1. 3-D foliations and type III factors
Connes mentioned 3-D foliations V which give rise to type III factors. Foliation property requires a slicing of V by a one-form v to which slices are orthogonal (this requires metric).
- The foliation property requires that v multiplied by suitable scalar is gradient. This gives the integrability conditions dv= w\wedge v, w=-dψ/ψ =-dlog(ψ). Something proportional to log(ψ) can be taken as a third coordinate varying along flow lines of v: the flow defines a continuous sequence of maps of 2-dimensional slice to itself.
- If the so called Godbillon-Vey invariant defined as the integral of dw\wedge w over V is non-vanishing, factor of type III is obtained using Schrödinger amplitudes for which the flow lines of foliation define the time evolution. The operators of the algebra in question are transversal operators acting on Schrödinger amplitudes at each slice. Essentially Schrödinger equation in 3-D space-time would be in question with factor of type III resulting from the exotic choice of the time coordinate defining the slicing.
In TGD light-like 3-surfaces are natural candidates for V and it is interesting to look what happens in this case. Light-likeness is of course a disturbing complication since orthogonality condition and thus contravariant metric is involved with the definition of the slicing. Light-likeness is not however involved with the basic conditions.
- The one-form v defined by the induced Kähler gauge potential A defining also a braiding is a unique identification for v. If foliation exists, the braiding flow defines a continuous sequence of maps of partonic 2-surface to itself.
- Physically this means the possibility of a super-conducting phase with order parameter satisfying covariant constancy equation Dψ=(d/dt -ieA)ψ=0. This would describe a supra current flowing along flow lines of A.
- If the integrability fails to be true, one cannot assign Schrödinger time evolution with the flow lines of v. One might perhaps say that 3-surface behaves like single quantum event not allowing slicing into a continuous Schrödinger time evolution.
- In TGD Schrödinger amplitudes are replaced by second quantized induced spinor fields. Hence one does not face the problem whether it makes sense to speak about Schrödinger time evolution of complex order parameter along the flow lines of a foliation or not. Also the fact that the "time evolution" for the modified Dirac operator corresponds to single position dependent generalized eigenvalue identified as Higgs expectation same for all transversal modes (essentially zn labelled by conformal weight) is crucial since it saves from the problems caused by the possible non-existence of Schrödinger evolution.
- The construction of quantum TGD at parton level using light-like 3-surfaces as basic objects forces the introduction of Lorentz invariant component of Kähler gauge potential Aa= constant, where a=(t2-r2)1/2 denotes light-cone proper time. The value of this component could depend on the sector of the generalized imbedding space partially characterized by the value of the Planck constant. The modification does not affect the Kähler form but has the highly non-trivial implication that Chern-Simons action is non-vanishing even when the CP2 projection of the light-like 3-surface is 2-dimensional. D=2 holds true for the extremals of Chern-Simons action.
- Non-vanishing Aa is necessary in order to modify the topological QFT defined by Chern-Simons action to an almost topological QFT. What is of utmost importance is that the Noether currents associated with the four-momentum are non-trivial and non-conserved whereas four-momentum squared is conserved and non-vanishing. The breaking of Poincare invariance does not however take place at the level of the world of classical worlds since the configuration space is a union of sub-configuration spaces for which a choice of preferred future light-cone has been made.
- Since the integrability conditions for A are not gauge invariant, the non-vanishing value of Aa implies that integrability conditions fail already for D=2 as is easy to see by taking two X3 coordinates to be the coordinates of geodesic sphere of CP2 and the remaining coordinate a light-like coordinate. The light-like 3-surfaces associated with all non-vacuum extremals would behave like quantum events rather than continuous evolutions of Schrödinger equation. This is in spirit of the zero energy ontology in which the ends of the space-time sheet carry positive and negative energy states defining physical state as zero energy state. It conforms also with the notion of time-like entanglement defined by Connes tensor product, which can be reduced only partially in quantum measurements. The failure of the integrability condition means that the flow lines of A define typically helical structures which means a non-trivial braiding. This brings strongly in mind the helical structures of living matter.
Some comments relating to the interpretation of the classification of the extremals of Kähler action by the dimension of their CP projection are in order. In the chapter Basic Extremals of the Kähler Action classical field equations of TGD are studied. It was found that the extremals can be classified according to the dimension D of the CP2 projection of space-time sheet in the case that Aa=0 holds true.
- For D=2 integrability conditions for the vector potential can be satisfied for Aa=0 so that one has generalized Beltrami flow and one can speak about Schrödinger time evolution associated with the flow lines of vector potential defined by covariant constancy condition Dψ=0 makes sense. Kähler current is vanishing or lightlike. This phase is analogous to a super-conductor or a ferromagnetic phase. For nonvanishing Aa the Beltrami flow property is lost but the analogy with ferromagnetism makes sense still.
- For D=3 foliations are lost even for Aa=0. The phase is dominated by helical structures as also D=2 phase with non-vanishing Aa. This phase is analogous to spin glass phase around phase transition point from ferromagnetic to non-magnetized phase and expected to be important in living matter systems.
- D=4 is analogous to a chaotic phase with vanishing Kähler current and to a phase without magnetization. The interpretation in terms of non-quantum coherent "dead" matter is suggestive.