## Science is Vital, again…

April 26th, 2013 by ajb
 As you may remember, back in 2010 Science is Vital organised a petition to protect the UK science budget. Thankfully funding for science and engineering was ring-fenced and frozen instead of being slashed.

However, inflation and cuts elsewhere in the UK government research budgets have eroded investment in science. This is making it difficult for the UK to maintain its reputation for scientific research. In June, the Government will announce its budget for 2015–16. As such Science is Vital has organised another petition to urge the Government to set a long-term target of raising R&D spending to at least 0.8% of GDP.

Science is Vital published a letter in the Daily Telegraph signed by over 50 leading scientists in the UK including Stephen Hawking and Brian Cox.

You do not have to be a scientist or similar to support this campaign, all you need to help is have some concern about how the UK may loose its world standing as a hub of scientific knowledge.

So on behalf of Dr Jenny Rohn (Chair Science is Vital) and all other interested parties I ask you to sign the petition and let the Government know that Science is Vital. (Follow the link below)

The current spend
According to Science is Vital [1]

• The current UK spend on public-funded research is 0.57% of GDP.
• The eurozone average is 0.74% of GDP, whilst the EU-27 average is 0.69%.

Reference
[1] Show me the numbers, Science is Vital Blog

## 97 per cent of climate scientists agree…

May 20th, 2013 by ajb
 John from wikipedia A study of 4000 abstracts of peer-reviewed papers published in the past 21 years that state a position on global has revealed an astonishing consensus; 97% of these endorsed the position that we are seeing anthropogenic global warming (AGW)[1].

Warning Science is not a democracy in the sense that if most scientists agree, then it must be correct.

Also note;

• 66.4% of abstracts expressed no position on AGW.
• 32.6% endorsed AGW.
• 0.7% rejected AGW.
• 0.3% were uncertain about the cause of global warming.

Reference
[1]John Cook et al (2013), Quantifying the consensus on anthropogenic global warming in the scientific literature, Environ. Res. Lett. 8 024024

## Hofstadter’s butterfly experimentally seen

May 17th, 2013 by ajb
 http://animatedcliparts.net/ Hofstadter’s butterfly is a fractal pattern that describes the behavior of electrons in a magnetic field. Such a fractal was predicted by Douglas Hofstadter, which he described in 1976 [1]. It is a very rare example of a fractal arising from quite fundamental physics.

However, all earlier attempts to experimentally observe see this pattern were unsuccessful. The wonder material graphene, first made by Andre Geim and Kostya Novoselov from the University of Manchester in 2004, came to the rescue…

For the first time ever a teams from Columbia University, the University of Manchester and MIT have experimentally observed this pattern [2,3,4].

Plot of electron density (horizontal axis) versus magnetic-field strength from data obtained by the Columbia team. (Courtesy: C R Dean et al. Nature 10.1038/nature12186)

Follow the link below to find out more.

Hofstadter’s butterfly spotted in graphene, PhysicsWorld.com

References
[1] Douglas R. Hofstadter (1976). “Energy levels and wavefunctions of Bloch electrons in rational and irrational magnetic fields”. Physical Review B 14 (6): 2239–2249.

[2] C. R. Dean et. al, Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices, Nature (2013) doi:10.1038/nature12186

[3] L. A. Ponomarenko et. al, Cloning of Dirac fermions in graphene superlattices, Nature (2013) doi:10.1038/nature12187

[4] B. Hunt et. al, Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure, arXiv:1303.6942 [cond-mat.mes-hall]

## Quantum physics and heavy metal

May 14th, 2013 by ajb
 I’m a middle-aged professor of physics and I love heavy metal. Professor Philip Moriarty.

Professor Moriarty of the University of Nottingham just loves heavy metal. In physicsfocus he explains that the quantum uncertainty is a byproduct of the inescapable fact that atoms have wave-like properties.

Follow the link below to find out more.

## G-torsors

May 14th, 2013 by ajb
 Let me quickly define a G-torsor.

Definition A G-torsor is a non-empty set X on which a group G acts freely and transitively.

A little more explicitly, X is a G-torsor if X is a nonempty set that is equipped with a map X × G → X such that
1) x·1 = x
2) x·(gh) = (x·g)·h

for all x in X, and h,g in G such that

(x,g) → (x, g·x)

is an isomorphism of sets.

Note that here we have picked the right action of G on X.

Remark One can modify these definitions to include categories other than sets, for example topological groups and spaces or even Lie groups and spaces.

Note that as we have an isomorphism, as sets, between X and G, they are equivalent objects. However, the subtlety is that there is no preferred identity point in X.

Ethos A G-torsor is a group that has lost its identity.

Once you have picked an identity element in X, you get an isomorphism as groups between X and G. This means that X and G are isomorphic as groups, but not canonically, a choice is needed.

What is the point of all this?
So it seems at first glance that torsors are very abstract objects far too complicated to be of much use to anyone. That is, until you realise that you have been using torsors without knowing it.

A good example of the use of a torsor is the potential difference in electromagnetism. When you measure a voltage, you in fact measure the difference of some voltage relative to some other fixed voltage. In practice one takes the ground to be zero, but this is a choice. Other values would work just as well. You can think of voltages as being elements of a torsor as there is no fixed identity voltage to measure against.

Energy in classical physics is very similar. The energy of a specified isolated system only really makes sense when one has set the “zero point energy”. One can only really measure energy differences relative to the “zero point energy”. This is why one can arbitrarily shift energies without effecting the physics. Actually, this is important when looking at the notion of energy in quantum field theory, but that is another story. Anyway, energies can be viewed as being elements of a torsor, you have no fixed “zero point energy” to measure all other energies against.

Physics is littered with similar examples.

A counter example would be temperature. We have a zero point temperature, that is absolute zero fixed for us.

Mathematics of course has lots of its own examples of torsors.

Consider a vector space V we can take G to be the general linear group GL(V) and X to be the set of all ordered bases of V. The group G acts transitively on X since any basis can be transformed via G to any other basis. In essence, one can take a specified basis and transform it into any other basis. Thus, one can consider all other bases as transformed versions of the initial basis. However, there is no natural choice of this “identity frame”. The set of bases do not form a group, but rather a torsor.

I will let the John Baez explain further here.

## Large meteor over South Wales

May 9th, 2013 by ajb

A meteor was spotted traveling across the night sky by people across many parts of the UK including Cornwall, Hampshire, Lancashire, Worcestershire and South Wales.

Of course, I missed it…

Large meteor spotted in night sky in England and Wales, BBC News

## Particle physics meets underground art!

May 9th, 2013 by ajb

The Institute of Physics has commissioned an artistic partnership between physicist Ben Still and artist Lyndall Phelps. A Victorian ice well beneath the London Canal Museum is to become a subterranean physics-inspired art installation this summer.

London Canal Museum, image by Oxyman.

Ben Still is a particle physicist based at Queen Mary, University London. He works on Super-Kamiokande, a neutrino detector which is part of the T2K experiment in Japan.

Victorian ice well to be home for detector-inspired art, IOP News

## Some photos of Staszic Palace

May 9th, 2013 by ajb

I bought a new camera the other day and here are a few pictures of Staszic Palace.

Outside the Staszic Palace, which seat the Polish Academy of Sciences. I am stood next to Bertel Thorvaldsen’s statue of Nicolaus Copernicus.

Now we have a closer picture of me next to Nicolaus Copernicus.

As you can see, Staszic Palace also houses the Warsaw Society of Friends of Learning.

Just a short walk down the road from Staszic Palace is the University of Warsaw.

If you visit Warsaw please take care with the wildlife… not really, this is one of the bears at Warsaw Zoo that are housed just outside the main zoo. You can see them from the street in Praga.

## Renaming the Higgs boson

April 23rd, 2013 by ajb
 Peter Higgs was treated as something of a rock star and the rest of us were barely recognised. It was clear that Higgs was the dominant name because his name has become associated with the boson. Prof Carl Hagen Rochester University, New York (BBC World News)

As you are all probably aware, on the 14th of March 2013 the ATLAS and CMS collaborations at CERN’s Large Hadron Collider (LHC) presented new results that further support the discovery of the so called Higgs boson [1].

This has all reignited the debate on the naming of the standard model scalar boson, as “Higgs” only reflects one of the physicists who made early contributions to the generation of mass within the standard model.

In 1964 Robert Brout & François Englert [2], Peter Higgs [4] and Gerald Guralnik , Carl Hagen & Tom Kibble [4] published papers proposing similar, but different mechanisms to give mass to particles in gauge theories, such as the standard model. All of the six physicists were awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics for their work on spontaneous symmetry breaking and mass generation.

2010 Sakurai Prize Winners – (L to R) Kibble, Guralnik, Hagen, Englert, and Brout

The trouble now is that the nomenclature Higgs boson has been around for a while now and remaining it could probably course unnecessary confusion. Also, it is not clear who should decide the new name. Nomenclature in physics, and indeed mathematics, arises largely due to popular usage. The initial name may come from the discoverer, but it still takes the community to use this nomenclature before it becomes standard. Thus, the community would have to change nomenclature and this cannot really be imposed from “outside”.

Naming the particle after the six physicists as BEHGHK-boson, which would be pronounced “berk-boson” is one one possible solution, but not a very nice one!

Pallab Ghosh (Science correspondent, BBC News) discusses this further here.

References
[1] New results indicate that particle discovered at CERN is a Higgs boson, CERN press office 14th March 2013.

[2] Englert, F.; Brout, R. (1964). “Broken Symmetry and the Mass of Gauge Vector Mesons”. Physical Review Letters 13 (9): 321.

[3] Higgs, P. (1964). “Broken Symmetries and the Masses of Gauge Bosons”. Physical Review Letters 13 (16): 508.

[4]Guralnik, G.; Hagen, C.; Kibble, T. (1964). “Global Conservation Laws and Massless Particles”. Physical Review Letters 13 (20): 585.

## Compatible homological vector fields

April 23rd, 2013 by ajb
 In an earlier post, here, I showed that the homological condition on an odd vector field $Q \in Vect(M)$, on a supermanifold $M$, that is $2Q^{2}= [Q,Q]=0$, is precisely the condition that $\gamma^{*}x^{A} = x^{A}(\tau)$, where $\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, be an integral curve of $Q$.

A very natural question to answer is what is the geometric interpretation of a pair of mutually commuting homological vector fields?

Suppose we have two odd vector fields $Q_{1}$ and $Q_{2}$ on a supermanifold $M$. Then we insist that any linear combination of the two also be a homological vector field, say $Q = a Q_{1} + b Q_{2}$, where $a,b \in \mathbb{R}$. It is easy to verify that this forces the conditions

$[Q_{1}, Q_{1}]= 0$, $[Q_{2}, Q_{2}]= 0$ and $[Q_{1}, Q_{2}]= 0$.

That is, both our original odd vector fields must be homological and they mutually commute. Such a pair of homological vector fields are said to be compatible. So far this is all algebraic.

Applications of pairs, and indeed larger sets of compatible vector fields, include the description of n-fold Lie algebroids [1,3] and Q-algebroids [2].

The geometric interpretation
Based on the earlier discussion about integrability of odd flows, a pair of compatible homological vector fields should have something to do with an odd flow. We would like to interpret the compatibility of a pair of homological vector fields as the integrability of the flow of $\tau = \tau_{1} + \tau_{2}$. Indeed this is the case;

Consider $\gamma^{*}_{\tau_{1} + \tau_{2}}(x^{A}) = x^{A}(\tau_{1} + \tau_{2}) = x^{A}(\tau_{1}, \tau_{2})$, remembering that we define the flow via a Taylor expansion in the “odd time”. Expanding this out we get

$x^{A}(\tau_{1}, \tau_{2}) = x^{A} + \tau_{1}\psi_{1}^{A} + \tau_{2}\psi_{2}^{A} + \tau_{1} \tau_{2}X^{A}$.

Now we examine the flow equations with respect to each “odd time”. We do not assume any conditions on the odd vector fields $Q_{1}$ and $Q_{2}$ at this stage.

$\frac{\partial x^{A}}{\partial \tau_{1}} = \psi_{1}^{A} + \tau_{2}X^{A} = Q_{1}^{A}(x(\tau_{1}, \tau_{2}))$
$= Q_{1}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}}$,

and
$\frac{\partial x^{A}}{\partial \tau_{2}} = \psi_{2}^{A} {-} \tau_{1}X^{A} = Q_{2}^{A}(x(\tau_{1}, \tau_{2}))$
$= Q_{2}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}}$.

Then equating coefficients in order of $\tau_{1}$ and $\tau_{2}$ we arrive at three types of equations

i) $\psi_{1}^{A} = Q_{1}^{A}$, $\psi^{B}_{1} \frac{\partial Q_{1}^{A}}{\partial x^{B}}=0$ and $\psi_{2}^{A} = Q_{2}^{A}$, $\psi^{B}_{2} \frac{\partial Q_{2}^{A}}{\partial x^{B}}=0$.

ii) $X^{A} = \psi^{B}_{2} \frac{\partial Q_{1}^{A}}{\partial x^{B}}$ and $X^{A} = {-}\psi^{B}_{1} \frac{\partial Q_{2}^{A}}{\partial x^{B}}$.

iii) $X^{B} \frac{\partial Q_{1}^{A}}{\partial x^{B}} =0$ and $X^{B} \frac{\partial Q_{2}^{A}}{\partial x^{B}} =0$.

It is now easy to see that;

i) implies that $[Q_{1}, Q_{1}] =0$ and $[Q_{2}, Q_{2}] =0$ meaning we have a pair of homological vector fields.

ii) implies that $[Q_{1}, Q_{2}]=0$, that is they are mutually commuting, or in other words compatible.

iii) is rather redundant and follows from the first two conditions.

Thus our geometric interpretation was right.

References
[1] Janusz Grabowski and Mikolaj Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59(2009), 1285-1305.

[2]Rajan Amit Mehta, Q-algebroids and their cohomology, Journal of Symplectic Geometry 7 (2009), no. 3, 263-293.

[3] Theodore Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys. 2012; 315:279-310.