^{0}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 Z

^{0}fields inducing mixing of different charge states.

- Induced W fields mean that the modes of Kähler-Dirac equation do not in general have well-defined em charge. The problem disappears if the induced W gauge fields vanish. This does not yet guarantee that couplings to classical gauge fields are physical in long scales. Also classical Z
^{0}field should vanish so that the couplings would be purely vectorial. Vectoriality might be true in long enough scales only. If W and Z^{0}fields vanish in all scales then electroweak forces are due to the exchanges of corresponding gauge bosons described as string like objects in TGD and represent non-trivial space-time geometry and topology at microscopic scale.

- The conditions solve also another long-standing interpretational problem. Color rotations induce rotations in electroweak-holonomy group so that the vanishing of all induced weak fields also guarantees that color rotations do not spoil the property of spinor modes to be eigenstates of em charge.

- The representation of the covariantly constant curvature tensor is given by

R_{01}= e^{0}∧ e^{1}-e^{2}∧ e^{3}, R_{23}= e^{0}∧ e^{1}- e^{2}∧ e^{3},

R

_{02}=e^{0}∧ e^{2}-e^{3}∧ e^{1}, R_{31}= -e^{0}∧

e^{2}+e^{3}∧ e^{1},

R

_{03}= 4e^{0}∧ e^{3}+2e^{1}∧ e^{2}

, R_{12 = 2e0∧ e3+4e1∧ e2 . }R

_{01}=R_{23}and R_{03}= R_{31}combine to form purely left handed classical W boson fields and Z^{0}field corresponds to Z^{0}=2R_{03}.

Kähler form is given by

J= 2(e

^{0}∧e^{3}+e^{1}∧ e^{2}) .

- The vanishing of classical weak fields is guaranteed by the conditions

e

^{0}∧ e^{1}-e^{2}∧e^{3}=0 ,

e

^{0}∧ e^{2}-e^{3}∧e^{1},

4e

^{0}∧ e^{3}+2e^{1}∧e^{2}.

- There are many manners to satisfy these conditions. For instance, the condition e
^{1}= a× e^{0}and e^{2}=-a×e^{3}with arbitrary a which can depend on position guarantees the vanishing of classical W fields. The CP_{2}projection of the tangent space of the region carrying the spinor mode must be 2-D.

Also classical Z

^{0}vanishes if a^{2}= 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 Z^{0}field or induced Kähler form is non-vanishing and deduce similar condition.

- The vanishing of the weak part of induced gauge field implies that the CP
_{2}projection of the region carrying spinor mode is 2-D. Therefore the condition that the modes of induced spinor field are restricted to 2-surfaces carrying no weak fields sheets guarantees well-definedness of em charge and vanishing of classical weak couplings. This condition does not imply string world sheets in the general case since the CP_{2}projection of the space-time sheet can be 2-D.

- Additional consistency condition to neutrality of string world sheets is that Kähler-Dirac gamma matrices have no components orthogonal to the 2-surface in question. Hence various fermionic would flow along string world sheet.

- If the Kähler-Dirac gamma matrices at string world sheet are expressible in terms of two non-vanishing gamma matrices parallel to string world sheet and sheet and thus define an integrable distribution of tangent vectors, this is achieved. What is important that modified gamma matrices can indeed span lower than 4-D space and often do so (massless extremals and vacuum extremals representative examples). Induced gamma matrices defined always 4-D space so that the restriction of the modes to string world sheets is not possible.

- String models suggest that string world sheets are minimal surfaces of space-time surface or of imbedding space but it might not be necessary to pose this condition separately.

- The vanishing conditions for induced weak fields allow also 4-D spinor modes if they are true for entire spatime surface. This is true if the space-time surface has 2-D projection. One can expect that the space-time surface has foliation by string world sheets and the general solution of K-D equation is continuous superposition of the 2-D modes in this case and discrete one in the generic case.

- If the CP
_{2}projection of space-time surface is homologically non-trivial geodesic sphere S^{2}, the field equations reduce to those in M^{4}× S^{2}since the second fundamental form for S^{2}is vanishing. It is possible to have geodesic sphere for which induced gauge field has only em component?

- If the CP
_{2}projection is complex manifold as it is for string like objects, the vanishing of weak fields might be also achieved.

- Does the phase of cosmic strings assumed to dominate primordial cosmology correspond to this phase with no classical weak fields? During radiation dominated phase 4-D string like objects would transform to string world sheets.Kind of dimensional transmutation would occur.

- Electroweak gauge potentials do not couple to ν
_{R}at all. Therefore em neutrality condition is un-necessary if the induced gamma matrices do not mix right handed neutrino with left-handed one. This is guaranteed if M^{4}and CP_{2}parts of Kähler-Dirac operator annihilate separately right-handed neutrino spinor mode. Also ν_{R}modes can be interpreted as continuous superpositions of 2-D modes and this allows to define overlap integrals for them and induced spinor fields needed to define WCW gamma matrices and super-generators.

- For covariantly constant right-handed neutrino mode defining a generator of super-symmetries is certainly a solution of K-D. Whether more general solutions of K-D exist remains to be checked out.

## 12 comments:

http://www.math.neu.edu/event/geometry-algebra-singularities-and-combinatorics/2013-11-18

Dear Matti,

If classical em field is associated with long ranged color field as TGD say it, i try to clear my imagination by ask my questions:

em wave has tensor product of spin group and color group?

Why every em wave with special wavelength has special color or in other words there is 1-1 correspondence between them?

Dear Hamed,

one can say that color field is associated with induced Kähler form, which does not always coincide with em field which contains also other term proportional to classical Z^0 field. If classical weak fields vanish as they should at string world sheets carrying fermions so that they have well-defined em charge and couple purely vectorially then Kähler form is proportional to em field. Em field is accompanied by classical long ranged color field.

What matters is whether classical color gauge potentials couple to induced spinor fields. The classical color field is not visible in the couplings of spinors since they couple directly only to the ew gauge potentials.

Color is visible only at quantum level. Conformal algebra generators carrying color and color partial waves assignable to imbedding space spinors and appearing as ground states of super-conformal representations. These imbedding space spinor modes correspond to the spinors of QFT and QCD appears as QFT approximation to the dynamics.

I would not assign to em waves color. Color is quale which in TGD inspired theory of consciousness corresponds to increment of color

quantum numbers (this is not a joke!) in quantum jump. Quite concretely the flow of color quantum numbers from subsystem in question or to it give rise to sensation of color. There are many manners to induce this flow. Photons have specialised to induce this flow but photons as such do not have any color. Qualia are not properties of physical objects but characterise their change in quantum jump. This saves from logical paradoxes.

Dear Matti,

Thanks,

this is correct? for example when a photon with the wavelength of 700nm come to my eyes, although it hasn't any color but it leads to increment of red color quantum number in the retina.

But if photons haven't any color, how they can induce the flow of color quantum numbers in retina? conservation of color quantum number is violated?

Maybe i can improve my last question in this form: Why does every photon with special wave length can increase special color quantum number in the retina?

Dear Hamed.

Color sensation can quite well induced the flow of color quantum numbers. The total color quantum numbers of system plus complement should vanish. It is also possible that color polarisation occurs so that the sensory receptor and its complement have opposite but increasing color quantum numbers. I have modelled sensory perception as analog of di-electric breakdown for the analog of capacitor: now color capacitor.

It is interesting that for instance an object with color is surrounded by narrow stripe with complementary color: does this follow from vanishing of total color quantum numbers.

One can of course ask whether gluons are generated in color perception by incoming photons and create the flow of color quantum numbers.

If one believes that dark QCD is present in scales of living matter (four Gaussian Mersennes (1+i)^n-1 in length scale range 10 nm, 2.5 micrometer which is number theoretical miracle, suggest that both scaled/dark versions of QCD and and ew interactions are there and correspond to these Mersennes).

An interesting thing is also that we can observe color only if there are at least two colours. Completely homogenous light to screen does not create sensation of color. I think here one must build a model for for color perception is. For years ago I considered this problem: do not remember anymore the explanation.

Dear Hamed,

as you say, an interesting question is why color

of certain wavelength (actually distribution of wavelengths) can induce color sensation.

Could one speak of Bose-Einstein condensation color receptors have Bose-Einstein condensate with definite color quantum numbers and this is amplified by polarisation induced by incoming photons around resonance energies. Photons give only the energy. Total color remains zero but color quantum numbers at plates of color capacitor increase. The color quantum numbers of color receptors would correlate with visual colours.

Could one consider the situation in terms

dark QCD in which string like objects having opposite colors and realised as color magnetic

flux tubes are generated between the plates

of color capacitor. Kind of resonant burst of mesons generated by incoming photon energy.

Could this relate to dark photons decaying to biophotons in turn providing energy for creating color polarisation? Biophotons are just ordinary photons in eneryy range including visible and part of UV (molecular excitation energies).

Dear Matti,

Thanks,

Does state function reduction, occurs for massless extremals too? as I understand, In standard QM, only wave function of electron reduce after measurement and it can’t describe what occurs for photons after the measurement. Although if we do the double-slit experiment for a single photon, we see the wave function of photons reduce at a point of screen. Hence one can say that it is a problem for standard QM that can’t describe wave function evolution and reduction of wave function of a photon.

After reduction of wave function of an electron in some boundary of CD(suppose in lower boundary),

if at this moment of consciousness we go from lower to upper boundary and observe the wave function of the electron(In other words we are observing geometric time evolution of electron in this process), what does we see? Is this correct? We see the electron wave function just over a small 3-surface in lower boundary and when we are going from lower to upper at the moment of consciousness we see electron wave function propagates over more regions or over more and more small 3-surfaces.

State function reduction is completely general process. All particles, not only electrons. For photons it occurs for instance in double slit experiments. When both slits are open the photons are in waves going through both slits. Measurement redoes to localised state in screen.In TGD this initial state would mean two brached 3-surface going through slits. When only single slit is open they are initially in superposition involving only single slit.

An important correction to the original picture about state function reduction in ZEO. I have mentioned about this few times but maybe I have not emphasized enough the change.

Originally I thought that reduction occurs alternately to the boundaries of CD. This might be the case in very short scales where time arrow is not yet present.In longer scales - larger CDs- this is not the case: microscopy and arrow of time enter into the picture.

Then I realised that state function reduction at given boundary of CD can occur arbitrary many times repeatedly. In ordinary QM this means repeated state function reduction giving again the reduced states ( quantum mechanical Zeno paradox relates to this and is empirical fact).

In TGD each repeated state function reduction leaves the state at already reduced boundary invariant but changes the state at other boundary. This gives rise to experience flow and arrow of time.

The changes at second boundary however occur and give to conscious experience something: the experience about flow of time at least. Interesting problem is to try to understand exactly what is this contribution! IS this just our experience about classical everyday world as opposed to experience of state function reduction which brings to my mind moment of birth about which I do not remember much;-).

Returning to your question. The information about electron's state is coded by the wave function at reduced boundary. Effective 2-dimensionality reduces this information to partonic 2-surfaces and tangent space data at then.

When we observe electron localisation it is a state function reduction to opposite boundary after sequence of reduction to same boundary. Boundary of CD is changed. What the scale of CD

is in this case. Scale of electrons CD of .1 seconds if one assumes that it corresponds to secondary p-adic time scale? Amusingly, this is also the time scale of our sensory perception. Maximal time resolution of sensory perception. In smaller scales time ordering can fluctuate.

The best manner to speak about state function is to speak about WCW spinor fields. Quantum superpositions of space-time surfaces. These change in reduction reducing them to eigenstates of measured observables at either boundary of CD.

*Superposition* of massless externals (MEs) changes to a new one for instance. It is easily misleading to talk about state function reduction for ME. If one wants to do this, one must assume that reduction occurs in scales shorter than that characterising ME so that ME is unchanged in the process. ME serves as an arena of quantum physics in this case.

Sorry I must stop. I am going to Helsinki! I continue later!

State function reduction is completely general process. All particles, not only electrons. For photons it occurs for instance in double slit experiments. When both slits are open the photons are in waves going through both slits. Measurement redoes to localised state in screen.In TGD this initial state would mean two brached 3-surface going through slits. When only single slit is open they are initially in superposition involving only single slit.

An important correction to the original picture about state function reduction in ZEO. I have mentioned about this few times but maybe I have not emphasized enough the change.

Originally I thought that reduction occurs alternately to the boundaries of CD. This might be the case in very short scales where time arrow is not yet present.In longer scales - larger CDs- this is not the case: microscopy and arrow of time enter into the picture.

Then I realised that state function reduction at given boundary of CD can occur arbitrary many times repeatedly. In ordinary QM this means repeated state function reduction giving again the reduced states ( quantum mechanical Zeno paradox relates to this and is empirical fact).

In TGD each repeated state function reduction leaves the state at already reduced boundary invariant but changes the state at other boundary. This gives rise to experience flow and arrow of time.

The changes at second boundary however occur and give to conscious experience something: the experience about flow of time at least. Interesting problem is to try to understand exactly what is this contribution! IS this just our experience about classical everyday world as opposed to experience of state function reduction which brings to my mind moment of birth about which I do not remember much;-).

Returning to your question. The information about electron's state is coded by the wave function at reduced boundary. Effective 2-dimensionality reduces this information to partonic 2-surfaces and tangent space data at then.

When we observe electron localisation it is a state function reduction to opposite boundary after sequence of reduction to same boundary. Boundary of CD is changed. What the scale of CD

is in this case. Scale of electrons CD of .1 seconds if one assumes that it corresponds to secondary p-adic time scale? Amusingly, this is also the time scale of our sensory perception. Maximal time resolution of sensory perception. In smaller scales time ordering can fluctuate.

The best manner to speak about state function is to speak about WCW spinor fields. Quantum superpositions of space-time surfaces. These change in reduction reducing them to eigenstates of measured observables at either boundary of CD.

*Superposition* of massless externals (MEs) changes to a new one for instance. It is easily misleading to talk about state function reduction for ME. If one wants to do this, one must assume that reduction occurs in scales shorter than that characterising ME so that ME is unchanged in the process. ME serves as an arena of quantum physics in this case.

Sorry I must stop. I am going to Helsinki! I continue later!

Thanks,

Well, Have a good time there.

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