More about the replacement of S-matrix with a generalized Kaehler metric in fermionic degrees of freedom

The following is almost as such a response to  a comment of Stephen Crowley relating to unitarity.

I explain first the origins for the postulate about unitary S-matrix (introduced 1937 by Wheeler and 1940 independently by Heisenberg) as an encoder of the predictions of quantum theory. I consider the problems associated with this notion and then briefly summarize the TGD based solution of the problems discussed in here.

1. Wave mechanics starts  from a non-relativistic situation: there is a preferred time coordinate and Hamiltonian time evolution. In the relativistic situation this is not the case.  

In wave mechanics, time evolution is classically obtained by exponentiating a Hamiltonian: one has a flow. In  wave-mechanics  Hamiltonian time evolution is replaced with that generated by a Hamiltonian as an operator. There is  of course normal ordering non-uniqueness but the  real  problem is that Hilbert space allows an endless variety of different unitary evolutions. Any hermitian operator generates such.  This looks really ugly.

The failure of locality is the second cornerstone assumption, which seems to fail in the twistor approach: Yangian algebras have multi-local generators and multilocal Noether charges are definitely in conflict with locality of QFT. In TGD, the replacement of point-like with 3-surface means giving up locality from the very beginning.

In the unitarity problem,   the geometrization of the quantum physics program  came   into rescue.

1.  Encode the physics, not by unitary S-matrix, but by  the Kähler geometry of  the  state space. K”ahler geometry has been already introduced for WCW but can can do it also for the state space: I proposed this  here.

It turned out that this is not  yet quite a correct  idea.  A more precise statement would be following. Encode the transition amplitudes which define zero energy states in the  fermionic degrees of freedom as the analog of Kähler geometry. Bosonic degrees of freedom would correspond to WCW. The resulting generalization of Kähler geometry would be somewhat analogous to what might be called super-WCW.

The fermionic  operator monomials consisting of creation (annihilation) operators creating positive (negative) energy parts of many fermion states would be  multiplied by complex coordinate Z_i (their conjugates) would  define  analogs of super fields. Monomials for theta parameters would be replaced by oscillator operator monomials.  The monomials with odd fermion  number need not be multiplied with anticommuting parameters since fermion number conservation is forced by vacuum expectation value.

The generalization of exponent exp(-K) of Kähler action  K obtained by adding to K this linear combination of these monomials would be formally analogous to QFT action expential containing also the fermionic part. What would matter,  would be the vacuum expectation value of the expansion of the exponential giving rise to scattering amplitudes at the limit Z_i=0. It is the Z_i→ 0 limit   that  one considers in QFT for the action to deduce n-point functions.  Z_i &neq;0 would be something different and in QFT interaction terms would correspond to this kind of terms.  Could  Z_i &neq;0 represent real physical situations?

It must have maximal symmetries and  is necessary a constant curvature geometry so that the generalization of Kähler metric, whose components are transition amplitudes, allow to code the entire  Kähler geometry for arbitrary values of complex coordinates Z_i.

Conserved probabilities for particle reactions still provide an empirical  source of information about the state in thermo-dynamical sense. This picture of course conforms with the TGD based view about consciousness as a continual recreation of reality.

In the “Buddhist” view there are only events A → B but no fixed reality exists.  The ratios of probabilities for the occurrence of  events A → B are fundamental from the experimental point of view and deduced at the limit Z_i they are enough in the constant curvature case.  From these probabilities  one can  in principle deduce  what one can know about the values of Z_i.

 This conforms with ZEO, where time evolution  associated  with a transition A → B becomes a key element: behavior in biology and neuroscience, computer program in computer science.  Could exact holography at classical level mean that the eastern and western views are nearly equivalent. In any case, the exact holography is broken: the minimal surfaces as preferred extremals are not completely deterministic but have singularities as analogs of frames and at frames determinism is violated.   Cosmologists  are  probably not happy about this  ontological relativism. In TGD cosmology indeed becomes a hierarchy of sub-cosmologies assignable to causal diamonds (CDs):  there is no absolute reality lasting from time=-&infty; eternity to time=+&infty;.

I am however sure that the extreme mathematical beauty,  elegance  and uniqueness of this view leave no other option than to accept it. This view only repeats  at the level of WCW and  at the level of  the fermionic state space  what Einstein did for space-time.

For a summary of earlier postings see Latest progress in TGD.

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