I just posted the slides for “Groupoidification and 2-Linearization”, the colloquium talk I gave at Dalhousie when I was up in Halifax last week. I also gave a seminar talk in which I described the quantum harmonic oscillator and extended TQFT as examples of these processes, which covered similar stuff to the examples in a talk I gave at Ottawa, as well as some more categorical details.

Now, in the previous post, I was talking about different notions of the “state” of a system – all of which are in some sense “dual to observables”, although exactly what sense depends on which notion you’re looking at. Each concept has its own particular “type” of thing which represents a state: an element-of-a-set, a function-on-a-set, a vector-in-(projective)-Hilbert-space, and a functional-on-operators. In light of the above slides, I wanted to continue with this little bestiary of ontologies for “states” and mention the versions suggested by groupoidification.

State as Generalized Stuff Type

This is what groupoidification introduces: the idea of a state in . As I said in the previous post, the key concepts behind this program are state, symmetry, and history. “State” is in some sense a logical primitive here – given a bunch of “pure” states for a system (in the harmonic oscillator, you use the nonnegative integers, representing n-photon energy states of the oscillator), and their local symmetries (the -particle state is acted on by the permutation group on elements), one defines a groupoid.

So at a first approximation, this is like the “element of a set” picture of state, except that I’m now taking a groupoid instead of a set. In a more general language, we might prefer to say we’re talking about a stack, which we can think of as a groupoid up to some kind of equivalence, specifically Morita equivalence. But in any case, the image is still that a state is an object in the groupoid, or point in the stack which is just generalizing an element of a set or point in configuration space.

However, what is an “element” of a set ? It’s a map into from the terminal element in , which is “the” one-element set – or, likewise, in , from the terminal groupoid, which has only one object and its identity morphism. However, this is a category where the arrows are set maps. When we introduce the idea of a “history “, we’re moving into a category where the arrows are spans, (which by abuse of notation sometimes gets called but more formally ). A span represents a set/groupoid/stack of histories, with source and target maps into the sets/groupoids/stacks of states of the system at the beginning and end of the process represented by .

Then we don’t have a terminal object anymore, but the same object is still around – only the morphisms in and out are different. Its new special property is that it’s a monoidal unit. So now a map from the monoidal unit is a span . Since the map on the left is unique, by definition of “terminal”, this really just given by the functor , the target map. This is a fibration over , called here for “phi”-bration, but this is appropriate, since it corresponds to what’s usually thought of as a wave mechanicsfunction .

This correspondence is what groupoidification is all about – it has to do with taking the groupoid cardinality of fibres, where a “phi”bre of is the essential preimage of an object – everything whose image is isomorphic to . This gives an equivariant function on – really a function of isomorphism classes. (If we were being crude about the symmetries, it would be a function on the quotient space – which is often what you see in real mechanics, when configuration spaces are given by quotients by the action of some symmetry group).

In the case where is the groupoid of finite sets and bijections (sometimes called ), these fibrations are the “stuff types” of Baez and Dolan. This is a groupoid with something of a notion of “underlying set” – although a forgetful functor (giving “underlying sets” for objects in a category ) is really supposed to be faithful (so that -morphisms are determined by their underlying set map). In a fibration, we don’t necessarily have this. The special case corresponds to “structure types” (or combinatorial species), where is a groupoid of “structured sets”, with an underlying set functor (actually, species are usually described in terms of the reverse, fibre-selecting functor , where the image of a finite set consists of the set of all “$\Phi$-structured” sets (such as: “graphs on set “, or “trees on “, etc.) The fibres of a stuff type are sets equipped with “stuff”, which may have its own nontrivial morphisms (for example, we could have the groupoid of pairs of sets, and the “underlying” functor selects the first one).

Over a general groupoid, we have a similar picture, but instead of having an underlying finite set, we just have an “underlying -object”. These generalized stuff types are “states” for a system with a configuration groupoid, in . Notice that the notion of “state” here really depends on what the arrows in the category of states are – histories (i.e. spans), or just plain maps.

Intuitively, such a state is some kind of “ensemble”, in statistical or quantum jargon. It says the state of affairs is some jumble of many configurations (which we apparently should see as histories starting from the vacuous unit ), each of which has some “underlying” pure state (such as energy level, or what-have-you). The cardinality operation turns this into a linear combination of pure states by defining weights for each configuration in the ensemble collected in .

2-State as Representation

A linear combination of pure states is, as I said, an equivariant function on the objects of . It’s one way to “categorify” the view of a state as a vector in a Hilbert space, or map from (i.e. a point in the projective Hilbert space of lines in the Hilbert space ), which is really what’s defined by one of these ensembles.

The idea of 2-linearization is to categorify, not a specific state , but the concept of state. So it should be a 2-vector in a 2-Hilbert space associated to . The Hilbert space was some space of functions into $mathbb{C}$, which we categorify by taking instead of a base field, a base category, namely . A 2-Hilbert space will be a category of functors into – that is, the representation category of the groupoid .

(This is all fine for finite groupoids. In the inifinte case, there are some issues: it seems we really should be thinking of the 2-Hilbert space as category of representations of an algebra. In the finite case, the groupoid algebra is a finite dimensional C*-algebra – that is, just a direct sum (over iso. classes of objects) of matrix algebras, which are the group algebras for the automorphism groups at each object. In the infinite dimensional world, you probable should be looking at the representations of the von Neumann algebra completion of the C*-algebra you get from the groupoid. There are all sorts of analysis issues about measurability that lurk in this area, but they don’t really affect how you interpret “state” in this picture, so I’ll skip it.)

A “2-state”, or 2-vector in this Hilbert space, is a representation of the groupoid(-algebra) associated to the system. The “pure” states are irreducible representations – these generate all the others under the operations of the 2-Hilbert space (”sum”, “scalar product”, etc. in their 2-vector space forms). Now, an irreducible representation of a von Neumann algebra is called a “superselection sector” for a quantum system. It’s playing the role of a pure state here.

There’s an interesting connection here to the concept of state as a functional on a von Neumann algebra. As I described in the last post, the GNS representation associates a representation of the algebra to a state. In fact, the GNS representation is irreducible just when the state is a pure state. But this notion of a superselection sector makes it seem that the concept of 2-state has a place in its own right, not just by this correspondence.

So: if a quantum system is represented by an algebra of operators on a Hilbert space , that representation is a direct sum (or direct integral, as the case may be) of irreducible ones, which are “sectors” of the theory, in that any operator in can’t take a vector out of one of these “sectors”. Physicists often associate them with conserved quantities – though “superselection” sectors are a bit more thorough: a mere “selection sector” is a subspace where the projection onto it commutes with some subalgebra of observables which represent conserved quantities. A superselection sector can equivalently be defined as a subspace whose corresponding projection operator commutes with EVERYTHING in . In this case, it’s because we shouldn’t have thought of the representation as a single Hilbert space: it’s a 2-vector in – but as a direct integral of some Hilbert bundle that lives on the space of irreps. Those projections are just part of the definition of such a bundle. The fact that acts on this bundle fibre-wise is just a consequence of the fact that the total is a space of sections of the “2-state”. These correspond to “states” in usual sense in the physical interpretation.

Now, there are 2-linear maps that intermix these superselection sectors: the ETQFT picture gives nice examples. Such a map, for example, comes up when you think of two particles colliding (drawn in that world as the collision of two circles to form one circle). The superselection sectors for the particles are labelled by (in one special case) mass and spin – anyway, some conserved quantities. But these are, so to say, “rest mass” – so there are many possible outcomes of a collision, depending on the relative motion of the particles. So these 2-maps describe changes in the system (such as two particles becoming one) – but in a particular 2-Hilbert space, say for some groupoid describing the current system (or its algebra), a 2-state is a representation of the of the resulting system). A 2-state-vector is a particular representation. The algebra can naturally be seen as a subalgebra of the automorphisms of .

So anyway, without trying to package up the whole picture – here are two categorified takes on the notion of state, from two different points of view.

I haven’t, here, got to the business about Tomita flows coming from states in the von Neumann algebra sense: maybe that’s to come.

Hello Reader,

So this week, so far, I’ve learned that I can measure time by distance and visa versa. Einstein’s theory of Relativity is quite interesting.

more at http://keithsmind.blogspot.com

Oops, I guess I dropped a minus sign!

This Wednesday, September 16th, 2009 at 5pm in PHSC 1105 Dr. Nikola Petrov, a professor in the math department at OU, will be giving a talk entitled:

“physics theory and Math for lazy people: From the non-existence of Godzilla to the energy of a nuclear explosion“

He’ll be talking about how a little common sense goes a long way when solving problems in the real world.  It’s bound to be a great talk!

And, as always, we’ll be eating Free Pizza!

If you’d like the flyer for this talk, you can find it here.

At least King Kong still exists, right?

I’ve been thinking about the uncertainty principle and trying to think of a classical answer as to how this apparently dual nature of subatomic structures can exist and came up with this thought experiment:
What if the “duality” of a particle is a false premise? What if there are no subatomic pointlike-particles such as the electrons, protons, neutrons, quarks, etc that we’ve been taught exists. What if instead of being pointlike-particles all these subatomic structures were all wave mechanicss? If this was the case then we mistakenly apply Heisenberg’s uncertainty principle for it only applies to pointlike-particles and not wave mechanicss.

Consider a wave mechanics that approaches a detection grid. As the wave mechanics nears the grid it is the wave mechanics front that shall contact the grid. The point where the wave mechanics-front contacts the grid is called the tangent for the grid touches the circumference of the wave mechanics-front at one point which it displays as a point of light leading (or rather misleading) us to believe it was a pointlike-particle that made contact and illuminated it. Remember that these subatomic structures are so small that they do indeed appear to be pointlike-particles and the wave mechanics-fronts which emanate/radiate from them while on a quantum level may be considered large are—in a real world scale—also extremely small so that even upon “seeing” these wave mechanicss—as in a cloud chamber—they still appear to be particles.

Borrowing the description of these structures from string theory—in which there are many parts I do not agree—we see “particles” are strings of energy which vibrate at various frequencies and which when coming into contact with other strings the sum totals of their combined frequencies add up to the various “particles” we know of. These frequencies interfere either constructively or destructively to give us the various frequencies of the various strings.
Now, consider just one such string traveling in a box toward a detection grid. As this string travels along its trajectory it is continuously radiating and so if a screen with one small hole is place between it and the detection grid the parts of the wave mechanicss that are radiated away from the central wave mechanics illuminates (though not necessarily in the visible spectrum) everything around it including the detection grid beyond the dividing screen which contains the small hole. The radiated wave mechanicss which all arrive at the detection grid ahead of the primary wave mechanics may not be detectable to the grid as these grids are not set to read such weak energy nevertheless these low energy wave mechanicss are there and causes the grid to act as a dark body which re-radiate these low energy wave mechanicss thereby laying the ground work which eventually lead/add to the interference patterns we see when we either add a second hole to the dividing screen or we move the hole in the screen to another position.

If we do not move the hole or add another hole then we do not see an interference pattern because the low energy wave mechanicss which are radiated away from the main wave mechanics-front are at intervals which do not cause interference with itself and so when each of them arrive at the detection grid they too touch the grid a manner which interfere constructively with the main wave mechanics-front.

If we simply change the position of the hole while leaving the trajectory of the string unchanged then the wave mechanicss which arrive at the detection grid arrive at a different point on the grid as a result of the change in the position of the hole. If the change in the position of the hole is sufficient enough (very likely as the only way it would not cause interference is if we moved the hole exactly one wave mechanics length away from the original position) it will cause an interference pattern.

Finally, if we add a second hole instead of moving the position of the hole and the trajectory of the string is exactly central to the position of the two holes then the string structure radiates through both holes at the same time and the wave mechanics-fronts which go through each of these holes—which are not exactly one wave mechanics length apart—interfere destructively to cause the wave mechanics patterns we witness on the detection grid.

Hypothetically, if we spaced (or moved) the holes exactly one wave mechanics length apart and set the string on a trajectory which would put it exactly central to the two holes then these separate wave mechanics-fronts should act constructively upon reaching the detection grid and we would not see any interference patterns.

Other thoughts:
There is no such thing as a perfect vacuum anymore than there is such a thing as absolute zero. You may get infinitesimally close to absolute zero just as you can get infinitely close to a perfect vacuum. I believe that if you can find a perfect vacuum there also you will find absolute zero.

Don’t forget that one of the first assingments you will need to complete is the Student Information Assignment on darkboard.  The purpose of this is to get the typical information from you that teachers often do on the 3×5 cards and also to get practice submitting assignments to me via darkboard.

The method we are using to submit assignements is called the Assignment Manager.  If you used the Digital DropBox in the past this is a little different and has some advantages.  It shouldn’t be any more difficult to do, however.

Please click on the link on the left entitled Assignments to take you to this assignment.  In there you will click on the “Student Information” Assignment and follow the directions.  You will download a MS Word Document that I provided, fill it out, and send it back to me via darkboard.

Now is the time to iron out any difficulties with this method.  Throughout the year we will be doing it just like this for some (not all) labs so it is good to learn how to do it now with a pretty low-key assignment.

Finally, I’ve noticed that sometimes in Firefox, there are errors submitting the file and you get a message like “please select a valid file…”  Unfortunately I do not know the solution to this and you may need to use Internet Explorer for this procedure.  Please see me with any questions, or, you can post a comment on this blog entry as well.

In one of my earliest posts I talked about linearly spanning the Hilbert-Schmidt space using physical states.  It turns out that to do this one needs a set projectors is larger than the set of orthogonal projections.  But orthogonality is a nice property for sake of calculations.  Therefore I (in the earlier post) relied on a set of matrices that I called the dual set satisfying , where is the set of projectors that span the space and is the dual set.

When talking to people about such a construction they always ask ‘can such dual set always be constructed?’  I will try to prove that the answer is yes by constructing the dual set in general.  This will probably take a few posts over next couple of weeks.

Conjecture: “Golden rule” of quantum-classic information. A gain in quantum algorithms is outweighed by losses in classical I/O and programing.

Los algoritmos cuánticos son sólo una parte de los ordenadores cuánticos. Además se requiere la entrada de datos (preparación de los cubits en el estado adecuado), programar (construir) la secuencia de puertas cuánticas que ejecuta el algoritmo y la salida de datos (lectura del estado final de los cubits). Estos procesos, hoy en día, son clásicos y requieren un alto costo en tiempo. Lo que se gana por un lado, se pierde por otro. Kisil conjetura que teniendo en cuenta el tiempo total los computadores cuánticos nunca serán más eficientes que los clásicos. Quizás se equivoque, quizás no. Nos lo cuenta en Vladimir V. Kisil, “Computation and Dynamics: Classical and Quantum,” ArXiv, Submitted on 8 Sep 2009.

Kisil nos recuerda que toda implementación física del algoritmo de factorización de Peter Shor requiere que reensamblar un circuito cuántico cada vez que en la entrada del algoritmo se introduce un número (pseudo)aletario. El coste de este reensamblaje (programación en palabras de Kisil) debe ser incluido en el coste total y en la práctica es muy alto. Lo mismo nos recuerda para el algoritmo de Grover para la búsqueda de números que requiere múltiples repeticiones en cada una de las cuales se destruye el contenido de la base de datos (cuando se mide, el estado cuántico colapsa). El resultado es que reescribir la base de datos (volver a preparar su estado cuántico) múltiples veces, con un costo mucho mayor que la ventaja obtenida con el algoritmo cuántico.

¿Podrán ser superadas estas barreras algún día? ¿Se podrán construir ordenadores cuánticos que no requieren de la intervención constante de procesos clásicos para su ejecución? Kisil es pesimista al respecto, de ahí su conjetura. La Mula Francis, por el contrario, se encuentra entre los optimistas.

Lorenzo Maccone’s argument (see here)  is on the hot list yet. Today, a paper by David Jennings and Terry Rudolph (Imperial College, London) appeared (see here) claiming Maccone’s argument being incorrect. Indeed, they write down Maccone’s argument as follows

“Any decrease in entropy of a system that is correlated with an observer entails a memory erasure of said observer”

but this erasure is provided by quantum correlations. The key point is the link between quantum correlations and local decrease of entropy as seen by classical correlations. Jennings and Rudolph interpret Maccone’s view as the reduction of information at a quantum level entails a reduction of information at a classical level and we do not observe such events. These authors show counterexamples where this does not happen arguing that Maccone’s argument does not explain rather worsens the problem as quantum correlations can decrease while classical ones can increase.

I guess that this comment will undergo the standard procedure of Physical Review Letters for it and Lorenzo Maccone will produce a counterargument facing in this way a review process. As it stand, it appears a substantial open problem to the original Maccone’s proposal but relies in an essential way on the interpretation Jennings and Rudolph attach to it.

Being this a really exciting matter, it will be really interesting to following the way events will take place.

Something about her veil, veiling her true smile

Smiling into my eyes, eyes trailing over the exposed shoulders

Of a tan student, taking notes idly, bored.

Student discounts, girls being grabbed while they want to dance,

I want to dance, untouched, unheeded, hands grabbing dark corners

Hands passing over curves , over mountains, feeling cavern walls

Where old tapestries of earth paint and blood paint mark

Old huntings and gatherings,

Winnings and losses.

Translucent coffee cup, invisible wall made by thick lenses

Surrounded by thick dark frames, my glasses,  my shield

My shield to deflect incoming stares,

My shield to reflect outgoing thoughts.

Something about the absence of friends, the absence of a group,

A thin veil of old memories, mostly forgotten, of kind smiles

Who pity the person you’ve become

But cherish the person you were.

Old promises like broken mirrors now, halfhearted and heartbroken

But still there, invisible, slicing into your skin every time you flex your muscles

And stretch out your arms, your back, side to side, in and out,

Shards of glass coating you, a thin exoskeleton of shimmering crystal

Reflecting the truth, hurting you more than hurting those hands

You tolerate with a heavy heart and an even heavier gaze.

Old photographs of a dog against the ocean,

Of the ocean against an even deeper ocean,

Blue, blue, blue, like his eyes, like the ocean, like the box

The box, the wooden painted box, blue, holding what’s left

Of brittle bones, sweet brown eyes, and a twisted, dark, dying stomach

I miss you too much, I couldn’t even voice a proper goodbye,

I just latched unto your ear, sure you could not feel through the pain,

Wishing you could hear me, that you could know my heart
And know that I would have been there when you woke up.

Shadows on a wall, shadows on the wood, in the carvings, in my heart

Not a hole, a shadow, a shadow of an abyss, where you are and I can’t whistle

To call you to me.

I am a turtle, la tortue, from the fairy tale,

I was the slow one, the slow one, slowly, slowly, marching forward

Sometimes, marching so slowly as to advance backwards.

The turtle whose shell makes it impossible to understand jokes

Like the others do, the turtle who wishes for a real solitude, occupied with cigarettes

And music, the heartbeat, my voice, the wind.

The wind is the melody of my voice, when I’m not really singing

But allowing the world to sing to me, the music of an empty house,

Listen into the silence, its obvious to me that me hates I.