# reperiendi

## Lazulinos

Posted in Borges, Fun links, General physics, Perception, Quantum by Mike Stay on 2010 April 27

Lazulinos are quasiparticles in a naturally occurring Bose-Einstein condensate first described in 1977 by the Scottish physicist Alexander Craigie while at the University of Lahore [3]. The quasiparticles are weakly bound by an interaction for which neither the position nor number operator commutes with the Hamiltonian. A measurement of a lazulino’s position will cause the condensate to go into a superposition of number states, and a subsequent measurement of the population will return a random number; also, counting the lazulinos at two different times will likely give different results.

Their name derives from the stone lapis lazuli and means, roughly, “little blue stone”. Lazulinos are so named because even though the crystals in which they arise absorb visible light, and would otherwise be jet black, they lose energy through surface plasmons in the form of near-ultraviolet photons, with visible peaks at 380, 402, and 417nm. Optical interference imparts a “laser speckle” quality to the emitted light; Craigie described the effect in a famously poetic way: “Their colour is the blue that we are permitted to see only in our dreams”. What makes lazulinos particularly interesting is that they are massive and macroscopic. Since the number operator does not commute with the Hamiltonian, lazulinos themselves do not have a well-defined mass; if the population is N, then the mass of any particular lazulino is m/N, where m is the total mass of the condensate.

In a recent follow-up to the “quantum mirage” experiment [2], Don Eigler’s group at IBM used a scanning tunneling microscope to implement “quantum mancala”—picking up the lazulino ‘stones’ in a particular location usually changes the number of stones, so the strategy for winning becomes much more complicated. In order to pick up a fixed number of stones, you must choose a superposition of locations [1].

1. C.P. Lutz and D.M. Eigler, “Quantum Mancala: Manipulating Lazulino Condensates,” Nature 465, 132 (2010).
2. H.C. Manoharan, C.P. Lutz and D.M. Eigler, “Quantum Mirages: The Coherent Projection of Electronic Structure,” Nature 403, 512 (2000). Images available at http://www.almaden.ibm.com/almaden/media/image_mirage.html
3. A. Craigie, “Surface plasmons in cobalt-doped Y3Al5O12,” Phys. Rev. D 15 (1977). Also available at http://tinyurl.com/35oyrnd.

## NY Times article on conditional probability

Posted in Uncategorized by Mike Stay on 2010 April 27

There’s actually very good justification for this method of reasoning: it maximizes entropy.

## Coends

Posted in Category theory, Math, Quantum by Mike Stay on 2010 April 11

Coends are a categorified version of “summing over repeated indices”. We do that when we’re computing the trace of a matrix and when we’re multiplying two matrices. It’s categorified because we’re summing over a bunch of sets instead of a bunch of numbers.

Let $C$ be a small category. The functor $\mbox{hom}:C^{\mbox{op}} \times C \to Set$ assigns

• to each pair of objects the set of morphisms between them, and
• to each pair of morphisms $(f:c \to c', h:d\to d')$ a function that takes a morphism $g \in \mbox{hom}(c', d)$ and returns the composite morphism $h \circ g \circ f \in \mbox{hom}(c, d')$, where $c, c', d, d' \in \mbox{Ob}(C).$

It turns out that given any functor $S:C^{\mbox{op}} \times D \to \mbox{Set},$ we can make a new category where $C$ and $D$ are subcategories and $S$ is actually the hom functor; some keywords for more information on this are “collages” and “Artin glueing”. So we can also think of $S$ as assigning

• to each pair of objects a set of morphisms between them, and
• to each pair of morphisms $(f:c \to c', h:d\to d')$ a function that takes a morphism $g \in S(c', d)$ and returns the composite morphism $h \circ g \circ f \in S(c, d')$, where $c,c' \in \mbox{Ob}(C)$ and $d,d' \in \mbox{Ob}(D).$

We can think of these functors as adjacency matrices, where the two parameters are the row and column, except that instead of counting the number of paths, we’re taking the set of paths. So $S$ is kind of like a matrix whose elements are sets, and we want to do something like sum the diagonals.

The coend of $S$ is the coequalizer of the diagram

$\begin{array}{c}\displaystyle \coprod_{f:c \to c'} S(c', c) \\ \\ \displaystyle S(f, c) \downarrow \quad \quad \downarrow S(c', f) \\ \\ \displaystyle \coprod_c S(c, c) \end{array}$

The top set consists of all the pairs where

• the first element is a morphism $f \in \mbox{hom}(c, c')$ and
• the second element is a morphism $g \in S(c', c).$

The bottom set is the set of all the endomorphisms in $S.$

The coequalizer of the diagram, the coend of $S,$ is the bottom set modulo a relation. Starting at the top with a pair $(f, g),$ the two arrows give the relation

$\displaystyle c \stackrel{f}{\to} c' \stackrel{g}{\multimap} c \stackrel{c}{\to} c \quad \sim \quad c' \stackrel{c'}{\to} c' \stackrel{g}{\multimap} c \stackrel{f}{\to} c',$

where I’m using the lollipop to mean a morphism from $S.$

So this says take all the endomorphisms that can be chopped up into a morphism $f$ from $\mbox{hom}$ going one way and a $g$ from $S$ going the other, and then set $fg \sim gf.$ For this to make any sense, it has to identify any two objects related by such a pair. So it’s summing over all the endomorphisms of these equivalence classes.

To get the trace of the hom functor, use $S = \mbox{hom}$ in the analysis above and replace the lollipop with a real arrow. If that category is just a group, this is the set of conjugacy classes. If that category is a preorder, then we’re computing the set of isomorphism classes.

The coend is also used when “multiplying matrices”. Let $S(c', c) = T(b, c) \times U(c', d).$ Then the top set consists of triples $(f: c\to c',\quad g:b \multimap c,\quad h:c' \multimap d),$ the bottom set of pairs $(g:b \multimap c, \quad h:c \multimap d),$ and the coend is the bottom set modulo

$(\displaystyle b \stackrel{g}{\multimap} c \stackrel{c}{\to} c, \quad c \stackrel{f}{\to} c' \stackrel{h}{\multimap} d) \quad \sim \quad (\displaystyle b \stackrel{g}{\multimap} c \stackrel{f}{\to} c', \quad c' \stackrel{c'}{\to} c' \stackrel{h}{\multimap} d)$

That is, it doesn’t matter if you think of $f$ as connected to $g$ or to $h$; the connection is associative, so you can go all the way from $b$ to $d.$

Notice here how a morphism can turn “inside out”: when $f$ and the identities surround a morphism in $S$, it’s the same as being surrounded by morphisms in $T$ and $U$; this is the difference between a trace, where we’re repeating indices on the same matrix, and matrix multiplication, where we’re repeating the column of the first matrix in the row of the second matrix.

## 5-axis mill

Posted in Uncategorized by Mike Stay on 2010 April 9

I really like the different tones on the metal, from polished to brushed, to whatever cool thing they did to get their name in lettering on the back.

## marginalia

Posted in Uncategorized by Mike Stay on 2010 April 8

Idea for an annotation engine:

• Annotation has the form (search query, regular expression, content)
• search query should be in a form where given the content and URL of a page you can tell if it ought to match the query.
• execute the search query; for each hit
• run the regex on the result; if it matches
• attach the content to \$1
• When an annotation is created, cache the first n URLs that the search query and regex succeed on
• When on a page and you want to know if it has annotations
• if the URL is in the cache or the current page’s content matches the search query,
• run the regexp; if it succeeds,
• attach the content
• cache the URL
• content post-processors
• media inclusion
• translation
• whatever
• notes on notes?
• rather than keep a local database of annotations, publish them on a blog tagged as an annotation
• then can annotate the blog
• concept pages?
• auto-linkify takes you to a concept page on the blog