Kea talks about topos, n-logos,... and their possibly existing quantum variants. I have used to visit Kea's blog in the hope of stealing some category theoretic intuition. It is also nice to represent comments knowing that they are not censored out immediately if their have the smell of original thought: this is quite too often the case in alpha male dominated blogs. It might be that I had luck this morning!
1. Locales, frames, Sierpinski topologies and Sierpinski space
Kea mentioned the notions of locale and frame . In Wikipedia I learned that complete Heyting algebras, which are fundamental to category theory, are objects of three categories with differing arrows. CHey, Loc and its opposite category Frm (arrows reversed). Complete Heyting algebras are partially ordered sets which are complete lattices. Besides the basic logical operations there is also algebra multiplication. From Wikipedia I learned also that locales and the dual notion of frames form the foundation of pointless topology. These topologies are important in topos theory which does not assume the axiom of choice.
So called particular point topology assumes a selection of single point but I have the physicist's feeling that it is otherwise rather near to pointless topology. Sierpinski topology is this kind of topology. Sierpinski topology is defined in a simple manner: set is open only if it contains a given point p. The dual of this topology defined in the obvious sense exists also. Sierpinski space consisting of just two points 0 and 1 is the universal building block of these topologies in the sense that a map of an arbitrary space to Sierpinski space provides it with Sierpinski topology as the induced topology. In category theoretical terms Sierpinski space is the initial object in the category of frames and terminal object in the dual category of locales. This category theoretic reductionism looks highly attractive to me.
2. Particular point topologies, their generalization, and finite measurement resolution
Pointless, or rather particular point topologies might be very interesting from physicist's point of view. After all, every classical physical measurement has a finite space-time resolution. In TGD framework discretization by number theoretic braids replaces partonic 2-surface with a discrete set consisting of algebraic points in some extension of rationals: this brings in mind something which might be called a topology with a set of particular algebraic points.
Perhaps the physical variant for the axiom of choice could be restricted so that only sets of algebraic points in some extension of rationals can be chosen freely. The extension would depend on the position of the physical system in the algebraic evolutionary hierarchy defining also a cognitive hierarchy. Certainly this would fit very nicely to the formulation of quantum TGD unifying real and p-adic physics by gluing real and p-adic number fields to single super-structure via common algebraic points.
There is also a finite measurement resolution in Hilbert space sense not taken into account in the standard quantum measurement theory based on factors of type I. In TGD framework one indeed introduces quantum measurement theory with a finite measurement resolution so that complex rays becomes included hyper-finite factors of type II1 (HFF, see this).
- Could topology with particular algebraic points have a generalization allowing a category theoretic formulation of the quantum measurement theory without states identified as complex rays?
- How to achieve this? In the transition of ordinary Boolean logic to quantum logic in the old fashioned sense (von Neuman again!) the set of subsets is replaced with the set of subspaces of Hilbert space. Perhaps this transition has a counterpart as a transition from Sierpinski topology to a structure in which sub-spaces of Hilbert space are quantum sub-spaces with complex rays replaced with the orbits of subalgebra defining the measurement resolution. Sierpinski space {0,1} would in this generalization be replaced with the quantum counterpart of the space of 2-spinors. Perhaps one should also introduce q-category theory with Heyting algebra being replaced with q-quantum logic.
This program, which I formulated only after this section had been written, might indeed make sense (ideas never learn to emerge in the logical order of things;-)). The lucky association was to the ideas about fuzzy quantum logic realized in terms of quantum 2-spinor that I had developed a couple of years ago. Fuzzy quantum logic would reflect the finite measurement resolution. I just list the pieces of the argument.
Spinors and qbits: Spinors define a quantal variant of Boolean statements, qbits. One can however go further and define the notion of quantum qbit, qqbit. I indeed did this for couple of years ago (the last section in Was von Neumann Right After All?).
Q-spinors and qqbits: For q-spinors the two components a and b are not commuting numbers but non-Hermitian operators. ab= qba, q a root of unity. This means that one cannot measure both a and b simultaneously, only either of them. aa+ and bb+ however commute so that probabilities for bits 1 and 0 can be measured simultaneously. State function reduction is not possible to a state in which a or b gives zero! The interpretation is that one has q-logic is inherently fuzzy: there are no absolute truths or falsehoods. One can actually predict the spectrum of eigenvalues of probabilities for say 1. q-Spinors bring in mind strongly the Hilbert space counterpart of Sierpinski space. One would however expect that fuzzy quantum logic replaces the logic defined by Heyting algebra.
Q-locale: Could one think of generalizing the notion of locale to quantum locale by using the idea that sets are replaced by sub-spaces of Hilbert space in the conventional quantum logic. Q-openness would be defined by identifying quantum spinors as the initial object, q-Sierpinski space. a (resp. b for dual category) would define q-open set in this space. Q-open sets for other quantum spaces would be defined as inverse images of a (resp. b) for morphisms to this space. Only for q=1 one could have the q-counterpart of rather uninteresting topology in which all sets are open and every map is continuous.
Q-locale and HFFs: The q-Sierpinski character of q-spinors would conform with the very special role of Clifford algebra in the theory of HFFs, in particular, the special role of Jones inclusions to which one can assign spinor representations of SU(2). The Clifford algebra and spinors of the world of classical worlds identifiable as Fock space of quark and lepton spinors is the fundamental example in which 2-spinors and corresponding Clifford algebra serves as basic building brick although tensor powers of any matrix algebra provides a representation of HFF.
Q-measurement theory: Finite measurement resolution (q-quantum measurement theory) means that complex rays are replaced by sub-algebra rays. This would force the Jones inclusions associated with SU(2) spinor representation and would be characterized by quantum phase q and bring in the q-topology and q-spinors. Fuzzyness of qqbits of course correlates with the finite measurement resolution.
Q-n-logos: For other q-representations of SU(2) and for representations of compact groups (see appendix of this) one would obtain something which might have something to do with quantum n-logos, quantum generalization of n-valued logic. All of these would be however less fundamental and induced by q-morphisms to the fundamental representation in terms of spinors of the world of classical worlds. What would be however very nice that if these q-morphisms are constructible explicitly it would become possible to build up q-representations of various groups using the fundamental physical realization - and as I have conjectured (see this) - McKay correspondence and huge variety of its generalizations would emerge in this manner.
The analogs of Sierpinski spaces: The discrete subgroups of SU(2), and quite generally, the groups Zn associated with Jones inclusions and leaving the choice of quantization axes invariant, bring in mind the n-point analogs of Sierpinski space with unit element defining the particular point. Note however that n≥3 holds true always so that one does not obtain Sierpinski space itself. Could it be that all of these n preferred points belong to any open set? Number theoretical braids identified as subsets of the intersection of real and p-adic variants of algebraic partonic 2-surface define second candidate for the generalized Sierpinski space with set of preferred points. Recall that the generalized imbedding space related to the quantization of Planck constant is obtained by gluing together coverings of M4×CP2→ M4×CP2/Ga×Gb along their common points. The topology in question would mean that if some point in the covering belongs to an open set, all of them do so. The interpretation could be that the points of fiber form a single inseparable quantal unit.
For more details see the chapter Was von Neumann Right After All?.
5 comments:
Great stuff! Of course, I have not the least intention of deleting your posts, but then of course I share your talent for being ignored.
Perhaps the physical variant for the axiom of choice could be restricted so that only sets of algebraic points in some extension of rationals can be chosen freely.
As you say, the physics of measurement really seems to require working with foundational axioms, and constructing numbers outside of set theory. I have worked a bit on the ordinary Hilbert space type of quantum logic, but the breaking of monoidal structure by the parity cube kills such ordinary logic as a foundation for n-logoses.
The physics inspired dynamical axioms for mathematics as an evolutionary structure having direct quantum physical correlates could be something between conventional mathematics with unrestricted axiom of choice and category theory with no axiom of choice.
We are after all making choices and maybe the restriction to rationals or algebraics in a given extension of rationals could characterize the restrictions to the completely free choice and would also make the axiom of choice number theoretically universal.
Someone claimed to me that category theory is not capable of reproducing the structures of classical mathematics, such as number theory and classical number fields. I am wondering if this true or not and how this possibly relates to axiom of choice.
Matti
Someone claimed to me that category theory is not capable of reproducing the structures of classical mathematics, such as number theory and classical number fields.
Well, very few people that work in Category Theory worry about number constructivism. This push is really coming from the physics. My guess is that once people better appreciate the new possibilities for attacking the Riemann hypothesis (for example) they will be more keen to think about physics this way. Recent work in AlgGeom about MZV algebras (using Motivic Cohomology) is very suggestive, since it is exactly how Veneziano amplitudes come out of M theory.
Good day, dear Mr. Pitkanen. :-)
I can't get it, what's the difference between ordinary quantum physics and p-adic quantum physics. They are equivalent, aren't they?
Here author says about some p-adic quantum effects but not about what they are exactly? :-S
Thanks. :-)
The real and p-adic variants of light-like 3-surfaces obey same algebraic equations being dictated by rational functions with algebraic coefficients. p-Adic worlds and real worlds meet at common algebraic points of real and p-adic "partonic" 3-surfaces, which can have arbitrary size. Stronger assumption is that same occurs for space-time sheets too.
p-Adic surfaces have necessarily infinite size in the real sense since rational points, q and q+ p^n, n goes to infinity are infinitesimally near p-adically but infinitely distance in real sense.
If real and p-adic space-time sheet have a lot of common points, the continuity and smoothness of p-adic physics induces long range correlations and fractality of real physics: also effective p-adic topology in some length scale range. This justifies p-adic thermodynamics used in mass calculations.
p-Adic physics is interpreted as physics of cognition and intentionality, which would be therefore present already at elementary particle level. Quantum jump in which p-adic space-time sheet becomes real is interpreted as a transformation of intention to action. Cognition and intentionality would be cosmic phenomena whereas sensory input from the real world would be localized to finite space-time region. Entire hierarchy of physics corresponding to algebraic extensions of rationals and p-adics is predicted and would correlate with the evolution of cognition and intentionality. This would include evolution of mathematical consciousness (with evolving axioms, say axiom of choice sharpened by increasing dimension for the extension of rationals).
Algebraic universality at the level of S-matrix could have forms of differing strengths.
a) In the weakest form it would only say that S-matrices between different number fields, say R-->Q_p (intentional action), are dictated completely by the data from points in the common intersection of the partonic two surfaces corresponding to the number fields in question. The discreteness of the intersection would correlate with the fact that cognitive representations are discrete (for instance, every numerical calculation involves discretization).
b) In the strongest sense all S-matrices, real-to-real Q_p-Q_p and non-diagonal ones would be dictated by the data in this intersection. This condition is not necessary but rather attractive. One might however wonder whether it implies that the information about number field disappears completely. This is not the case since in R-R case all algebraic points in a given algebraic extension of rationals can contribute whereas in R-Q_p this set is much smaller and depends on p. It might be also possible perform completion in the arguments of S-matrix (say in momenta) to the entire number field.
Post a Comment