Panthalassa

Panthalassa is a hypothesised vast ocean that accompanied the land of ancient Pangaea. But did it really exist? Matti Pitkanen brings our attention to a video by Neal Adams, which you simply must watch. Here also is an interview with Neal Adams. I'm really not sure what I think of it all, because I have not checked the data and there is no doubt that some of his polemic (see example below) is outrageous, but it is an absolutely fascinating picture of the quantum Earth.

...cause all elements are manufactured on the inside Earth, all elements. Yes. All. What else? All!! Some? No, all! ALL!

Bigger and Bigger?

For centuries our view of the cosmos has grown in size, from a focus on the exploration of Earth's oceans, to a few stars in the firmament, to the recognition of spiral galaxies beyond our own in an almost endless array. The rational extension of this progress has brought us now to a notion of multiverse, which is often viewed rather pragmatically as a simple classical extension to even more stuff out there somewhere. In the news, one can see the Big Hole touted as evidence for a parallel universe.

But in quantum gravity, an effectively endless range of classical spaces is a very dull construct. If this was all there was, how could spacetime observables take on different values depending on experiment in this sector of the multiverse? How could we derive the particle masses and other parameters of the Standard Model? How could we really understand the cosmological horizon? Fortunately, an extension in size is not the only way to extend the realm of possibility.

M Theory Lesson 131

Speaking of the j invariant, recall that

$J(q) = j(q) - 744$

where $744 = 248 \times 3$, so in terms of the modular forms $\theta_0$ and $\theta_1$,

$J(q) = -3 [9 \frac{(\theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3})^{3}}{(\theta_{1}^{4} - \theta_{0}^{3} \theta_{1})^{3}} + 248]$

M theorists will recognise the number 248 as the dimension of the Lie group E8. Thus polynomials in $J(q)$ are a bit like polynomials in $\Theta$ and $\Pi$ (otherwise known as Eisenstein series $E_4$ and $E_6$). In fact, a classical result (mentioned in this paper) says that the modular discriminant of the denominator is given by

$\Delta = \frac{1}{1728} (\Theta^{3} - \Pi^{2})$

whereas the numerator is simply $\Theta^{3}$. The series $\Theta$ is sometimes written, for $q = e^{2 \pi i \tau}$ and $y = \textrm{Im} (\tau)$,

$\Theta (q) = 1 - \frac{3}{\pi y} - 24 \sum_{1}^{\infty} \frac{r q^{r}}{1 - q^{r}}$

For $y = 1$ (and ignoring the summation, which roughly equals 0.0018779 at $\tau = i$) this equals $1 - \frac{3}{\pi}$, which Louise Riofrio often likes to tell us is equal to 4.507034% (the observed fraction of baryonic matter). Is this another crackpot coincidence? Note that $0.0018779 \times 24$ is also equal to 4.507%, showing that $\Theta (q)$ is close to zero for $\tau = i$.

Mass Gap Again

Speaking of particle masses, let us recall the famous Mass Gap problem.

I am curious about something. As Distler pointed out in June, in a post discussing Witten's new $Z(q)$, which is polynomial in the $J$ invariant, Witten emphasized in his talk [on 2+1D gravity] that the gauge theory approach is wrong. Does this mean that the Mass Gap problem needs to be rephrased? After all, the official statement begins with the words: prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists. Why would we care about this nonsensical question if it's the wrong question? Surely the Mass Gap of interest is the physical one.

By the way, the Hoehn paper mentioned in Distler's post looks like an interesting application of lattices and codes.

Quick Hello

Carl Brannen has reminded us that we need to spend some time at the Particle Data Group website. As Michael Rios said a while back, if you wake up at night in a cold sweat having forgotten your particle data, there's nothing for it but to put some study hours in. Meanwhile, for the quote of the week we have this gem from Yatima at Woit's blog:

I think these discussions should be put on ice until the moment Theological Engineering finally manages to disburse Mana from vending machines or until we have regressed to the Dark Ages, whatever comes first.

M Theory Lesson 130

On page 149, Ebeling mentions that the characteristic polynomial (weight enumerator) of a self-dual ternary code is always a polynomial in $\Theta$ and $\Pi^{2}$, where $\Pi = \theta_{0}^{6} - 20 \theta_{0}^{3} \theta_{1}^{3} - 8 \theta_{1}^{6}$.

Now the functions $\theta_{0}$ and $\theta_{1}$ give a mapping of a hyperbolic quotient space for $\Gamma (3)$ to the Riemann sphere. The modular group (quotiented by $\Gamma (3)$) acts on this Riemann sphere by the symmetries of a tetrahedron, where fixed points are the vertices of the tetrahedron.

Note also that the usual generators $S$ and $T$ of the modular group act on polynomials in $\theta_{0}$ and $\theta_{1}$ by simple $2 \times 2$ matrices. In particular, $T$ is given by the phase gate matrix

1 0
0 exp$(\frac{2 \pi i}{3})$

All of this extends to interesting facts about a cube made of two tetrahedrons. Modular forms go from being weight 1 to being weight $\frac{1}{2}$ for the group $\Gamma (4)$. Using $\Gamma (4)$ we obtain the symmetries of a cube from the modular quotient. One can have hours of fun studying Ebeling's course in lattice theory.

M Theory Lesson 129

Jacques Distler bemoans the pitiable standards of the physics blogosphere in his post on Lisi's paper. Apparently, triality for the generations is a no starter. If one sticks to ordinary classical representation theory, no doubt this is correct.

But in M Theory one does not do this. The term triality is clearly a ternary analogue of the ubiquitous stringy term duality. Now let's look at a bit more lattice theory.

Recall (see Ebeling) that the subgroup $\Gamma (3)$ of $SL(2, \mathbb{Z})$ has modular forms $\theta_0$ and $\theta_1$ of the form, for $q = e^{2 \pi i z}$,

$\theta_0 = 1 + 6(q + q^3 + q^4 + 2 q^7 + q^9 + \cdots)$
$\theta_1 = 3 q^{\frac{1}{3}} (1 + q + 2q^2 + 2q^4 + \cdots)$

The theta function for the $E8$ lattice takes the form

$\Theta = \theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3}$

and this function appears three times in the celebrated j-invariant

$j = \frac{1728 (\theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3})^{3}}{- 4^{3} (\theta_{1}^{4} - \theta_{0}^{3} \theta_{1})^{3}}$

The group $\Gamma (3)$ appears naturally when studying ternary codes. The quotient of $SL(2, \mathbb{Z})$ by this group gives the group $PSL(2, F_{3})$, which is otherwise known as the group $A_4$, studied by Ernest Ma in his derivation of the tribimaximal mixing matrix.

M Theory Lesson 128

In Geometric Representation Theory lecture 13 you can hear James Dolan talking about braid diagrams and Hecke operators.

First, we think of Hecke operators as magic matrices with sets as elements. Composition of operators is sort of like a renormalised matrix multiplication. These can be redrawn as braid diagrams where we don't worry too much about the crossings. Matrix multiplication will be replaced by braid compositions. Dolan gave an example like this one: The dots on the top line come from the cardinalities $R$ and $B$, whereas the dots on the bottom line come from $R'$ and $B'$. The left hand strands represent the set in the top left box. Note that the total number of dots on the top and bottom is always the same. Braids with real crossings supposedly come in handy when considering not sets but vector spaces, or rather representations of groups like $GL(n, F_{q})$ over a finite field with $q$ elements (Langlands, anyone?). This links a simple set cardinality with a knot parameter $q$. But the knotty $q$ can take many complex values, most notably a complex root of unity. Fortunately, we already know that cardinalities can also take such values.

Another example, this time for a $3 \times 3$ matrix, shows how to associate an element of $B_3$ with a matrix whose entries sum to $3$. Sticking to the set interpretation, a zero is given by an empty square. M theorists will find such diagrams familiar by now. If you enjoyed lecture 13, in lecture 14 you can see John Baez write up the three matrices $\mathbf{1}$, $(231)$ and $(312)$ which underlie the mass Fourier transform.

M Theory Lesson 127

The Lie group $E8$ (which appeared in Lisi's TOE) is just a classical manifold, so at some point we will want to consider deformations of it. Returning to a much simpler group to begin with, recall that the $S_3$ permutation $(231)$ generates a basis for the $3 \times 3$ Fourier transform. That is, a general circulant is expressed in the form

$a_0 + a_1 (231) + a_2 (312)$

where $(312) = (231)^{2}$. The usual non-commutative replacement for $S_3$ is the braid group on three strands, $B_3$. The elements $(231)$ and $(312)$ are replaced by $\sigma_2 \sigma_{1}^{-1}$ and $\sigma_{1}^{-1} \sigma_2$ respectively. Thus a squaring of the element $(231)$ is replaced by a switching of the order of $B_3$ generators. Note also that an example of a $(231)$ braid is the Bilson-Thompson diagram for a left electron, whereas the right electron is represented by $(312)$. This suggests that a mass circulant is indeed associated to an operator ordering for left and right handed particles, if one takes seriously the utility of a braid analogue to the vector $(a_0, a_1, a_2)$.

Measurement E8

Carl Brannen's E8 applet now has a sidebar link. Also check out his new preon E8 blog post and don't forget the oodles of E8 facts at Tony Smith's site. For the best kindergarten introduction to this kind of symmetry see the Neighbourhood of Infinity.

Ternary Logic I

Alice stumbles into a garden where she comes across a strange looking man with writhing multicoloured hair, which is tied back in a braid with three strands.
Had Matter: Cup of tea? How do you like it?
Alice: Oh, yes thank you. Milk and no sugar. [Pauses] Excuse me, sir, but how did your hair get like that?
Had Matter: What's wrong with it?
Alice: I didn't say there was anything wrong with it. I was just wondering how it got like that.
Had Matter: Two sugars, was it?
Alice: No, no sugar, thank you. You are rather odd. What is it like being a mad hatter?
Had Matter: I'm not a mad hatter. Have you ever met a hatter? I'm Mr Matter. What's it like to matter, you ask? Would you like to try? I'd quite like to be an Alice.
Alice: How did you know my name was Alice?
Had Matter: No information escapes me! Here's your tea, with three sugars, as you like it.
Alice: Er, thank you very much. Yes, I'd like to try being a Mr Matter. Maybe we could exchange places?
Had Matter: Maybe we just did! Maybe I was Alice just a moment ago, only I don't remember being Alice, else I wouldn't be Mr Matter. I think I'm Mr Matter. Hmm.
Alice: Where's Bob? I need to send him a message.
Had Matter: I never invited Bob to my party. Come to think of it, I never invited you either. And you can't send a message, 'cause my butler's on a picnic.
Alice: Oh, I don't need a butler. It's all arranged...

Lisi Update II

Heh, Garrett just got named surfer dude in another headline, this time on Fox news. Cool, dude.

M Theory Lesson 126

As Ebeling explains, by taking the seven rows of the Hamming circulant and adding a check bit to each row we can write down the seven vectors

$v_1 = \frac{1}{\sqrt{2}} (0,1,1,0,1,0,0,1)$
$v_2 = \frac{1}{\sqrt{2}} (0,0,1,1,0,1,0,1)$
$v_3 = \frac{1}{\sqrt{2}} (0,0,0,1,1,0,1,1)$
$v_4 = \frac{1}{\sqrt{2}} (1,0,0,0,1,1,0,1)$
$v_5 = \frac{1}{\sqrt{2}} (0,1,0,0,0,1,1,1)$
$v_6 = \frac{1}{\sqrt{2}} (1,0,1,0,0,0,1,1)$
$v_7 = \frac{1}{\sqrt{2}} (1,1,0,1,0,0,0,1)$

in $\mathbb{R}^{8}$. These satisfy the rule for a root lattice, $v^{2} = 2$. With the change of variables $e_1 = v_1$, $e_2 = v_2 - v_1$, $e_3 = v_3 - v_2$, $e_4 = v_4 - v_3$, $e_5 = v_5 - v_4$, $e_6 = v_6 - v_5$ and $e_7 = v_7 - v_6$, and the addition of the vector

$e_8 = \frac{1}{\sqrt{2}} (-1,-1,0,0,1,0,-1,0)$

we have a basis for the $E8$ lattice. Since $u^{2} = 2$ for such vectors, it follows that $u \cdot v \in \{ 0, \pm 1, \pm 2 \}$. Hopefully these numbers are familiar.

Peirce and de Morgan

In Augustus de Morgan's Trigonometry and Double algebra from 1849, he replaces numbers by axiomatic algebraic symbols.

In the Single Algebra a symbol denotes a process either forwards or backwards, pictured as segments on a line. In order to deal with numbers of the form $a + ib$ de Morgan introduced the Double Algebra, where a line is given a direction in the plane. For a long time de Morgan tried to develop a Triple Algebra in the same spirit of pure logic, even after Hamilton showed that the Quadruple Algebra of quaternions was the next natural geometric step.

It was the originator of Category Theory, C. S. Peirce, who remained inspired by de Morgan's ideas and went on to develop the Theory of Signs and the triad philosophy. The under-appreciated Peirce is now recognised to have, amongst many other things, axiomatised arithmetic before Peano and to have discovered the ability of electrical circuits to do Boolean algebra.

M Theory Lesson 125

The $D_4$ triality in Garrett's paper (slide 16) leaves invariant the $W^{3}$. Deleting this component of the $4 \times 4$ rotation matrix reduces it to the matrix

010
001
100

which is the $(231)$ Fourier basis $3 \times 3$ circulant familiar to M theorists. Observe that this matrix permutes the $\frac{1}{2} \omega_{R}^{3}$, $\frac{1}{2} \omega_{L}^{3}$ and $B_{1}^{3}$ gravity fields.

Lisi Update

Not to be outdone by The Telegraph or New Scientist, The Sun uses the headline Computer Geek: I know it all. Intriguingly, despite the introduction

An internet nerd has solved the most baffling riddle of modern physics that stumped even Einstein – the theory of everything in the universe.

the Sun's readers did not find the story fascinating enough to push it onto the most-read stories column alongside Jessica Alba and a story on bra wars. Then again, let's give it a few hours. Motl also reports.

M Theory Lesson 124

In coding theory, a Tanner graph is a means of dealing with error correcting codes. Nodes on the graph are of two types: digit and subcode. For a simple linear code, the subcode nodes represent rows of the parity check matrix. Mao et al consider the utility of such graphs and their duality, as a kind of Fourier transform duality. The diagram shows two Tanner graphs: the first a multiplicative graph for the $(7,4)$ Hamming code and the second a convolution graph for the dual code. The $+$ bits denote the single bit parity check. Observe how similar the placing of these bits is to the Time vertices of the Space-Time hexagon. The full Hamming code generator matrix is then a $(7,1)$ matrix, with the $7 \times 7$ circulant spatial part.

I am now reading this article on the triality of a Hamming code VOA. This paper embeds the Hamming code operad into the $D_4$ one so that the $D_4$ triality (which appears in the graviweak sector of Garrett's work as the generation structure) restricts to the triality that permutes the mutually orthogonal conformal vectors for the Hamming code.

In his book, Quantum Fluctuations of Spacetime, L. B. Crowell discusses the Hamming code in the context of a Galois field representation associated to General Relativity. That is, the parallel transport of a spacetime vector in a finite element analysis is expressed in terms of field extensions associated to sets of points.

M Theory Lesson 123

A while back we saw how the E8 lattice in eight dimensions arises from the 7 bit Hamming code with a check bit. Lattices have theta functions, such as the j invariant, which exhibits an interesting triality. The quantum E8 Dynkin diagram of Coquereaux also exhibits a kind of triality on its 24 vertices, which we think of as three copies of a cube's 8 vertices. Three cubes make up the off diagonal elements of the $3 \times 3$ octonionic exceptional Jordan algebra. The three real diagonal elements bring the dimension up to 27. As Garrett says, maybe triality has something to do with the generation structure of standard model particles and why they have different masses.

Hanging In There

Heh, one minute he's banned from hep-th and next minute he's in the Telegraph under the heading Surfer dude stuns physicists with theory of everything. Go, Garrett. Update: Heh, and here's the New Scientist top story!

M Theory Lesson 122

Speaking of Garrett Lisi's paper, recall the pretty picture from lesson 119. This picture turns up on slide number 30 from Garrett's recent talk (slides here).

Lisi points out that in this picture we see the 27 elements of the exceptional Jordan algebra, converging for each color and anti-color. The central cluster of leptons and gauge fields are the 72 roots of the E6 subgroup of E8. It is also interesting that Lisi links the Dark Force directly to the fairy field.

Doctor Who?

At my age the years just flow by and sometimes very little seems to happen, but today I must fear that even Pioneer One may blacklist me, because I am now officially fudded. Actually, the graduation isn't until December 21, but whatever. Given how much paperwork seems to occur behind the brief email exchanges that I get, I've given up trying to interpret the technical meanings of terms like conferment or transcript. Meanwhile, the number of interesting links around here continues to grow and Garrett Lisi's paper has mysteriously reappeared on the hep-th arxiv. Wonders never cease.

M Theory Lesson 121

Ma continues with a consideration of $S_4$, the permutations on four letters. Recall that this group describes a 24 vertex permutohedron polytope in three dimensions, which is a hexagonal version of the Stasheff polytope for the pentagon. Ma thinks in terms of a SUSY seesaw model for mass matrices. On reduction of the parameters, his neutrino matrix now becomes a degenerate 1-circulant with $a$ on the diagonal.

We associated $3 \times 3$ 1-circulants with vertices of a hexagon, which comes from the cube. A cube has the symmetry group $S_4$, but in operad land it is more natural to relate the cube to the permutohedron via the Loday-Ronco maps.

M Theory Lesson 120

A non-blacklisted professional physicist from Riverside, Ernest Ma, has a 2006 paper on the arxiv about neutrino mixing matrices. He discusses a derivation of tribimaximal mixing from the groups $S_3$ and $A_4$, the even permutations on four objects.

This relies on describing the mass eigenvalues using parameters $a$, $b$ and $d$ for which the masses become $a - b + d$, $a + 2b$ and $-a + b + d$. Setting $d = 0$ looks wrong, because the first and third masses would be equal and of opposite sign, but note the similarity between $a - b$ and $a + 2b$ and the expressions $1 - r$ and $1 + 2r$ which appear in the circulant matrices. In fact, in the $d = 0$ case Ma's mass matrix reduces to a 2-circulant in $a$, $b$ and $b$.

NP2008

After the removal of Garrett Lisi's paper from the hep-th arxiv, Carl Brannen pointed out on Louise's blog that any publicity was good publicity, and that we should now announce a conference for the New Physics. So I hereby offer my Christchurch garden as an excellent venue for NP2008: a half hour bus ride into town, nearby walks, internet connection (I might need to borrow a computer, but people will have laptops anyway) and good weather outside the months June to September. What do you think?

Amongst those I would like to invite (as plenary speakers) are Carl Brannen, Michael Rios, Matti Pitkanen, Tony Smith, Louise Riofrio, Garrett Lisi, Laurent Freidel, Urs Schreiber, Bob Coecke and Lubos Motl. Unfortunately, you will all have to pay your own way, but a sleeping bag and a tent is a cheap option.

M Theory Lesson 119

Carl Brannen has related the Koide mass formula to a field energy expression, from which he has derived the tribimaximal mixing matrix. Just another unpublishable crackpot coincidence. This matrix (more or less) also appears in the new and much discussed paper by Garrett Lisi on an E8 theory of everything. Of course, TOEs are only theories of what little we knew before, but you must look at the pretty hexagonal pictures in the paper.

Quote of the Year

Talking to a brick wall with a PhD hanging from it is still talking to a brick wall.

Louise Riofrio

M Theory Lesson 118

Duality is everywhere, whether $T$ or $S$, categorical or Fourier. The process of integration hinges on a pairing between dual entities. For a simple monomial, the rule of integration $x^{n} \rightarrow \frac{1}{n + 1} x^{n + 1}$ looks a bit like the operator for (left) multiplication by $x$, except for the coefficient depending on $n$. Similarly, differentiation is like (left) multiplication by a factor of $\frac{1}{x}$.

An ordinary momentum operator $p$ is also like differentiation, up to a factor of $i \hbar$, which is more or less an integer $m$ when $\hbar$ belongs to a hierarchy. So if monomials in $x$ represent quantized paths in one dimension, then $p$ is a kind of dual to the operator that increases the path length by 1. That is, $p$ decreases the path length by 1. We could take care of the annoying factors by rescaling $x$ appropriately, as an operation is performed. Then for large values of $n$ (long paths) a momentum operation rescales $x$ almost by infinity!

What about noncommutative polynomials in two variables or more? Now one must specify whether left or right multiplication is being performed, but these operators are easy enough to define. For two variables $x$ and $y$ there would be four such operators for integration: $L_x$, $L_y$, $R_x$ and $R_y$. These operators also have simple multiplicative inverses, but a path may only be decreased by a step if the polynomial happens to start with the right letter. For example, the monomial $xyx^{3}$ can be multiplied by $\frac{1}{x}$ on the left, but not by $\frac{1}{y}$. That is, unless we allow negative components in the paths, but we didn't allow that in one dimension, where $\frac{d}{dx}$ applied to a constant just gives zero.

A duality between expansion and contraction may remind M theorists of either $T$ duality, which interchanges large and small scales, or the $S$ duality which turned up in our musings of the plane of Space and Time.

Wordpress II

All right. The latex on Wordpress is just so nice. But the new Arcadian Functor has a very limited upload capacity (for free) so I will be staying here with Blogger.

Blogger and Wordpress

The ability to upload pdf files has finally convinced me to start up a Wordpress blog. At present, however, posts will continue to be written here at Blogger, because I can customise things for free. Files for public viewing will be made available at Wordpress.

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.

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.

Connes-onical Time

In his latest post, Alain Connes comments on the canonical nature of time evolution for noncommutative spaces. In M Theory, the analogue should be a whole heirarchy of Planck constants. For example, the Weyl relations of the quantum torus depend on the parameter $\hbar$. These spaces are studied in a very nice paper from 1993 by Alan Weinstein, where $\hbar$ is associated to time evolution for a particle on an ordinary torus. The particle is initially concentrated at a point, for $\hbar = 0$, but quickly becomes non-localised.

Louise Riofrio also has many fascinating posts on an emergent thermodynamic Time associated to Planck's number. In M theory we may view this as an approximation to a 3-Time picture, which brings to mind the twistor triality of Sparling, or the 2-Time theory of Itzhak Bars. It seems that wherever we look, the canonical Arrow raises its head.

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

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

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)

Breeze

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.

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.