# Science is Vital, again…

 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

# Renaming the Higgs boson

 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

 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

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

# Lee Smolin in London

Lee Smolin is to mark the publication of his new book, “Time Reborn: From the Crisis of Physics to the Future of the Universe” [1] by giving a talk in London, hosted by the Institute of Physics. The talk will he held at the Institute’s offices in London from 6pm, 22 May 2013.

 In his new book Smolin suggests the laws of physics are not fixed, but rather they evolve in time. This hypothesis maybe a way of resolving some of the open questions in physics, such as the nature of the quantum mechanics and its unification with space-time and cosmology.

A poster for the talk can be downloaded from here (opens PDF)

If interested in attending, you need to register online here.

 Lee Smolin is a theoretical physicist who has been since 2001 a founding and senior faculty member at Perimeter Institute for Theoretical Physics. His main contributions have been so far to the quantum theory of gravity, to which he has been a co-inventor and major contributor to two major directions, loop quantum gravity and deformed special relativity Read more at Smolin’s homepage here.

Reference
[1]Lee Smolin, TIME REBORN: From The Crisis in Physics to the Future of the Universe, April 23, 2013

# Crackpot science at conferences

 There is an interesting article on the BBC website about crackpots at scientific conferences. The likes of Bayard Peakes are mentioned, as is the interesting case of Prof Schwartz who won a Nobel prize.

I have fortunately, not seen many crackpots at conferences that I have attended, but I have seem some…

# Odd curves and homological vector fields

 On a supermanifold one has not just even vector fields but also odd vector fields. Importantly, the Lie bracket of an odd vector field with itself does not automatically vanish.

This is in stark contrast to the even vector fields on a supermanifold and indeed all vector fields on a classical manifold. Odd vector fields that self-commute under Lie bracket are known as homological vector fields and a supermanifold equipped with such a vector field is known as a Q-manifold.

In the literature one is often interested in homological vector fields from an algebraic perspective. Indeed, the nomenclature “homological” refers to the fact that on a Q-manifold one has a cochain complex on the algebra of functions on the supermanifold. You should have in mind the de Rham differential and the differential forms on a manifold in mind here.

In fact, if we think of differential forms as functions on the supermanifold $$\Pi TM$$, then the pair $$(\Pi TM, d)$$ is a Q-manifold.

But can we understand the geometric meaning of a homological vector field?

Odd curves and maps between supermanifolds
Consider a map $$\gamma : \mathbb{R}^{0|1} \longrightarrow M$$, for any supermanifold $$M$$. We need to be a little careful here as we take $latex\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, that is we include odd maps here. Informally, we will use odd parameters at our free disposal. More formally, we need the inner homs, which requires the use of the functor of points, but we will skip all that.

Let us employ local coordinates $latex(x^{A})$ on $$M$$ and $$\tau$$ on $$\mathbb{R}^{0|1}$$. Then

$$\gamma^{*}(x^{A}) = x^{A}(\tau) = x^{A} + \tau \; \; v^{A}$$,

where $latex\widetilde{v^{A}} = \widetilde{A}+1$. This is why we need to include odd variables in our description. Note that as $$\tau$$ is odd, functions of this variable can be at most linear.

Aside One can now show that $$\Pi TM = \underline{Map}(\mathbb{R}^{0|1}, M)$$. Basically we have local coordinates $$(x^{A}, v^{A})$$ noting the shift in parity of the second factor. One can show we have the right transformation rules here directly.

Odd Flows
Now consider the flow on odd vector field, that is the differential equation

$$\frac{d x^{A}(\tau)}{d \tau} = X^{A}(x(\tau))$$,

where in local coordinates $$X = X^{A}(x)\frac{\partial}{\partial x^{A}}$$.

From our previous considerations the flow equation becomes

$$v^{A} = X^{A}(x) + \tau v^{B} \frac{\partial X^{A}}{\partial x^{B}}$$.

Thus equating the coefficients in order of $$\tau$$ shows that

$$X^{A}(x) = v^{A}$$ and $$v^{B} \frac{\partial X^{A}}{\partial x^{B}}=0$$.

Then we conclude that

$$X^{B} \frac{\partial X^{A}}{\partial x^{B}}=0$$, which implies that $$[X,X]=0$$ and thus we have a homological vector field.

Conclusion
The homological condition is the necessary and sufficient condition for the integrability of an odd vector field. Note that in the classical case there are no integrability conditions on vector fields.

# Leonhard Euler's birthday

Today, the 15th of April, is Euler’s birthday. Euler, a pioneer of modern mathematics, was born on April 15 1707, in Basel, Switzerland. His work introduced much of today’s modern notation. He worked on quite diverse areas such as mathematical analysis, geometry, number theory, graph theory and so on, as well as making massive impact in the world of physics in areas such as mechanics and optics.

Portrait by Johann Georg Brucker (1756)

Euler Biography (The MacTutor History of Mathematics archive)

# Faces Of Mathematics

The London Mathematical Society has a webpage in which they “take a look at some of the people who’ve made major contributions in the world of mathematics”. Indeed, it is always nice to put a face to the names we read about. My only comment about the page is that I do not think all the information is up to date, but never mind.

Faces of Mathematics, London Mathematical Society

# Einstein's spooky action at a distance in space.

The International Space Station, image courtesy of NASA

Scheidl, Wille and Ursin [1] have proposed using the International Space Station to test the limits of spooky action at a distance. These experiments could help develop global quantum communication systems.

Part of their plans include a Bell test experiment which probes the theoretical contradiction between quantum mechanics and classical physics. A pair of entangled photons would be generated on the Earth. One of these would then be sent to a detector aboard the International Space Station, while the other photon would be measured locally on the ground for comparison.

According to quantum physics, entanglement is independent of distance. Our proposed Bell-type experiment will show that particles are entangled, over large distances — around 500 km — for the very first time in an experiment…

Professor Ursin

It is also not really known if gravity plays any role in quantum entanglement. These experiments would be the first to really probe the potential effects of gravity.

References
[1] T Scheidl, E Wille and R Ursin, Quantum optics experiments using the International Space Station: a proposal, 2013, New J. Phys. 15 043008 (online here)