Sunday, January 02, 2022

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.

  2. Unitarity reduces to a belief already in quantum field theory based on path integral formalism and the situation is not made easier by the divergence problems. One can try to build unitarity by hand using unitarity conditions. Also here there is non-uniqueness since one is forced to take the discontinuity of the amplitude and use dispersion relations to deduce the entire amplitude.
  3. When space-time itself becomes topologically non-trivial and 3-space is dynamic changing its topology, the idea about Hamiltonian unitary evolution in a fixed spacetime becomes totally obsolete. This and the failure of the path integral approach, were the reasons why I ended up with the extensions of Einstein's program: geometrize entire quantum theory. Later the number theoretic dual of this program emerged and both are crucial in the twistorial construction of scattering amplitudes in the TGD framework.
  4. Also the twistor approach suffers from the unitarity problem: there is no proof for the unitarity. Second problem is that only planar amplitudes can be constructed. Could it be that these problems could have a common solution?
Something seems to go wrong with the entire QFT, and the natural guess is that the notion of unitarity is wrong. Also Nima Arkani-Hamed challenges the notion but cannot provide anything, which would replace it.

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 Zi (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 Zi=0. It is the Zi→ 0 limit that one considers in QFT for the action to deduce n-point functions. Zi &neq;0 would be something different and in QFT interaction terms would correspond to this kind of terms. Could Zi &neq;0 represent real physical situations?

  2. What is of extreme importance is that the situation is infinite-D. The experience with WCW geometry (already Freed noticed that loops spaces have unique Kähler geometry with maximal isometries from the mere existence of Riemann connection) strongly suggests that a non-trivial "super-Kaehler" geometry is unique if it exists at all.

    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 Zi.

  3. Unitarity conditions are replaced by almost identical conditions stating simply that the contravariant Kähler metric is the inverse of the covariant metric as a matrix. There is however an important difference: probabilities are in general complex. Probability conservation however holds true for the real parts of probabilities so that physics is OK.
  4. Allowance of complex probabilities creates however an interpretational problem and the solution became clear now the Kähler metric itself has an interpretation as Fisher information matrix. The usual probability interpretation is secondary but overall important for the testing of the theory. The information theoretic interpretation is more fundamental. It is accompanied by geometric interpretation so that infinite-D Kähler geometry both at the level of WCW and Hilbert space, number theory, information theory and probability theory meet and lead to a generalization of the notion of the fermionic Hilber space.
Zero energy ontology (ZEO) allows us to interpret the situation.
  1. In ZEO, the interpretation of quantum theory changes from "western" to "eastern". One gives up the western idea about a fixed reality. In ZEO only events are real both as zero energy states and as quantum jumps identified as moments of consciousness. To me this looks like Buddhism.

    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.

  2. The exponent of Kähler function with a fermionic part determined by a superposition of operator monomials creating the positive (negative) energy parts of zero energy states multiplied by complex coordinates Zi (their conjugates) becomes the analog of thermodynamic state. I have used to speak of a complex square root of thermodynamics. The complex coordinates Zi parametrize a given state as an analog for a phase of a quantum theory or of a coherent state.

    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 Zi they are enough in the constant curvature case. From these probabilities one can in principle deduce what one can know about the values of Zi.

    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.

  3. 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; 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.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

Articles and other material related to TGD.

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