Saturday, February 27, 2016

Three good reasons for the localization of fundamental fermions to string world sheets

There are now three good reasons for the modes of the induced spinor fields to be localized to 2-D string world sheets and partonic 2-surfaces - inf fact, to the boundaries of string world sheets at them defining fermionic world lines. I list these three good reasons in the same order as I became aware of them.

  1. The first good reason is that this condition allows spinor modes to have well-defined electromagnetic charges - the induced classical W boson fields and perhaps also Z field vanish at string world sheets so that only em field and possibly Z field remain and one can have eigenstates of em charge.

  2. Second good reason actually a set of closely related good reasons. First, strong form of holography implied by the strong form of general coordinate invariance demands the ocalization: string world sheets and partonic 2-surfaces are "space-time genes". Also twistorial picture follows naturally if the locus for the restriction of spinor modes at the light-like orbits of partonic 2-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian is 1-D fermion world line. Thanks to holography fermions behave like point like particles, which are massless in 8-D sense. Thirdly, conformal invariance in the fermionic sector demands the localization.

  3. The third good reason emerges from the mathematical problem of field theories involving fermions: also in the models of condensed matter systems this problem is also encountered - in particular, in the models of high Tc superconductivity. For instance, AdS/CFT correspondence involving 10-D blackholes has been proposed as a solution - the reader can decide whether to take this seriously.

    Fermionic path integral is the source of problems. It can be formally reduced to the analog of partition function but the Boltzman weights (analogous to probabilities) are not necessary positive in the general case and this spoils the stability of the numerical computation. One gets rid of the sign problem if one can diagonalize the Hamiltonian, but this problem is believed to be NP-hard in the generic case. A further reason to worry in QFT context is that one must perform Wick rotation to transform action to Hamiltonian and this is a trick. It seems that the problem is much more than a numerical problem: QFT approach is somehow sick.

    The crucial observation giving the third good reason is that this problem is encountered only in dimensions D≥3 - not in dimensions D=1,2! No sign problem in TGD where second quantized fundamental fermions are at string world sheets!

A couple of comments are in order.
  1. Although the assumption about localization 2-D surfaces might have looked first a desperate attempt to save em charge, it now seems that it is something very profound. In TGD approach standard model and GRT emerge as an approximate description obtained by lumping the sheets of the many-sheeted space-time together to form a slightly curved region of Minkowski space and by identifying gauge potentials and gravitational field identified as sums of those associated with the sheets lumped together. The more fundamental description would not be plagued by the mathematical problem of QFT approach .

  2. Although fundamental fermions as second quantized induced spinor fields are 2-D character, it is the modes of the classical imbedding space spinor fields - eigenstates of four-momentum and standard model quantum numbers - that define the ground states of the super-conformal representations. It is these modes that correspond to the 4-D spinor modes of QFT limit. What goes wrong in QFT is that one assigns fermionic oscillator operators to these modes although second quantization should be carried out at deeper level and for the 2-D modes of the induced spinor fields: 2-D conformal symmetry actually makes the construction of these modes trivial.

To conclude, the condition that the theory is computable would pose a powerful condition on the theory. As a matter fact, this is not a new finding. The mathematical existence of Kähler geometry of "world of classical worlds" fixes its geometry more or less uniquely and therefore also the physics: one obtains a union of symmetric spaces labelled by zero modes of the metric and for symmetric space all points (now 3-surfaces) are geometrically equivalent meaning a gigantic simplification allowing to handle the infinite-dimensional case. Even for loop spaces the Kähler geometry is unique and has infinite-dimensional isometry group (Kac-Moody symmetries).

For a summary of earlier postings see Links to the latest progress in TGD.

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