Gauge symmetries in TGD framework
Lubos Motl is defending string theory by arguing that gauge symmetries, which are put by hand in gauge theories, emerge in string theories. As a matter of fact, in the case of open string theory these symmetries are put in as Chan-Paton factors: to me this looks very much like "just by hand". The great triumph of string theory was believed to be the emergence of five basic string theories but landscape problem destroyed the hopes about predictive theory: at best one might hope of reproducing standard model gauge group but even this has not been shown to be possible yet. Here Lubos makes a rhetoric twist and says that just producing standard model gauge symmetries would not be very much. As a physicists I cannot but disagree: one must produce much more- but also the standard model symmetries!
Gauge symmetry as such is something rather passive. It has been often argued that it is only apparent symmetry. By considering everything modulo gauge transformations, one gets rid of gauge symmetries altogether. This is not quite true. Global gauge transformations which are not trivial in asymptotic regions correspond to non-vanishing gauge charges which vanish only in theories with confinement. Furthermore, gauge transformations act also in many fields: gauge potentials and spinors and this means that with gauge potential kept fixed (say) the gauge transforms of spinor fields yield something not gauge equivalent with the original configuration.
How gauge symmetries emerge in TGD?
In TGD framework the view about gauge symmetries differs from that of GUTs and also from that of string models. No ad hoc assumptions concerning gauge group are tolerated. Also the addition of Chan-Paton factors to the real or effective boundaries of the orbit of 3-surface is seen as an unacceptable ad hoc trick. And indeed, there is no need for ad hoc tricks.
- The geometrization of classical gauge field in terms of sub-manifold geometry allows to identify electroweak gauge potentials in terms of CP2 spinor connection projected to the space-time surface and therefore inheriting the geometro-dynamics of 3-surfaces. If one knows the space-time surface, one knows the evolution of induced gauge fields. The induction of metric and spinor structure is the umbrella concept and also fundamental in fiber bundle theory but for some reason colleagues have not shown interests towards it. What makes fashion unpredictable is its difficult-to-predict butterfly dynamics.
- The choice of CP2 is unique in that the absence of ordinary spinor structure forces to introduce so called Spinc structure which brings in full electroweak gauge group and leads to correct couplings to various gauge potential. This realizes Einstein's dream of reducing dynamics of both gravitational and em fields to geometry and also extends it.
- What about color gauge fields? Here the key observation is that CP2 has isometry group SU(3). The idea is to generalize Kaluza-Klein approach and identify classical gauge potentials as projections of Killing vector fields of CP2 to space-time surface. What is however important is that color gauge potentials do not couple to fermions. The resolution of the problem is the identification of color - not as a spin-like quantum number- but as orbital angular momentum like quantum number. Colored states correspond to color partial waves: center of mass motion of partonic 2-surface in its CP2 degrees of freedom: quantal rigid body motion in CP2. The difference from spin like quantum numbers does not show itself in the scales where we are able to measure but predicts many deviations from standard model: for instance, the existence of lepto-hadron physics in which color octet excitations of leptons form hadron like states. Second implication is the understanding
of separate conservation of B and L in terms of 8-D chiral invariance generalizing that of 4-D gauge theories.
- Even this is not enough: one must obtains a correct correlation between electroweak and color quantum numbers. One obtains correct correlation between em charge and triality of color partial waves but the detailed correspondence between color quantum numbers and electroweak quantum numbers for color partial waves is not that for the observed fermions. The presence of color Kac-Moody symmetries saves the situation, and one obtains the correlation that is observed for physical fermions as p-adic mass calculations demonstrate. By bosonic emergence (bosons having wormhole contacts with fermion and antifermion at its opposite wormhole throats) allows to obtain bosonic color quantum numbers correctly.
This is nice but one can raise several critical questions.
- Doesn't the ability to express all gauge fields in terms of just 4 imbedding space coordinates and their gradients mean an enormous - in fact fatal - reduction in the number of degrees of freedom? Are these kind of correlations between classical gauge fields acceptable? CAn one obtain approximate superposition of classical gauge fields in this framework? The notion of many-sheeted space-time together with the observation that it is the effects of fields that superpose than fields, resolves the problem. Very briefly: the test particle topologically condenses (froms topological sum contacts) simultaneously to all space-time sheets present and experiences the sum of forces.
- What about quantum counterpart of electroweak and color gauge fields? Also these should be identified. The recent progress in the understanding of the preferred extremals of Kähler action and solutions of modified Dirac equation has led to a dramatic progress in this respect (see this).
The mere conservation of em charge defined in spinorial sense requires that the solutions of the modified Dirac equation are restricted to string world sheets. This is not a new result and follows from many other arguments. The expressions for the generators of color and electroweak Kac-Moody algebra plus 2-D Minkowskian Kac-Moody algebra of transversal excitations can be derived. Also a proposal that quantum gauge potentials are 2-D Hodge duals of Kac-Moody currents follows naturally. Also super-generators of Kac-Moody are present and bosons are predicted to have spartners. SUSY is however badly broken and super-space approach does not seem to have a natural application in TGD framework. The picture conforms with the previous vision that 2-D conformal symmetry corresponds to large N SUSY in which all modes of induced spinor field at the string world sheet are generators of super-symmetries.
- Also the view about gauge symmetries as something much more than usually thought emerges. Quantum criticality is the basic postulate of TGD and implies that Kähler coupling strength is quantized as the analog of critical temperature. Conformal invariance realizes quantum criticality in 2-D theories and in TGD framework its generalization to 4-D analog (not ordinary conformal invariance in 4-D sense) is expected to realize quantum criticality in 4-D case. This conformal symmetry reduces to 2-D one for string world sheets carrying fermions with non-vanishing electroweak quantum numbers but right handed neutrino generating N=2 SUSY allows the full 4-D super-conformal symmetry with two analogs of complex coordinate z. One also obtains new insights about what the TGD counterpart of space-time SUSY is.
- By definition, critical deformations of space-time surface do not affect the value of Kähler function (or its analog Morse function in Minkowskian space-time regions) defined by Kähler action. The natural assumption is that also the solutions of the modified Dirac equation analogous to massless Dirac equation remains solutions. This is however not enough: this condition is always true and can be realized explicitly. The stronger condition is that the critical deformation acts as a mere electroweak gauge transformation at least at the string world sheet. The condition implies that the action is that of conformal scaling on the modified gamma matrices (defined as contractions of the canonical momentum currents for Kähler action with imbedding space gamma matrices). 4-D conformal invariance acting as a key symmetry of gauge theories emerges but as a symmetry of the effective metric defined by the modified gamma matrices. There are also many other juicy details related to the relation of TGD with the twistor approach. Clearly gauge symmetry in this framework is much more than in computerized version of gauge theories.