The New Functor

The latex is so nice here that I almost changed my mind about staying at Blogger. That is, until I found out that there is only a limited amount of free upload space on WordPress. The Old Functor still rules.

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Now Here

Now at another version of Arcadian Functor. All old posts have been imported, but the latex on them will be unreadable, so please continue to visit the Blogger blog. At present, this is a test site only. It is useful for uploading pdf files for public view, such as the example below. I will continue to use Blogger for now, because it allows a custom site for free.

Fourier Transform paper

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M Theory Lesson 117

A while back we looked at associating the number of strands in number theoretic braids with the size of the matrix operators in the Fourier transform, which is the same as the number of points on the circle (3 for mass). In the two strand case, the braids are easy to classify: m copies of the only generator, \sigma_{1}. In other words, an integer m labels all possible knots.

The homflypt polynomial for the torus knot \sigma_{1}^{2k + 1} is

p^{k} (1 + q^{2} (1 - p) \frac{(1 - q^{2k})}{(1 - q^{2})})

for two parameters q and p. Specialisations include p = q^{2} which results (effectively) in the Jones polynomial

q^{2k} (1 + q^{2} (1 - q^{2k}))

Since the endpoints of braid diagrams lie on a circle, there are two circles bounding a diagram. For the 3-strand case, there are two sets of three points defining the boundary, which thus looks like a 6 point torus.

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The Empire

I have been attempting to get a very concise and simple 2 page paper on the Fourier transform uploaded to the arxiv. First, I established electronically that I appeared to have posting rights only to the physics arxiv. Since I thought that hep-th might be more appropriate, I requested an endorsement from 2 people last week. One has yet to reply but the other, a highly respected professional theoretical physicist from the northern hemisphere, replied very promptly and sent an email to the arxiv that same day confirming his wish to act as my endorser.

Alas, the arxiv rules now require that endorsers be active users of hep-th, so my potential endorser was sent an email explaining that he wasn’t qualified to endorse for hep-th. If anyone who is qualified to endorse for hep-th would like to take up this case, it would be greatly appreciated. Anyway, people brilliant enough to spell my name correctly may obtain a copy of the paper here.

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Art and Science

On browsing the Serpentine Gallery of equations one can find the predictable Einstein equations and Standard Model actions, but my favourite was by Neil Gershenfeld: Other beauties include A = A, habitable planets and of course Dyson on the tau function. This last item is really cool: Dyson rediscovered for himself the identity

$\tau (n) = \sum \frac{(a – b)(a – c)(a – d)(a – e)(b – c)(b – d)(b – e)(c – d)(c – e)(d – e)}{1!2!3!4!}$

where $a,b,c,d,e$ are all possible numbers (respectively) equal to $1,2,3,4,5$ mod 5, satisfying

$a + b + c + d + e = 0$
$a^2 + b^2 + c^2 + d^2 + e^2 = 10n$

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M Theory Lesson 116

In quantum computation [1] the Fourier transform for $2^{n}$ basis states on $n$ qubits is implemented with Hadamard gates and unitary gates $G_{k}$ of the form

$1$ $0$
$0$ $\textrm{exp} \frac{2 \pi i}{2^{k}}$

which add a phase factor to $| 1 >$. A ternary analogue of such gates would be a $3 \times 3$ diagonal matrix with entries $1$, $\textrm{exp} \frac{2 \pi i}{3^{k}}$ and $\textrm{exp} \frac{4 \pi i}{3^{k}}$, responsible for adding phases to each of the three basis states. For example, for two ternary objects ($n = 2$) the central phase factor in $G_{2}$ is $\textrm{exp} \frac{2 \pi i}{9}$. Any number from 1 to 9 is expressable in base 3 as a combination of 1, 3 and 9, so the product representation of the qubit transform has a ternary analogue. This ability to switch from an additive expression to a product expansion in superpositions of qutrits is quite reminiscent of Euler’s relation for zeta functions. It is no surprise then that the quantum qubit transform is used in the algorithm for integer factorization.

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge (2000)

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A positive mention by Carl Brannen might have inspired a change in attitude from one of our esteemed colleagues. Mottle’s latest post includes the statement

Wegener’s wisdom about continents was very analogous to Darwin’s wisdom about evolving species that was formulated half a century earlier. Nevertheless, it was still hard for most people to swallow. Are we doing a similar error in another discipline today?

Well, it is probably obvious where I am going. The convoluted properties of the particle spectrum we observe may also be a result of some historical evolution, as eternal inflation combined with the landscape may suggest. But it doesn’t have to be so.

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Geon Eon

Carl Brannen’s guest post at Tommaso’s blog contains a link to the papers of Mark Hadley. In particular, there is a 1997 paper which considers particles as non-trivial topologies in spacetimes with closed timelike curves (4-geons).

It takes seriously the idea of obtaining non-classical logic from General Relativity. That is, propositions determining states do not necessarily obey the distributive law. Typical 2-valued propositions ask whether or not a certain region contains a particle. This is an argument for the requirement of higher dimensional toposes in gravity, because the logic of a 1-topos is always distributive, whether or not it is Boolean. Recall that distributivity in general is naturally expressed by a map $ST \Rightarrow TS$ for two monads $S$ and $T$.

The basic idea bears a little resemblance to Louis Crane’s geometrization proposal for four dimensional spin foam models, which allows for a relaxation of the manifold condition at a point. Hadley only discusses manifolds, for ordinary GR, as if quantization of gravity were unnecessary, but no analysis of solutions to Einstein’s equations is actually given. His conclusion is that gravitons do not exist, because as 3-geons they would lack the topological structure needed to localise them.

Note that Hadley’s later papers have even more grandiose titles, without much accompanying mathematical analysis.

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Tic Tac Toe II

A Slashdot report says that Vaughan Pratt, a well known computer scientist who works in Category Theory, claimed to have found an elementary error in Smith’s purported proof of the Wolfram Turing machine conjecture. The Wolfram response is available here. They claim the proof stands, although Smith did need to alter the definition of universality. Even better, Alex Smith himself replies to Pratt.

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Riemann Riddles

Given a preference for surreals over the reals as arguments for the Riemann zeta function, one cannot help but wonder about the association of 2-branchings with the parity cubes in all dimensions. At the third level, for instance, there are 8 nodes on the tree correponding to (a) 8 vertices of a cube, or (b) a set of zeta values.

Does this set of zeta values combine to obey a Koide type relation, just like the primary 3 faces of the space generation cube? This sounds like an idea to generate endless hours of play, but that may have to wait until I am elsewhere! The good news is that I’ve found the best internet cafe in the city, which opens early and has good cheap coffee. And on the short walk from the bus station to work, there is a garden by the river where I can sit in the sun by a statue of Captain Scott, engraved with the words, “I do not regret this journey…”

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