what is involved.

a) Imbed CP_2 isometrically as a vacuum extremal to M^4xCP_2: there is infinite number of warped imbeddings with M^4 projection a lightlike random curve. In this case induced metric is just CP_2 metric and has Euclidian signature. These surfaces are identified as elementary particles.

b) Imbed M^4 in canonical manner to H. In this case you have Minkowskian signature.

c) You can get also huge number of vacuum extremals by assuming that the projection CP_2 Kahler form vanishes: any four-surface whose CP_2 projection is Lagrange manifold (at most 2-dimensional submanifold of CP_2) is vacuum extremal. Already for these surface induced metric can change to Euclidian signature. The physical interpretation of Euclidian regions is an interesting question.

d) More general surfaces are obtained by gluing CP_2 type extremals to space-time sheets with M^4 signature. Lightlike causal horizon is unavoidable and identified as a carrier of elementary particle quantum numbers. CP_2 type extremal itself is vacuum extremal. Same happens when you connect space-time sheets by wormhole contacts which are simply pieces of CP_2 type extremal.

Comment concerning your worries about Lorentz invariance. Euclidian signature of the induced metric does not mean breaking of Poincare invariance which is not a symmetry of space-time but of imbedding space (M^4xCP_2). Poincare symmetries do not shift particle in space-time but the entire space-time sheet, which is like 4-D rigid body. This also resolves the energy problem of general relativity which was the starting point of TGD.

Best,

Matti

I see, the situation is indeed different.

Interesting.

Is there any maths in your books about this change of signature on the induced metric ? I uploaded so many chapters, I'm a bit lost... :-)

The change of signature of induced metric is actually simpler than one might think. The induced metric is obtained by projection ds^2 of imbedding space to space-time surface: just express coordinate differentials dh^k for H in terms of dx^alpha for X^4: writing the partial derivatives you get for the induced metric tensor. [By the way, there is old book of Eisenhart about differential geometry of submanifolds, also Spivak has book about this.]

At the 3-surface where signature changes the determinant of the induced metric must vanish since the metric tensor diagonalized in the local coordinates must have a vanishing eigenvalue. This gives the general criterion.

A very simple example: consider a vacuum extremal obtained as a map from M^4 to M^4xS^1 subset M^4xCP_2, where S^1 is geodesic circle. Assume that the angle coordinate phi of S^1 is function of M^4 time coordinate: phi= phi(t), dphi/dt==omega. The imbeddings of Robertson-Walker cosmologies are actually very much analogous to this vacuum extremal (also vacuum extremals so that the densities of conserved inertial momenta and all conserved quantum numbers must vanish in cosmological length scales).

Use standard M^4 coordinates for X^4. The induced metric is dt^2-dx.dx -R^2 omega^2 dt^2.

For 1-R^2 omega^2>0 (<0) you have Minkowskian (Euclidian) signature. The change of signature occurs for some value of time t everywhere for this simple solution.

This example does not represent the typical situation but one can say that when CP_2 coordinates, typically angle coordinate like phi which does not appear in CP_2 metric, changes too rapidly as function of time, the induced metric changes signature.

Note that the vacuum solution considered is flat. There is no gravitational field but there is an arbitrary large time dilatation. Thus a new kind of time dilation effect not present in general relativity and due to the warping of space-time surface is predicted and can be very large. There are some reports about large time dilations but no one takes them seriously. A more familiar example about warping is the instability of thin sheets of paper against warping involving bending but no stretching.

Best,

Matti

Thanks very much, Matti. I now see much clearer what it's all about.

As I said, that's very interesting indeed.

But you need a fibration over M^4 to be able to change the signature of the INDUCED metric.

This leaves the fundamental problem of the signature of base space-time M^4 itself complete, understand ? :-)

The question is : ON CLASSICAL SPACE-TIME M^4 itself, how to get rid of the minus sign before (dx^0)² WITHOUT CHANGING THE LORENTZIAN CHARACTER OF THE DYNAMICS, i.e. THE LIGHT CONE STRUCTURE ? 8-)

(and of course, without using wormholes or the Wick rotation)

That's why I said this seems to be not just paradoxal, but completely incompatible.

If you find the trick...

... keep it for yourself. :-)

Work with it and you'll have no problem with "space-like" and "time-like" directions anymore.

The (3,1) signature of space-time is an illusion.

That's what it all amounts to. :-)

I think that there is some trivial misundersanding involved here.

*There is no need to get rid of

signature of M^4: it is Lorentzian and fixed to what it is: this is just the key element behind TGD as Poincare invariant theory of gravitation. It is signature of *space-time surface* X^4 subset M^4xCP_2 rather than M^4 that I am considering. This is completely dynamical being determined by the induced metric (distances being measured at space-time using the meter sticks H).

*If you have 8-D space with signature 1,-1,...-1 of metric you can have 2-D sub-manifolds with signatures 1,-1,-1,-1 and -1,-1,-1,-1. The simplest possible example is provided by 1-D surfaces, that is curves consisting of time/space/lightlike portions in M^4: these are 1-dimensional surfaces of M^4 with a dynamical signature.

*I am not sure about what you mean with "fibration" in M^4. No fibration in the sense that CP_2 would be fiber space of non-trivial bundle is needed. H is simply Cartesian product of M^4 and CP_2. Actually this is absolutely essential for the theory to work but this is a separate story.

Best,

Matti

Well, if there's no need to get rid of the Lorentzian signature, then so much the better !

Sure, from a 7+1 dimensional space, you can build 2D submanifolds of many different signatures, including full euclidian ones.

But, as I said, you need to add dimensions to the ordinary 4 ones.

the end of your reply is a bit surprising : how can you build a gauge theory if your total space is not fibered ? And even if it's a conventional euclidian product of spaces, you can always get fibers. For instance, M^4 can be fibered with respect to the time-like x^0 coordinate : you then obtain 3-space sections parametrized by x^0 and x^0-curves appear as fibers in M^4. The structure group of the fibration is the 1-parameter dynamical group parametrized by x^0.

Am I wrong ?

H possesses tangent bundle with metric and spinor bundle structure. In particular, CP_2 is curved space so that spinor bundle is non-trivial and possesses non-flat spinor connection with symmetry structure of electroweak gauge fields (holonomy group is U(2).

There is completely standard process known as induction of bundles structure, which for some reason is not too well-known for

physicists.

If you have bundle T-->N and map M-->N then induction gives bundle T-->M: you just move the fiber over point of N over its inverse image.

In the recent case you have tangent bundle TH-->H and spinor bundle SH-->H and imbedding X^4--> H so that tangent and spinor bundles are induced to X^4 and you can induce metric and spinor connection.

At the level of formulas this means projecting of metric/gamma matrices/spinor connection/ curvature to X^4. Projection makes them dynamical fields inheriting their dynamics from that of 3-surface. One exotic looking feature is that gamma matrices become dynamical. Any general coordinate invariant action respecting isometries of H can be taken as action.

In the bosonic sector Kahler action is unique choice because of its completely exceptional symmetries.

The first guess for Dirac action for induced spinors is obvious but not super-symmetric. The requirement that the action allows conserved super currents leads to so called modified Dirac action where Gamma matrices are replaced by modified ones:

Gamma^alpha = partial L/\partial h^k_alpha Gamma^k, where

L is bosonic action density, say Kahler action density,

h^k_alpha = partial h^k/partial x^alpha

h^k is H coordinate, x^alpha is X^4 coordinate, and Gamma^k is gamma matrix of H.

Although all good ideas are said to become part of superstring theory sooner or later (and often forgetting to mention who of the quantum gravity crackpots censored out from arXiv.org discovered the idea first) string model people have not discovered this universal supersymmetrization yet: perhaps they should study my homepage more carefully;-).

Best Regards,

Matti

Okay, I think I get it this time : you first double dimensions by coupling space-time with spinor space CP_2, getting 8D space-time H as a normal euclidian product H = M^4 x CP_2. Then, your work on the tangent bundle of H, collecting geometrical datas on both TM^4 (the bosonic metric) and TCP_2 (the fermionic metric). What follows are projections on space X^4. Is that it ?

Well, this construction is a bit different from the one I'm used to in Gauge theory, where you fiber over M^4 with gauge group G and fibers of dimension dim(G). The total space is a manifold that doesn't even need to be riemannian. However, a beautiful mathematical theorem from Thiry (1951), established in the realm of real geometry and probably totally unknown of today's physical community, shows that any Finsler manifold of dimension n can be transformed into a Riemann manifold of dimension n+1. 8-)

Yes. It seems that we understand each other now. The construction indeed differs from Kaluza-Klein decisively. In string theory the replacement of K-K view about space-time with the identification of 3-brane as space-time surface would be the analog of TGD picture Here people bring however in independent fields living on branes and believed to describe phenomenologically the fundamental states of strings: this is extremely messy and uneconomical picture full of poorly justified steps.

I had to check from Wikipedia what Finsler geometry means and learned that tangent space norm is defined by Banach norm so that you do not have metric. Somehow the increase of dimension by 1 should allow to introduce Riemann metric giving Banach norm. I would guess that 2a.b== ||a+b||^2+||a-b||^2 is involved. The dimensional transmutation sounds strange: it is not clear to me in what sense can one say that two manifolds with different dimensions are equivalent.

Wikipedia says rubbish. You'd better ask Mathworld !

Of course, Finsler spaces are metric ! But there are not in the physicists' common sense : for a physicist, a metric space satisfies the Riemann axiom. Thus, any space with torsion is not "metric" in the physical sense, but it obviously is in the much wider mathematical sense ! :-))

Thiry's construction has nothing to do with the Banach norm, also. :-)))) It's a pure geometrical construction based on Lagrangian dynamics (after his thesis, Thiry worked at the Bureau des Longitudes, where he specialized in the dynamics of the Solar System). Very roughly, it shows how to go from an asymmetric linear form ds on n-dimensional space to a symmetric quadratic form ds² (the famous "metric" of physicists) on an (n+1)-dimensional space. The aim of it was to send the Finsler line element ds' = ds + (q/mc)A_i.dx^i of a moving particle with mass and electric charge coupled to an EM potential onto a fully Riemannian element dS² in dimension 5, thus unifying the gravitational part (ds or rather ds² at Einstein's) and the EM one into a single unified field, classical of course.

I understood. Starting point is ds rather than ds^2 and ds can contain also linear term in coordinate differentials. I think that the representation of Wikipedia was formally correct but somewhat misleading since it did not stress this aspect.

By the way, also Dirac's gamma matrices define an analog of ds as an operator acting on spinors "ds"= gamma_mu dx^mu. Square is ds^2.

Matti

Yes ! And that's why you can be metric while having torsion : look, isn't Kähler geometry the best example ? :-)

The thing is, Finsler geometry is rather delicate to handle, mostly because you have to consider TWO directions instead of a single one in the Riemannian case. So, in the general situation, it's not easy at all to calculate lenghts, areas and volumes in a Finsler geometry.

In a word, Finsler spaces are ANISOTROPIC spaces. Hence the need to define TWO Levi-Civita linear connexions D1 and D2 with the relation D2 = D1 + S, where S is the torsion (a 3-tensor, here). When S vanishes, you retrieve D2 = D1 = D and the Riemann axiom.

As for the definition Wikipedia gives, it's more an analytical one than a purely geometrical one.

Also notice hermitian complex manifolds (Kähler being a special case) are ISOTROPIC with respect to the hermitian structure (you can define a real-valued ds² and a metric tensor g_ik such that

(g_ik)* = g_ki), but ANISOTROPIC with respect to real geometry (the imaginary part of g_ik generates torsion).

IN PRINCIPLE, applying Thiry's result on supersymmetric models would send the ambiant superspaces onto purely Riemannian ones, thus eliminating torsion, that is, MATTER FIELDS. The physical result in dimension n+1 would be PURE RADIATION.

I never had the idea to check Thiry's theorem in the complex domain, so I don't know if it still works there. A priori, it should, at least in the hermitian situation that is of physical interest, but there may appear mathematical subtleties.

Post a Comment