Weak Bimonoids
Without going into the diagrams, which unfortunately is most of the story, I will say a little about the recent AusCat talk of Ross Street on weak bimonoids in braided monoidal categories, which he also spoke about at ANU recently. The study of these new diagrams was partly motivated by the work of the mathematical physicist Robert Coquereaux.
A monoid is given by an object A and arrows m and eta for multiplication and unit. The braiding enters in considering AoB to be a monoid. Dually, a comonoid has a comultiplication and counit. A bimonoid is a monoid A which also has a comonoid structure. The trick to defining a weak bimonoid is to carefully choose a self dual set of diagrams such that bimonoids are always weak bimonoids.
The aim is to relate this definition to a concept of quantum category. In this setting the term quantum involves a linearisation process (so this is not really a quantum gravity kind of quantum). A category is usually specified by source and target maps from C1 to C0, the arrow and object sets. On linearisation one moves into a category of vector spaces rather than sets, and the objects C1 and C0 are comonoids in this category.
How about working with more general monoidal categories? A quantum category is by definition such a category V, with maps s and t into opp(C0) and C0 respectively. Note that the condition of oppositeness is null in the case of vector spaces. This is a natural definition because it looks like the dual of a diagram defining a bialgebroid (I looked on Wiki but they haven't quite got this far). In this dual diagram we consider objects A and R, where A is a weak bimonoid and R is the analogue of C0.
The nice diagrams allow one to show that R can be recovered from the structure of A. The proof crucially uses the idempotents t and s, and the splitting of idempotents in the sense of a Karoubi envelope.
A monoid is given by an object A and arrows m and eta for multiplication and unit. The braiding enters in considering AoB to be a monoid. Dually, a comonoid has a comultiplication and counit. A bimonoid is a monoid A which also has a comonoid structure. The trick to defining a weak bimonoid is to carefully choose a self dual set of diagrams such that bimonoids are always weak bimonoids.
The aim is to relate this definition to a concept of quantum category. In this setting the term quantum involves a linearisation process (so this is not really a quantum gravity kind of quantum). A category is usually specified by source and target maps from C1 to C0, the arrow and object sets. On linearisation one moves into a category of vector spaces rather than sets, and the objects C1 and C0 are comonoids in this category.
How about working with more general monoidal categories? A quantum category is by definition such a category V, with maps s and t into opp(C0) and C0 respectively. Note that the condition of oppositeness is null in the case of vector spaces. This is a natural definition because it looks like the dual of a diagram defining a bialgebroid (I looked on Wiki but they haven't quite got this far). In this dual diagram we consider objects A and R, where A is a weak bimonoid and R is the analogue of C0.
The nice diagrams allow one to show that R can be recovered from the structure of A. The proof crucially uses the idempotents t and s, and the splitting of idempotents in the sense of a Karoubi envelope.
posted by Kea | 3:21 PM
9 Comments:
Hee hee, nice to get you back on CV. Someday physicists will realise where the good research is, and even SC will be on our side.
Re CV. Kipling is always appropriate:
Though we called your friend from his bed this night,
he could not speak for you,
For the race is run by one and one
and never by two and two.
That is, souls have to be saved (or lost) one at a time. I wrote Koide a two page letter that spoke to what he alone was doing.
My tastes are a little darker than Kipling, say maybe William Blake:
As I was walking among the fires of Hell, delighted with the enjoyments of Genius; which to Angels look like torment and insanity, I collected some of their Proverbs: thinking that as the sayings used in a nation, mark its character, so the Proverbs of Hell, shew the nature of Infernal wisdom better than any description of buildings or garments.
When I came home; on the abyss of the five senses, where a flat sided steep frowns over the present world, I saw a mighty Devil folded
in black clouds, hovering on the sides of the rock, with corroding
fires he wrote the following sentence now percieved by the minds of men, and read by them on earth:
How do you know but ev'ry Bird that cuts the airy way,
Is an immense world of delight, clos'd by your senses five?
from The Marriage of Heaven and Hell (1790)
On idempotency:
Quantum states are usually represented by state vectors, and these can't be put into the idempotency relation. However, density matrices can, and pure states satisfy the equation.
As an operator, idempotency is another word for "projection operator", so you can see it as a way of picking out a subset of the available states.
I like to think of quantum mechanics as defining a relationship between an initial state and a final state (while refusing to define what happens in betweeen). In this sense, an idempotent can be thought of as something that stays the same; the final is the same as the initial.
When I told Dr. Hestenes that I was modeling particles based on idempotents, he said that most people used nilpotents instead. He went on to say that there is a close relation between nilpotents and idempotents.
You need niplotents in fermion state vectors because you want the Pauli exclusion principle to apply. I'm convinced that the Pauli exclusion principle is just a low energy (i.e. << Planck mass) approximation, but at our energies that means that it is perfect.
In my understanding of physics, there is no quantum vacuum, it's just a mathematical convenience used for calculations. So there is no foundational need to require nilpotency in order to enforce the Pauli exclusion principle among the creation operators. I see the nilpotency as resulting only from the whole thing being an effective theory only.
On the subject of parity. The usual parity operator is a mapping from states to states. For this, I don't think that P^2 = 1 is always obtained. Instead, you will end up with an arbitrary phase.
There is another way of thinking about parity (mentioned in Landau and Lipshitz QED I think), and that is to define the parity operator as taking an eigenvalue of +1 on particles with an even internal parity, and -1 on particles with odd internal parity. Then parity does satisfy P^2 = 1, and the observed internal states are eigenstates of parity.
The operators that satisfy Q^2 =1 are closely connected to the idempotents by the following prescription:
\rho = 0.5(1 +- Q)
Then \rho (for either sign) is idempotent. And the idempotents are eigenstates of Q with eigenvalues +- 1. And you can take Q as the rare definition of parity.
12 13 06
Hello Kea and Carl:
I could not post earlier so I responded to Carl's statement here.
12 13 06
Carl Basically, Parity is an idempotent operator. See Sakurai for more elucidation.
Hi Kea:
Is the bimonster a bimonoid?
I found a definition for wreath product but am having difficulty visualizing this entity. Are you familiar with it?
I have become interested in error correcting Golay codes.
More at PF RE Witten paper.
Golay codes [GC12 and GC24] and the bimonster: [especially Appendix A]
The complex Lorentzian Leech lattice and the bimonster by Tathagata Basak
Comments: 24 pages, 3 figures, revised and proof corrected. Some small results added. to appear in the Journal of Algebra
Subj-class: Group Theory; Number Theory
MSC-class: 11H56, 20F55
Abstract: We find 26 reflections in the automorphism group of the Lorentzian Leech lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the refletion group.
http://arxiv.org/abs/math.GR/0508228
Hi Doug
That certainly sounds like an interesting paper! Thanks for the link.
Carl
In 1953, Freudenthal noticed that rank one projections are a special case of operators that satisfy Q*Q=0 (up to nonzero scalar multiple), where ' * ' is a generalized cross product (Freudenthal product). In fact, the more modern definitions of projective space use this nilpotent relation to describe points.
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