The exact details of the quantization of fermions have remained open in TGD framework. The basic problem is the possibility of divergences coming from anti-commutators of fermions expected to involve delta functions in continuum case. In standard framework normal ordering saves from these divergences for the "free" part of the action but higher order terms give the usual divergences of quantum field theories. In supersymmetric theories the normal ordering divergences however cancel.
What happens in TGD?
One can start by collecting a list of reasonable looking conditions possibly leading to the understanding of the fermionic quantization, in particular anticommutation relations.
- The replacement of point like particles with 3-surfaces replaces the dynamics of fields with that of surfaces. The resulting non-locality in the scale of 3-surfaces gives excellent hopes about the cancellation of divergences in the bosonic sector. The situation is very similar to that in super-string models.
- What about fermions? The TGD counterpart of Dirac action - modified Dirac action - is dictated uniquely by the bosonic action which is induced from twistor lift of TGD as sum of Kähler action analogous to Maxwell action and of volume term (see this). Supersymmetry in TGD sense is proposed here.
In the second quantization based on cognitive representations (see this) as unique discretization of the space-time surface for an adele defined by extension of rationals superpartners would correspond to local composites of quarks and anti-quarks. TGD variant of super-space of SUSY approach so that space-time as 4-surface is replaced with its super-variant identified as union of surfaces associated with the components of super coordinates. Fermions are correlates of quantum variant of Boolean logic which can be seen as square root of Riemann geometry. There is no need for Majorana fermions.
This approach replaced the earlier view in which right-handed neutrinos served a role as generators of N=2 SUSY. In the approach to be discussed their counterparts as local 3-quarks composites make comepack in a more precise formulation of the picture first discussed in .
The simplest option involves only quarks as fundamental fermions and leptons would be local composites of 3 quarks: this is possibly by the TGD based view about color. Quark oscillator operators are enough for the construction of gamma matrices of "world of classical worlds" (WCW, see this) and they inherit their anti-commutators from those of fermionic oscillator operators. Even the super-variant of WCW can be considered. The challenge is to fix these anti-commutation relations for oscillator operator basis at 3-D surface: the modified Dirac equation would dictate the commutation relations later. This is not a trivial problem. One can also wonder whether one avoid the normal ordering divergences.
- In a discretization the anti-commutators of fermions and antifermions by cognitive representations (see this, this, this, and this) do not produce problems but in the continuum variant of this approach one obtains normal ordering divergences. Adelic approach (see this) suggests that also continuum variant of the theory must exists as also that of WCW so that one should find a manner to get rid of the divergences by defining the quantization of fermions in such a manner that one gets rid of divergences.
Induction procedure plays a key role in the construction of classical TGD. The longstanding question has been whether the induction of spinor structure could be generalized to the induction of second quantization of free fermions at the level of 8-D imbedding space to the level of space-time so that induced spinor field Ψ (x) would be
identified as Ψ(h(x)), where h(x) corresponds to the imbedding space coordinates of the space-time point. One would have restrictions of free fermion theory from imbedding space H to space-time surface.
- The quantization should be consistent with the number theoretic vision implying discretization in terms of cognitive representations. Could one assume that anti-commutators for the quark field for discretization is just Kronecker delta so that the troublesome squares of delta function could be avoided already in Dirac action and expressions of conserved quantities unless one performs normal ordering which is somewhat ad hoc procedure.
The anti-commutators of induced spinor fields located at opposite boundaries of CD and quite generally, at points of H=M4xCP2 (or in M8 by MH duality) with non-space-like separation should be determined by the time evolution of induced spinor fields given by modified Dirac equation.
In the case of cognitive representation could fix the anti-commutators for given time slice in M4× CP2 as usual Kronecker delta for the set of points with algebraic coordinates so that if anti-commutators of fermionic operators between opposite boundaries of CD were not needed, everything would be well-defined. By solving the modified Dirac equation for the induced spinors one can indeed express the induced spinor field at the opposite boundary of CD in terms of its values at given boundary. Doing this in practice is however difficult.
- Situation gets more complex if one requires that also the continuum variant of the theory exists. One encounters problems with fermionic quantization since one expects delta function singularities giving rise to at least normal ordering singularities. The most natural manner to quantize quarks fields is as a free field in H= M4 × CP2 expanded as harmonics of H. This however implies 7-D delta functions and bad divergences from them. Can one get rid of these divergences by changing the standard quantization recipes based on ordinary ontology in which one has initial value problem in time= constant snapshot of space-time to a quantization more appropriate in zero energy ontology (ZEO)?
The problem is that the anticommutators are 8-D delta functions in continuum case and could induce rather horrible divergences. It will be found that zero energy ontology (ZEO) (see this and the new view about space-time and particles allow to modify the standard quantization procedure by making modified Dirac action bi-local so that one gets rid of divergences. The rule is simple: given partonic 2-surface or even more generally given point of partonic surfaces contains either creation operators or annihilation operators but not both. Also the multi-local Yangian algebras proposed on basis of physical intuition to be central in TGD emerge naturally.
See the chapter Could ZEO provide a new approach to the quantization of fermions? or the article article with the same title.