M Theory Lesson 51

The gallant kneemo has pointed out that a locale is an important concept in topos theory. A locale is a simple generalisation of the lattice of open sets for a topological space. Localic toposes are discussed in the book of Mac Lane and Moerdijk.

By restricting attention to sober topological spaces, there is an equivalence of categories between spaces and a suitable collection of locales. Alternatively, there is a duality between spaces and frames, where a frame is just a locale in the opposite category. Now the initial object in the category of frames is a two point set {0,1}, because lattices always have a top element 1 and a bottom element 0.

When considering spaces, the most basic space is really the Sierpinski space $S$ and not a single point, which is often used to describe points in spaces via maps from the one point space. This space has the property of being both a space and a frame, the initial frame. An open set in a space $M$ is considered as a continuous map $M \rightarrow S$, whereby the inverse of 1 picks out the open set.

The existence of such a self-dual object in a categorical duality turns out to be very useful. Another example is the circle $U(1)$ in Pontrjagin duality for locally compact (Hausdorff) abelian groups. This is why kneemo’s remark about a ternary analogue is interesting. We have seen that generalised Fourier transforms are important in M Theory. It is not enough, however, to simply replace the two element set with the three element set {0,1,2}. The inclusion of classical locales into the ‘quantum’ topos theory would require a higher categorical setting, for which the three possible two point subsets are perhaps rather subcategories of the triangle category on three points. This is suggestive of the need to consider three inclusions for 2-logoses into 3-logoses.

Aside: I have inserted a button to Carl’s gravity simulator on the left sidebar.

1 Response so far »

  1. 1

    Matti Pitkanen said,

    Dear Kea,

    a funny association related to quantum 2-spinors. They might allow to define quantum variant of locale and a nice connection with q-quantum measurement theory (quantum measurements with finite resolution) and with Jones inclusions of hyper-finite factors of type II_1 (HFF) would emerge.

    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 2 components a and b are not commuting numbers. ab= qba, q root of unity. This means that one cannot measure both simultaneously, only either of them. aa^+ and bb^+ however commute so that probabilities for bits 1 and 0 can be measured simultaneously. Only the phase information about both qbits is unavailable. Note that state function reduction is not possible to state in which a or b gives zero! The interpretation is that one has q-logic is inherently fuzzy: there are now absolute truths or falses. One can actually predict the spectrum of eigenvalues of probabilities for say 1.

    Q-locale: Could one think of generalizing the notion of locale to quantum locale by using the idea that sets are replaced by subspaces 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 qcounterpart 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: In this framework finite measurement resolution (Q-quantum measurement theory;- in which complex rays are replaced by sub-algebra rays) would force the Jones inclusions associated with SU(2) spinor representation and 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 one would obtain which might have something to do with quantum n-logos. 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 possibly to build up q-representations of various groups using the fundamental physical realization – and as I have conjectured – Mc Kay correspondence and huge variety of its generalizations would emerge in this manner.

    Best, Matti

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