## Geometric view about Higgs mechanismThe improved understanding of the generalization of the imbedding space concept forced by the hierarchy of Planck constants led to a considerable progress in TGD. For instance, I understand now how fractional quantum Hall effect emerges in TGD framework. I have also a rather satisfactory understanding of the notion of number theoretic braid: in particular the question how the cutoff implying that the number of strands is finite, emerges from inherent geometry of the partonic 2-surface. Also a beautiful geometric interpretation of the generalized eigenstates and eigenvalues of the modified Dirac operator and understanding of super-canonical conforma weights emerges. It became already earlier clear that the generalized eigenvalue of Dirac operator which are scalar fields correspond to Higgs expectation value physically. The problem was to deduce what this expectation value is and I have now very beautiful geometric construction of Higgs expectation value as a coder of rather simple but fundamental geometric information about partonic surface. This leads also to an expression for the zeta function associated with number theoretic braid and understanding of what geometric information it codes about partonic 2-surface. Also the finiteness of the theory becomes manifest since the number of generalized eigenvalues is finite. In the following I describe the arguments related to the geometrization of Higgs expectation. I attach the text which can be also found from the chapter Construction of Quantum Theory Symmetries of "Towards S-matrix".
Geometrization of Higgs mechanism in TGD framework The identification of the generalized eigenvalues of the modified Dirac operator as Higgs field suggests the possibility of understanding the spectrum of D purely geometrically by combining physical and geometric constraints. The standard zeta function associated with the eigenvalues of the modified Dirac action is the best candidate concerning the interpretation of super-canonical conformal weights as zeros of ζ. This ζ should have very concrete geometric and physical interpretation related to the quantum criticality. This would be the case if these eigenvalues, eigenvalue actually, have geometric based on geometrization of Higgs field.
Before continuing it its convenient to introduce some notations. Denote the complex coordinate of a point of X
The eigenvalues of the modified Dirac operator have a natural interpretation as Higgs field which vanishes for unstable extrema of Higgs potential. These unstable extrema correspond naturally to quantum critical points resulting as intersection of M Quantum criticality suggests that the counterpart of Higgs potential could be identified as the modulus square of Higgs
V(H(s))= -|H(s)|
which indeed has the points s with V(H(s))=0 as extrema which would be unstable in accordance with quantum criticality. The fact that for ordinary Higgs mechanism minima of V are the important ones raises the question whether number theoretic braids might more naturally correspond to the minima of V rather than intersection points with S
Geometric interpretation of Higgs field suggests that critical points with vanishing Higgs correspond to the maximally quantum critical manifold R
The question is whether one can assign to a given point pair (h(w),h
This guess turns need not be quite correct. An alternative guess is that M
The value should be in general complex and invariant under the isometries of δ H affecting h and h
The phase factor should relate close to the Kähler structure of CP
H(w)= d(h,h
d(h,h
U(s,s
This gives rise to a holomorphic function is X
One can ask whether one should include to the phase factor also the phase obtained using the Kähler gauge potential associated with S
In each coordinate patch Higgs potential would be simply the quadratic function V= -ww*. Negative sign is required by quantum criticality. Potential could indeed have minima as minimal distance of X
An important constraint comes from the condition that the vacuum degeneracy of Käahler action should be understood from the properties of the Dirac determinant. In the case of vacuum extremals Dirac determinant should have unit modulus.
Suppose that the space-time sheet associated with the vacuum parton X
It seems however difficult to understand how to obtain non-trivial phase in the generic case for all points if the phase is evaluated along geodesic line in CP
One must add the condition that curve is not shorter than the geodesic line between points. For a given curve length s
S= ∫ A
This gives for the extremum the equation of motion for a charged particle with Kähler charge Q
D D The magnitude of the phase must be further minimized as a function of curve length s.
If the extremum curve in CP
The construction gives also a concrete idea about how the 4-D space-time sheet X
The definition of Dirac determinant should be independent of the choice of complex coordinate for X
The physical intuition based on Higgs mechanism suggests strongly that the Dirac determinant should be defined simply as products of the eigenvalues of D, that is those of Higgs field, associated with the number theoretic braid.
If only single kind of braid is allowed then the minima of Higgs field define the points of the braid very naturally. The points in R One would define the Dirac determinant as the product of the values of Higgs field over all minima of local Higgs potential
det(D)= [∏
Here w
This definition would be general coordinate invariant and independent of the topology of X
Number theoretical constraints require that the numbers w
The proposed picture supports the identification of the eigenvalues of D in terms of a Higgs fields having purely geometric meaning. The identification of Higgs as the inverse of ζ function is not favored. It also seems that number theoretic braids must be identified as minima of Higgs potential in X The question is then how to understand super-canonical conformal weights for which the identification as zeros of a zeta function of some kind is highly suggestive. The natural answer would be that the eigenvalues of D defines this zeta function as
ζ(s)= ∑
The number of eigenvalues contributing to this function would be finite and H(w
The ζ function would directly code the basic geometric properties of X
The zeros of this ζ function would in turn define natural candidates for super-canonical conformal weights and their number would thus be finite in accordance with the idea about inherent cutoff also in configuration space degrees of freedom. Note that super-canonical conformal weights would be functionals of X
The zeta function should exist also in p-adic
sense. This requires that the numbers λ:s at
the points s of S
The generalized eigenvalue λ(w) is only
proportional to the vacuum expectation value of
Higgs, not equal to it. Indeed, Higgs and gauge
bosons as elementary particles correspond to
wormhole contacts carrying fermion and antifermion
at the two wormhole throats and must be
distinguished from the space-time correlate of its
vacuum expectation as something proportional to
λ. In the fermionic case the vacuum
expectation value of Higgs does not seem to be even
possible since fermions do not correspond to
wormhole contacts between two space-time sheets but
possess only single wormhole throat (p-adic mass
calculations are consistent with this). Gauge
bosons can have Higgs expectation proportional to
λ. The proportionality must be of form
<H> propto λ/p
Suppose that that Dirac determinant is defined as a product of determinants associated with various points z
Since Dirac determinant is not real and is not invariant under isometries of CP
The objection is that Chern-Simons action depends not only on X - The first manner to circumvent this objection is to restrict the consideration to maxima of Kähler function which select preferred light-like 3-surfaces X
^{3}_{l}. The basic conjecture forced by the number theoretic universality and allowed by TGD based view about coupling constant evolution indeed is that perturbation theory at the level of configuration space can be restricted to the maxima of Kähler function and even more: the radiative corrections given by this perturbative series vanish being already coded by Kähler function having interpretation as analog of effective action. - There is also an alternative way out of the
difficulty: define the Dirac determinant and zeta
function using the minima of the modulus of the
generalized Higgs as a function of coordinates of
X
^{3}_{l}so that continuous strands of braids are replaced by a discrete set of points in the generic case.
The fact that general Poincare transformations fail to be symmetries of Dirac determinant is not in conflict with Poincare invariance of Kähler action since preferred extremals of Kähler action are in question and must contain the fixed partonic 2-surfaces at δ M
One can exclude the possibility that the exponent of the stringy action defined by the area of X The condition that the number of eigenvalues is finite is most naturally satisfied if generalized ζ coding information about the properties of partonic 2-surface and expressible as a rational function for which the inverse has a finite number of branches is in question.
Is the construction of space-time correlate of
Higgs as λ really unique? The replacement
of H with its power H
The map H→ H
One can therefore ask whether the powers of H could
define a hierarchy of quantum phases labelled by
different values of k and α For more details see the chapter Overall View about Quantum TGD. |