About physical interpretation of non-commutative geometry and discreteness

Characteristic for the recent day strongly methodology based approach to theoretical physics is that many people are working with refined mathematical notions but very few of them seems to be asking whether these notions could have some clear physical content.

The ideas about non-commutativity and discreteness of the space-time geometry at Planck length scale are good examples in this respect. Archives are filled with quantum deformations of all imaginable algebras and discrete theories of quantum gravity. The fact that both non-commutativity of space-time coordinates and discreteness are inconsistent with general coordinate invariance does not seem to bother these pragmatists. I have nothing against methological virtuosity but a more profound view about what the the use of the notion of Planck length actually implies might help to gain a more balanced view.

In TGD framework non-commutativity and discreteness have much less mysterious origin. The progress in quantum physics and mathematics has been to high degree based on an explicit articulation of our limitations (Heisenberg and Gödel) and I believe that an approach based on systematic attempt to identify the inherent limitations of perception and thought could be a useful strategy.

Indeed, non-commutativity emerges naturally when one constructs quantum measurement theory based on finite measurement resolution in terms of Jones inclusions N subset M of von Neumann algebras. Quantum non-commutativity in this sense occurs only at the level of operator algebras and quantum spinors, that is for inherently linear structures, rather than at level of space-time so that there are no problems with general coordinate invariance.

Discreteness in turn could reflect the limitations of our ability in representing our cognitions in the real world (consider only numerical computations contra analytic work), and leads naturally to an expression of S-matrix elements involving a discrete set of points defining a number theoretic braid. Space-time itself remains boringly continuous and only the "infinite-dimensional world of classical worlds" brings in some sexy elements but is probably something too abstract for an average shopper in the free market of ideas.

1. Non-commutativity and quantum measurement theory with resolution

Von Neumann algebras known as hyper-finite factors of type II₁ (HFFs shortly) play a key role in TGD Universe since the Clifford algebra of the world of classical worlds is direct sum of these factors. Recall that this infinite-dimensional space is a union of subworlds consisting of lightlike 3-surfaces of H=M⁴_+/-×CP₂. Subworlds of .... in the union are labelled by positions of tips of lightcones in Minkowski space.

Jones inclusions N subset M for HFF:s provide a very natural algebraization of quantum measurement theory with a finite measurement resolution (this is something new!). The space of resolvable degrees of freedom is quantum space N/M and corresponds in the simplest case to quantum version of 2×2 matrices and thus to quantum group SU(2)_q having fractal dimension M:N= 4cos²(π/n) from the quantization of the Jones index. That dimension M:N is below 4 reflects the non-commutativity of matrix elements implying that they are not completely independent. Corresponding quantum spinors have fractal dimension d=(M/N)^1/2: the interpretation of non-commutativity as reducer of effective dimension is completely analogous.

M/N is N-module and this means that N takes the role of complex numbers in "quantum quantum theory". One must generalize the notions of unitarity, hermiticity, eigenvalue, etc.. so that N replaces complex numbers. For instance, hermitian operators have as eigenvalues hermitian operators in N. Unitary operator is unitary matrix with N valued elements (exponent of N-hermitian operator).

The measurement resolution defined by N means following things.

1. In ordinary quantum measurement ideal state function reduction occurs to a 1-dimensional complex ray of state space. Now the reduction occurs to a 1-dimensional N-ray equivalent with N. Therefore a sequence of quantum measurements defines a sequence M contains M₁ contains M₂ contains... of Jones inclusions rather than leading to a fixed point in the first step. This in the case that it is possible to improve the resolution and quantum jumps do not interfere with the process.

2. Quantum measurement never fully reduces the entanglement. The reader is encouraged to consider a quantum version of Zeno's paradox from this point of view. In quantum theory of consciousness this implies that we share a pool of fused mental images correspond to entanglement below the resolution. Stereo vision is example of shared and fused mental images which now correspond to right and left visual fields. At collective level of consciousness our selves could fuse during sleep to a huge stereo view about human condition. TGD Universe would be infinitely sized conscious organism (sincere apologies for skeptics and the additional message that I am only bringing the message;-)).

3. The measurement of S-matrix (or reaction rates) corresponds to a measurement of unitary time like entanglement (Tr(SS⁺)=Tr(Id)=1!) for positive and negative energy components of zero energy states (all states of the Universe are creatable from vacuum in zero energy ontology natural in TGD Universe). A hierarchy of S-matrices identified as characterizers of time like entanglement results corresponding to a hierarchy of Jones inclusions with improving measurement resolution. S-matrix must be consistent with the N-module structure which is guaranteed by crossing symmetry with respect to the transfer of elements of N between initial and final states combined with hermitian conjugation. This follows from a more general crossing symmetry. Hermitian conjugation modifies sligtly the basic property of entanglement defined by Connes tensor product. Together with huge super-conformal symmetries extending those of super string models (S-matrix as super-conformal invariant) this could imply uniqueness of S-matrix with a given measurement resolution.

4. The measurement of spin J₃ for quantum spinor means that a physical state results for which either component of quantum spinor annihilates the state. This leads to a contradiction if one requires that the physical state is annihilated by second spinor component and is ordinary eigen state of second spinor component. The problem disappears if second component acts like an element of N and thus does not leave the quantum state invariant (C is indeed replaced with N!). The non-commutativity of different N-valued eigenvalues implies that non-commutativity of elements of J₃ does not lead to contradiction.

5. The moduli squared for quantum spinor commute and their eigenvalue spectrum is universal depending only on integer characterizing quantum phase. Eigen values for moduli are rational numbers. For M:N<4 fuzzy logic and fuzzy beliefs result.

To me this picture about quantum measurement with finite resolution replacing the usual horribly ugly cutoff description looks very beautiful and elegant. Non-commutativity would not be a fundamental property of space-time but would characterize finite measurement resolution and would apply at the level of quantum spinors rather than space-time coordinates (sorry for my spontaneous "auch!" which I cannot hide when I even imagine "[x^μ,x^ν]=...";-)). Quantum classical correspondence however encourages to think that this picture has also space-time correlate.

2. New view about discretization

The second misty idea of Planck length mystics is that space-time somehow becomes discrete in Planck scales and that continuum is an approximation emerging at long length scales. For some reason (not difficult to guess!) the alternative view that space-time is continuous and that discreteness might reflect the limitations of our cognition, has not gained popularity. In non-materialistic TGD Universe, where thinking is allowed, the latter view emerges naturally when one looks at space-time correlates of quantum measurement.

1. The generalized eigenvalue spectrum of the modified Dirac operator contains the scaling factor log(p), p prime, as an overall scaling factor, which means that one can assign to a given lightlike partonic 3-surface a unique p-adic prime. One implication is that coupling constant evolution in TGD framework can be identified as a discrete p-adic coupling constant evolution induced by this log(p)-proportionality of the generalized eigenvalue spectrum of the modified Dirac operator (recall that p-adic length scale is proportional to p^1/2). It is now possible to write renormalization group equations explicitly at the level of eigenspinors of modified Dirac operator so that coupling constant evolution emerges at the level of free field theory rather than from calculation of radiative corrections.

2. If partonic 3-surface is represented by algebraic equations then its p-adic counterpart is also well-defined and has a discrete set of common points with its real counterpart. Algebraic equations are possible since light-likeness is the only condition on partonic 3-surfaces and extremality property in the case of Chern-Simons action for induced Kähler gauge potential means only to at most 2-D CP₂ projection.

3. Number theoretic universality requires that S-matrix elements are algebraic numbers. This can be achieved if only a finite subset of points in the algebraic intersection of p-adic and real space-time sheet contributes to the definition of S-matrix elements. The definition of S-matrix assigns a number theoretic braid to each incoming particle with strands carrying sub-partonic quantum numbers, essentially operators of the super-conformal algebras involved. The emergence of number theoretic braids is very natural for hyper-finite factors of type II₁ whose inclusion sequences define inclusion sequences of Temperley-Lieb algebras associated with braids.

4. Discretization by number theoretic braids has nothing to do with the discreteness of space-time at fundamental level but reflects basically the limitations of cognition if p-adic space-time sheets are identified as representations of cognition and intentionality. p-Adic bosonic partons would correspond to intentions transforming in quantum jump to actions represented by real partons. Pair of real and p-adic fermionic parton would define state together with its cognitive representation analogous to hole and negative energy fermion kicked out of Dirac sea.

To me all this looks nice, elegant, and rational but probably I am correct in guessing that colleagues will stubbornly continue their exercises in Planck length mystics. In the so called free market of ideas "Planck length scale" and "black hole" are much sexier buzz words than "quantum measurement theory" and "measurement resolution". "Cognition" and "intention" are of course mere crack-talk for those fascinated by reductionism extended down to Planck length.