Saturday, April 23, 2005

Cognitive entanglement as Connes tensor product

In the construction of the cognitive hierarchy of S-matrices that I discussed few days ago, the lowest level N represents matter and higher levels give cognitive representations. The ordinary tensor product S \x S and its tensor powers define a hierarchy of S-matrices and the two-sided projections of these S-matrices in turn define entanglement coefficients for positive and negative energy states at various levels of hierarchy. The following arguments show that the cognitive tensor product restricted to projections of S-matrix corresponds to the so called Connes tensor product appearing naturally in the hierarchy of Jones inclusions. A slight generalization of earlier scenario predicting matter-mind type transitions is forced by this identification and a beautiful interpretation for these transitions in terms of space-time correlates emerges.

1. Connes tensor product

Connes has introduced a variant of tensor product allowing to express the union cup_i M_i, where M_i the inclusion hierarchy as infinite tensor product M\x_N M \x_N M\x_N... The Connes tensor product \x_N differs from the standard tensor product and is obtained by requiring that in the Connes tensor product of Hilbert spaces H_1 and H_2 the condition n a\x_N b= a \x_N nb for all n in N holds true. Connes tensor product means forces to replace ordinary statistics with braid statistics. The physical interpretation proposed by Connes is that this tensor product could make sense when N represents observables common to the Hilbert spaces H_1 and H_2. Below it will be found that TGD suggests quite different interpretation. Connes tensor product makes sense also for finite-dimensional right and left modules. Consider the spaces M_{nxq} of nxq-matrices and M_{pxn} of pxn matrices for which nxn matrix algebra M_{nxn} acts as a left resp. right multiplier. The tensor product x_N for these matrices is the ordinary matrix product of m_{pxn}\times m_{nxq} and belongs to M_{pxq} so that the dimension of tensor product space is much lower than mxqx n^2 and does not depend on n. For Jones inclusion N takes the role of M_{nxn} and since M can be regarded as beta=N:M-dimensional N-module, tensor product can be said to give sqrt{beta}x sqrt{beta}-dimensional matrices with N-valued entries. In particular, the inclusion sequence is an infinite tensor product of sqrt{beta} x sqrt{beta}-dimensional matrices.

2. Does Connes tensor product generate cognitive entanglement?

One can wonder why the entanglement coefficients between positive and negative energy states should be restricted to the projections of S-matrix. The obvious guess is that it gives rise to an entanglement equivalent with Connes tensor product so that the action of N on initial state is equivalent with its action on the final state. This indeed seems to be the case. The basic symmetry of Connes tensor product translates to the possibility to move an operator creating particles in initial state to final state by conjugating it: this is nothing but crossing symmetry characterizing S-matrix. Thus Connes tensor product generates zero energy states providing a hierarchy of cognitive representations.

3. Do transitions between different levels of cognitive hierarchy occur?

The following arguments suggest that the proposed hierarchy of cognitive representations is not exhaustive.
  • Only tensor powers of S involving (2^n)^{th} powers of S appear in the cognitive hierarchy as it is constructed. Connes tensor product representation of cup_iM_i would however suggest that all powers of S appear.
  • There is no reason to restrict the states to positive energy states in TGD Universe. In fact, the states of the entire Universe have zero energy. Thus much more general zero energy states are possible in TGD framework than those for which entanglement is given by a projection of S-matrix, and they occur already at the lowest level of the hierarchy.
On basis of these observations there is no reason to exclude transitions between different levels of the cognitive hierarchy transforming ordinary tensor product of positive and negative energy states with vanishing conserved quantum numbers to a Connes tensor product involving only the projection of S-matrix as entanglement coefficients. These transitions would give rise to S-matrices connecting different levels and thus fill the gaps in the spectrum of allowed tensor powers of S.

4. Space-time correlates for the matter-to-mind transitions

The scatterings in question would represent kind of matter-to-mind transitions, enlightment, or transition to a Buddha state. At space-time level zero energy matter would correspond to positive and negative energy states with a space-like separation whereas cognitive states would correspond to positive and negative energy states with a time-like separation. By the failure of the complete classical determinism time like entanglement makes sense but due to the fact determinism is not completely lost, entanglement could be of a very special kind only, and S-matrix could appear as entanglement coefficients. Light-like causal determinants (CDs) identifiable as orbits of both space-like partonic 2-surfaces and light-like stringy surfaces, can be said to represent both matter and mind. According to the earlier proposal, light-like CDs would correspond to both programs and computers for topological quantum computation, and matter-mind transformation would be also involved with the realization of the genetic code both as cognitive and material structures. This would support the view that the stringy 2-surfaces in the foliation of the space-time surface are time-like in the interior of the space-time sheet (or more generally, outside light-like causal determinants) and light-like causal determinants correspond to critical line between matter and mind. For more details see the chapter Was von Neumann Right After All? of TGD.

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