Von Neumann inclusions, quantum groups, and quantum model for beliefs

Configuration space spinor fields live in "the world of classical worlds", whose points correspond to 3-surfaces in H=M⁴×CP₂. These fields represent the quantum states of the universe. Configuration space spinors (to be distinguished from spinor fields) have a natural interpretation in terms of a quantum version of Boolean algebra obtained by applying fermionic operators to the vacuum state. Both fermion number and various spinlike quantum numbers can be interpreted as representations of bits. Once you have true and false you have also beliefs and the question is whether it is possible to construct a quantum model for beliefs.

1. Some background about number theoretic Clifford algebras

Configuration space spinors are associated with an infinite-dimensional Clifford algebra spanned by configuration space gamma matrices: spinors are created from vacuum state by complexified gamma matrices acting like fermionic oscillator operators carrying quark and lepton numbers. In a rough sense this algebra could be regarded as an infinite tensor power of M₂(F), where F would correspond to complex numbers. In fact, also F=H (quaternions) and even F=O (octonions) can and must(!) be considered although the definitions involve some delicacies in this case. In particular, the non-associativy of octonions poses an interpretational problem whose solution actually dictates the physics of TGD Universe.

These Clifford algebras can be extended local algebras representable as powers series of hyper-F coordinate (hyper-F is obtained by multiplying imaginary part of F number with a commuting additional imaginary unit) so that a generalization of conformal field concept results with powers of complex coordinate replaced with those of hyper-complex numerg, hyper-quaternion or octonion. TGD could be seen as a generalization of superstring models by adding H and O layers besides C so that space-time and imbedding space emerge without ad hoc tricks of spontaneous compactification and adding of branes non-perturbatively.

The inclusion sequence C in H in O induces generalization of Jones inclusion sequence for the local versions of the number theoretic Clifford algebras allowing to reduce quantum TGD to a generalized number theory. That is, classical and quantum TGD emerge from the natural number theoretic Jones inclusion sequence. Even more, an explicit master formula for S-matrix emerges consistent with the earlier general ideas. It seems safe to say that one chapter in the evolution of TGD is now closed and everything is ready for the technical staff to start their work.

2. Brahman=Atman property of hyper-finite type II₁ factors makes them ideal for realizing symbolic and cognitive representations

Infinite-dimensional Clifford algebras provide a canonical example of von Neumann algebras known as hyper-finite factors of type II₁ having rather marvellous properties. In particular, they possess Brahman= Atman property making it possible to imbed this kind of algebra within itself unitarily as a genuine sub-algebra. One obtains what infinite Jones inclusion sequences yielding as a by-product structures like quantum groups.

Jones inclusions are ideal for cognitive and symbolic representations since they map the fermionic state space of one system to a sub-space of the fermionic statespace of another system. Hence there are good reasons to believe that TGD universe is busily mimicking itself using Jones inclusions and one can identify the space-time correlates (braids connecting two subsystems consisting of magnetic flux tubes). p-Adic and real spinors do not differ in any manner and real-to-p-adic inclusions would give cognitive representations, real-to-real inclusions symbolic representations.

3. Jones inclusions and cognitive and symbolic representations

As already noticed, configuration space spinors provide a natural quantum model for the Boolean logic. When you have logic you have the notions of truth and false, and you have soon also the notion of belief. Beliefs of various kinds (knowledge, misbelief, delusion,...) are the basic element of cognition and obviously involve a representation of the external world or part of it as states of the system defining the believer. Jones inclusions for the mediating unitary mappings between the spaces of configuration spaces spinors of two systems are excellent candidates for these maps, and it is interesting to find what one kind of model for beliefs this picture leads to.

The resulting quantum model for beliefs provides a cognitive interpretation for quantum groups and predicts a universal spectrum for the probabilities that a given belief is true following solely from the commutation relations for the coordinates of complex quantum plane interpreted now as complex spinor components. This spectrum of probabilities depends only on the integer n characterizing the quantum phase q=exp(iπ/n) characterizing the Jones inclusion. For n < ∞ the logic is inherently fuzzy so that absolute knowledge is impossible. q=1 gives ordinary quantum logic with qbits having precise truth values after state function reduction.

One can make two conclusions.

1. Quantum logics might have most interesting applications in the realm of consciousness theory and quantum spinors rather than quantum space-times seem to be more natural for the inclusions of factors of type II₁.

2. For n< ∞ inclusions quantum physical constraints pose fundamental restriction on how precisely it is possible to know and are reflected by the quantum dimension d<2 of quantum spinors telling the effective number of truth values smaller than one by correlations between non-commuting spinor components representing truth values. One could speak about Uncertainty Principle of Cognition for these inclusions.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of the book "Mathematical Aspects of Consciousness".

Matti Pitkänen