Quantum Mathematics

The comment of Pesla to previous posting contained something relating to the self-referentiality of consciousness and inspired a comment which to my opinion deserves a status of posting. The comment summarizes the recent work to which I have associated the phrase "quantum adeles" but to which I would now prefer to assign the phrase "quantum mathematics".

To my view the self referentiality of consciousness is the real "hard problem". The "hard problem" as it is usually understood is only a problem of dualistic approach. My hunch is that the understanding of self-referentiality requires completely new mathematics with explicitly built-in self-referentiality. During last weeks I have been writing and rewriting chapter about quantum adeles and end up to propose what this new mathematics might be. The latest draft is here .

1. Replace of numbers with Hilbert spaces and + and × with direct sum and tensor product

The idea is to start from arithemetics : + and × for natural numbers and generalize it .

1. The key observation is that + and x have direct sum and tensor product for Hilbert spaces as complete analogs and natural number n has interpretation as Hilbert space dimension and can be mapped to n-dimensional Hilbert space.

   So: replace natural numbers n with n-D Hilbert spaces at the first abstraction step. n+m and n×m go to direct sum n⊕m and tensor product n⊗m of Hilbert spaces. You calculate with Hilbert spaces rather than numbers. This induces calculation for Hilbert space states and sum and product are like 3-particle vertices.

2. At second step construct integers (also negative) as pairs of Hilbert spaces (m,n) identifying (m⊕r,n⊕r) and (m,n). This gives what might be called negative dimensional Hilbert spaces! Then take these pairs and define rationals as Hilbert space pairs (m,n) of this kind with (m,n) equivalent to (k⊗m,k⊗n). This gives rise to what might be called m/n-dimensional Hilbert spaces!

3. At the third step construct Hilbert space variants of algebraic extensions of rationals. Hilbert space with dimension sqrt(2) say: this is a really nice trick. After that you can continued with p-adic number fields and even reals: one can indeed understand even what π-dimensional Hilbert space could be!

   The essential element in this is that the direct sum decompositions and tensor products would have genuine meaning: infinite-D Hilbert spaces associated with transcendentals would have different decompositions and would not be equivalent. Also in quantum physics decompositions to tensor products and direct sums (say representations of symmetry group) have phyiscal meaning: abstract Hilbert space of infinite dimension is too rough a concept.

4. Do the same for complex numbers, quaternions, and octonions, imbedding space M⁴×CP₂, etc.. The objection is that the construction is not general coordinate invariant. In coordinates in which point corresponds to integer valued coordinate one has finite-D Hilbert space and in coordinates in which coordinates of point correspond to transcendentals one has infinite-D Hilbert space. This makes sense only if one interprets the situation in terms of cognitive representations for points. π is very difficult to represent cognitively since it has infinite number of digits for which one cannot give a formula. "2" in turn is very simple to represent. This suggests interpretation in terms of self-referentiality. The two worlds with different coordinatizations are not equivalent since they correspond to different cognitive contents.

Replace also the coordinates of points of Hilbert spaces with Hilbert spaces again and again!

The second key observation is that one can do all this again but at new level. Replace the numbers defining vectors of the Hilbert spaces (number sequences) assigned to numbers with Hilbert spaces! Continue ad infinitum by replacing points with Hilbert spaces again and again.

You get sequence of abstractions, which would be analogous to a hierarchy of n:th order logics. At lowest levels would be just predicate calculus: statements like 4=2². At second level abstractions like y=x². At next level collections of algebraic equations, etc....

Connection with infinite primes and endless second quantization

This construction is structurally very similar to - if not equivalent with - the construction of infinite primes which corresponds to repeated second quantization in quantum physics. There is also a close relationship to - maybe equivalence with - what I have called algebraic holography or number theoretic Brahman=Atman identity. Numbers have infinitely complex anatomy not visible for physicist but necessary for understanding the self referentiality of consciousness and allowing mathematical objects to be holograms coding for mathematics. Hilbert spaces would be the DNA of mathematics from which all mathematical structures would be built!

Generalized Feynman diagrams as mathematical formulas?

I did not mention that one can assign to direct sum and tensor product their co-operations and sequences of mathematical operations are very much like generalized Feynman diagrams. Co-product for instance would assign to integer m all its factorizations to a product of two integers with some amplitude for each factorization. Same for co-sum. Operation and co-operation would together give meaning to 3-particle vertex. The amplitudes for the different factorizations must satisfy consistency conditions: associativity and distributivity might give constraints to the couplings to different channels- as particle physicist might express it.

The proposal is that quantum TGD is indeed quantum arithmetics with product and sum and their co-operations. Perhaps even something more general since also quantum logics and quantum set theory could be included! Generalized Feynman diagrams would correspond to formulas and sequences of mathematical operations with stringy 3-vertex as fusion of 3 -surfaces corresponding to ⊕ and Feynmannian 3-vertex as gluing of 3-surfaces along their ends, which is partonic 2-surface, corresponding to ⊗! One implication is that all generalized Feynman diagrams would reduce to a canonical form without loops and incoming/outgoing legs could be permuted. This is actually a generalization of old fashioned string model duality symmetry that I proposed years ago but gave it up as too "romantic": see this.

For details see the new chapter Quantum Adeles.