Neutrino Boom

New Scientist has stories on both the new MINOS results and the new MiniBooNE antineutrino results.

The MiniBooNE experiment has confirmed old results from the LSND experiment, namely that muon antineutrinos oscillate into electron antineutrinos at a higher rate than expected. Extra sterile neutrinos are often proposed as an explanation for this anomaly, but many other explanations are possible. It is interesting to note that the antineutrino mass triplet has a different ordering of masses to the neutrino triplet. The heaviest antineutrino corresponds to the central eigenvalue of the triplet. This is the heaviest of all neutrino and antineutrino mass states.

Working Facts

According to Human Rights experts:

It is (also) unlawful for employers to discriminate on grounds of sex against a worker, or subject the worker to harassment:
* In the terms of employment provided
* In the way they make opportunities for training, promotion or transfer, and other benefits/facilities available
* By refusing access to such opportunities or benefits/facilities

Quote of the Month

From Lubos Motl on Tommaso's blog:

Even today, we wouldn't have the right theory if we were only using experiments and the old conceptual ways of thinking.

MINOS Neutrinos IV

Recall that the scale $\mu$ for the Koide matrix of both neutrinos and antineutrinos was the same value. This can be seen in the eigenvalue relations where the selected minus sign is that used by Carl Brannen in his old paper on neutrino masses. This term is the lightest (at $0.00038$ eV) of all neutrinos and antineutrinos. Similarly, both mass sums $\Sigma m_{i}$ are close to $0.06$ eV. The Koide ratio $3/2$ is now obtained from $0.3^{2}/0.06$, and so it applies to both neutrinos and antineutrinos.

Quote of the Week

Thanks to FSP, from the article The End of Men:

For a long time, evolutionary psychologists have claimed that we are all imprinted with adaptive imperatives from a distant past ... this kind of thinking frames our sense of the natural order. But what if men and women were fulfilling not biological imperatives but social roles, based on what was more efficient throughout a long era of human history? What if that era has now come to an end?

FFP10 Proceedings

The FFP10 conference proceedings are now available online from AIP, at least to those fortunate enough to have access. There is a paper by me and one by Carl. Happy reading.

MINOS Neutrinos III

A summary plot from MINOS: Observe that the neutrino angle is consistent with $\pi / 4$, whereas the antineutrino angle is closer to $\textrm{sin} \theta = 1/ \sqrt{3}$.

The blogosphere is being fairly quiet about this, presumably because almost nobody seriously believes the result, but other random reports come from a press release, the Physics World blog and the ICHEP blog.

MINOS Neutrinos II

Expanding on my last comment, recall that Carl Brannen's phase for the neutrino Koide mass matrix is

$\delta = \frac{2}{9} + \frac{\pi}{12}$

which results in the eigenvalue set

$\sqrt{m_{i}} = \mu (1 + \sqrt{2} \textrm{cos} (\delta + \omega_i))$

for $i = 1,2,3$ and $\omega_i$ the three cubed roots of unity. The value $2/9$ and the factor $\sqrt{2}$ are shared by both neutrinos and charged leptons. This $\delta$ gives three neutrino masses and a $\Delta m^2$ in agreement with known measurements. If, on the other hand, we took a phase

$\delta = \frac{2}{9} - \frac{\pi}{12}$

for the same scale $\mu$ and same eigenvalue equation, we obtain three masses with a $\Delta m^2$ in perfect agreement with the new MINOS results. Note that the $\overline{m}_{2}$ mass dominates the $\Delta$, so it does not much matter if we are considering $\overline{m}_{1}$ or $\overline{m}_{3}$.

It is not yet clear why conjugation on the $\pi /12$ should correspond to antineutrinos, but recall that conjugation is a standard feature of phases for antiparticles in the braid spectrum. The $2/9$ component appears to be shared by all particles, irrespective of their quantum numbers.

MINOS Neutrinos

As reported by Philip and many others, the MINOS experiment has been releasing results at Neutrino 2010. They claim a $2$ sigma observation of a difference between (electron, muon) $\Delta m^2$ for neutrinos and antineutrinos. That is,
neutrinos have $\Delta m^2 = 2.35 \times 10^{-3} \pm 0.11$ $eV^2$
antineutrinos have $\Delta m^2 = 3.35 \times 10^{-3} \pm 0.45$ $eV^2$
with
neutrino parameter $\textrm{sin}^2 (2 \theta) > 0.91$
antineutrino parameter $\textrm{sin}^2 (\overline{2 \theta}) = 0.86 \pm 0.12$
where $\theta$ is the $2 \times 2$ matrix phase for two neutrino mixing. But the plot shows that one should not be too enthusiastic yet about such a radical result. The data points are for antineutrinos and the dashed line is the neutrino result. Given that such a result could demolish almost everything that has been written about quantum gravity, we live in interesting times.

Polynomial Functors

Over at the cafe, the mathematician Joachim Kock has been talking about polynomial functors. Think of these as polynomials in sets, although we could draw the diagrams in any suitable category. For example, a one variable polynomial $\Sigma_{a \in A} x^{B(a)}$ is given by a function $f: B \rightarrow A$, as in the diagram where $*$ is a terminal one point set and $!$ the unique map to this set. Allowing several variables (indexed by $I$) and families of polynomials (indexed by $J$) the diagram becomes where we use $s$ and $t$ to stand for source and target, since triangles like the one above are supposed to make us think of comma categories. In a topos like Set, some polynomials with $f$ monic (one to one) arise from the characteristic squares where we have chosen $J$ to be the two point set and $I$ the one point set. Observe that for finite sets, the one to one condition says that the allowed powers in the polynomial (i) only occur once and (ii) are restricted by the cardinality of $A$. Property (i) essentially says that only coefficients of $1$ or $0$ are allowed, which means we are looking at something like the two element field. Property (ii) relates to the number of copies of two, that is a $2^{n}$ element field, as in two qubits or four qubits and so on. The object $\Omega$ gives us a pair of such polynomials, which are complementary to each other in the sense that one characterises $B$ and the other the complement of $B$.

Thus polynomial functors are one hint that sets with a prime number of elements play a special role in categorified arithmetic, just as the prime ordinals are special, although in the usual category Set one does not worry about primes.

Around the House

My broadband account is running low again. Misty the cat comes to visit me every day now. She does not like being left alone when the ice is an inch thick in the garden. Misty is a talented bird hunter, but not too good at catching the genius mice. People are more reluctant to pick up hitchhikers these days, with the ski season about to begin. Of course, I will not be going skiing this year, or probably any other year. The winter jobs have all been taken, but I might be able to find a cafe or cleaning job if I move towns again soon. Then again, I am not as young as I once was, and people ask too many questions about where I've come from. My accent is such a mixed up antipodean one that I am a foreigner where ever I go, and besides, I cannot honestly call anywhere home.

The last few days have been sunny and still, for a nice change. Right on the winter cue, the dam is pumping high volumes and the power company doubled its prices. Luckily I still have a little mountain gear. The minimum wage is becoming quite unlivable here, but I would feel relatively rich if I actually had a job. For decades people have lectured me on what I should do to get ahead. Wow, look at the world they have created.

QPL 2010

Talks from QPL 2010 are now online. I am unable to watch any of them at present, but talks that might be interesting include those by:
Bruce Bartlett
Sanjeevi Krishnan
Chris Heunen
John Barrett
Louis Crane

Unstrings for Dummies

Picture Perfect

Universe Today reports on evidence that planets can form more quickly than previously thought. Beta Pictoris b is a Jupiter class exoplanet whose star is around $12$ million years old. News at ESO.

M Theory Lesson 336

Recall that the two factor mixing products corresponded to a new kind of magic matrix. For example, the tribimaximal values may be rearranged as where the number $6$ replaces the zero entry of the matrix. Such magic squares have both sum and product rules. For example, the tribimaximal values give us
$\frac{2}{3} \times \frac{1}{2} = \frac{1}{3} $
$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6} $.
Similarly, the CKM values give us the product rules
$(ud) \times (tb) = (cs) $
$(us) \times (cb) = (td)$.
The probability matrices are obtained from these magic squares by rearranging only a few values. One way that we might draw such a magic square is:

Quote of the Decade

From the current session of U.S. Congress:

Quote of the Week

Eileen Byrne points out that if a plant doesn't succeed in a garden, we ask what it is about the soil, water, sun or fertilizer that is causing the problem. We don't blame the plant first.

M Theory Lesson 335

Carl Brannen's spin path integral paper is now available online at Foundations of Physics.

The spin path integral is closely associated with ideas of the mathematician Kapranov about noncommutative integrals, as discussed in his 2006 paper. Recall that Kapranov uses noncommuting directions to build paths, just as in Carl's paper. At the Streetfest in 2005, Kapranov discussed categorical aspects of noncommutative integrals. Now Carl's paper directly links the particle generations to the third categorical dimension. Long live the parity cube and broken monoidal structures.

Twistors Again

Last year at Oxford, I was fortunate enough to attend a number of interesting String Theory seminars in the mathematics department. Actually, the seminars were all about twistor theory, because the local string theorists said that twistor theory was where all the interesting stuff was happening. A typical seminar would begin with a T duality transformation into a twistor space, and after this neither string theory nor the usual local formulation of the Standard Model made any appearance whatsoever.

Recall that algebraic aspects of twistors include the relation between its geometry and Jordan algebra. As kneemo points out, the Koide mass matrices may be viewed as elements of a Jordan algebra. Similarly, the $R_2$ factors used to build mixing matrices may be associated to Jordan algebras.

Some time ago, we discussed the relevance of the three moduli spaces of twistor dimension (namely $M(0,6)$, $M(1,3)$ and $M(2,0)$) with a total of $12$ degrees of freedom. Euler characteristics and the index theorem for the six point sphere then tell us that the number of generations must be $3$. This is a ternary analogue of the pair $M(0,3)$ and $M(1,1)$ of the Grothendieck tower, an idea associated to the ribbon graph papers of Mulase et al.

New Rios Paper

On the subject of information theoretic black holes, Michael Rios has a new arxiv preprint: Jordan $C^{*}$ algebras and Supergravity. From the conclusion:

Surely there are further applications for Jordan algebraic structures based on the bioctonions, and it is interesting to consider the direct physical interpretations of such structures in M Theory.

Best Maths Blogs

Lieven Le Bruyn bemoans not being selected one of the Best 50 Maths Blogs, for maths majors, at onlinedegree. But Arcadian Functor made the list!