What are the basic equations of quantum TGD?

After 32 years of hard work it is finally possible to proudly present the basic equations of quantum TGD. There are two kinds of equations.

  1. Purely classical equations define the dynamics of space-time sheets as preferred extremals of Kähler action. Preferred extremals are quantum critical in the sense that second variation vanishes for critical deformations. They can be also regarded as hyper-quaternionic surfaces. What these statements precisely mean has become clear during this year.

  2. The purely quantal equations are associated with the representations of various super-conformal algebras and with the modified Dirac equation. The requirement that there are deformations of the space-time surface -actually infinite number of them- giving rise to conserved fermionic charges implies quantum criticality at the level of Kähler action in the sense of critical deformations. The precise form of the modified Dirac equation is not however completely fixed without a further input.

Quantum classical correspondence requires a coupling between quantum and classical and this coupling should also give rise to a generalization of quantum measurement theory. The big question mark is how to realize this coupling. Few weeks ago I realized that the addition of a measurement interaction term to the modified Dirac action does the job.

In the previous posting about how the addition of measurement interaction term to the modified Dirac actions solves a handful of problems of quantum TGD I was not yet able to decide the precise form of the measurement interaction. There is however a long list of arguments supporting the identification of the measurement interaction as the one defined by 3-D Chern-Simons term assignable with wormhole throats so that the dynamics in the interior of space-time sheet is not affected. This means that 3-D light-like wormhole throats carry induced spinor field which can be regarded as independent degrees of freedom having the spinors fields at partonic 2-surfaces as sources and acting as 3-D sources for the 4-D induced spinor field. The most general measurement interaction would involve the corresponding coupling also for Kähler action but is not physically motivated. Here are the arguments.

  1. A correlation between 4-D geometry of space-time sheet and quantum numbers is achieved by the identification of exponent of Kähler function as Dirac determinant making possible the entanglement of classical degrees of freedom in the interior of space-time sheet with quantum numbers.

  2. Cartan algebra plays a key role not only in quantum level but also at the level of space-time geometry since quantum critical conserved currents vanish for Cartan algebra of isometries and the measurement interaction terms giving rise to conserved currents are possible only for Cartan algebras. Furthermore, modified Dirac equation makes sense only for eigen states of Cartan algebra generators. The hierarchy of Planck constants realized in terms of the book like structure of the generalized imbedding space assigns to each CD preferred Cartan algebra: in case of Poincare algebra there are two of them corresponding to linear and cylindrical M4 coordinates.

  3. Quantum holography and dimensional reduction hierarchy in which partonic 2-surface defined fermionic sources for 3-D fermionic fields at light-like 3-surfaces Y3l in turn defining fermionic sources for 4-D spinors find an elegant realization. Effective 2-dimensionality is achieved if the replacement of light-like wormhole throat X3l with light-like 3-surface Y3l "parallel" with it in the definition of Dirac determinant corresponds to the U(1) gauge transformation K→ K+f+f* for Kähler function of WCW ("world of classical worlds") so that WCW Kähler metric is not affected. Here is arbitrary holomorphic function of WCW complex coordinates and zero modes.

  4. An elegant description of the interaction between super-conformal representations realized at partonic 2-surfaces and dynamics of space-time surfaces is achieved since the values of Cartan charges are feeded to the 3-D Dirac equation which also receives mass term at the same time. Almost topological QFT at wormhole throats results at the limit when four-momenta vanish: this is in accordance with the original vision.

  5. A detailed view about the physical role of quantum criticality results. Quantum criticality fixes the values of Kähler coupling strength as the analog of critical temperature. Quantum criticality implies that second variation of Kähler action vanishes for critical deformations and the existence of conserved current except in the case of Cartan algebra of isometries. Quantum criticality allows to fix the values of couplings (gravitational coupling, gauge couplings, etc..) appearing in the measurement interaction by using the condition K→ K+f+f*. p-Adic coupling constant evolution can be understood also and corresponds to scale hierarchy for sizes of causal diamonds (CDs).

  6. CP breaking, irreversibility, and the space-time description of dissipation are closely related. What is interesting that dissipation does not make itself visible at the level of configuration space metric since it only induces the gauge transformation K→ K+f+f*. Space-time sheet is however affected. Also the interpretation of preferred extremals of Kähler action in regions where DC-S=0 holds true as asymptotic self organization patterns makes sense. Here DC-S denotes the 3-D modified Dirac operator associated with Chern-Simons action and DC-S,int to the corresponding measurement interaction term expressible as superposition of couplings to various observables to critical conserved currents.

  7. A radically new view about matter antimatter asymmetry based on zero energy ontology emerges and one could understand the experimental absence of antimatter as being due to the fact antimatter corresponds to negative energy states. The identification of bosons as wormhole contacts is the only possible option in this framework.

  8. Almost stringy propagators and a consistency with the identification of wormhole throats as lines of generalized Feynman diagrams is achieved. The notion of bosonic emergence leads to a long sought general master formula for the M-matrix elements. The counterpart for fermionic loop defining bosonic inverse propagator at QFT limit is wormhole contact with fermion and cutoffs in mass squared and hyperbolic angle for loop momenta of fermion and antifermion in the rest system of emitting boson have a precise geometric counterpart in the fundamental theory.

My overall feeling is that TGD is finally a mature physical theory with a clear physical interpretation and precise equations. As I started this business my optimistic belief was that it would be a matter of few years to write the Feynman rules. The continual trial and error process made it soon obvious that standard recipes fail and that deep conceptual problems must be solved before one can even dream about defining S-matrix in TGD framework. This forced a construction of TGD inspired theory of consciousness and vision about quantum biology as a by-product. During last half decade (zero energy ontology, the notion of finite measurement resolution, the hierarchy of Planck constants, bosonic emergence,...) it has become clear how dramatic a generalization of existing ontology and epistemology of physics is needed before it is possible to write the generalized Feynman rules. But it seems that they can be written now!

For details see the new chapter Does modified Dirac action defined the fundamental variational principle?.