Weighted Lie groupoids

February 24th, 2015 by ajb
 In collaboration with K. Grabowska and J. Grabowski, we have examined the finite versions of weighted algebroids which we christened ‘weighted Lie groupoids’.

Groupoids capture the notion of a symmetry that cannot be captured by groups alone. Very loosely, a groupoid is a group for which you cannot compose all the elements, a given element can only be composed with certain others. In a group you can compose everything.

Groups in the category of smooth manifolds are known as Lie groups and similarly groupoids in the category of smooth manifolds are Lie groupoids.

It is well-known every Lie groupoid can be ‘differentiated’ to obtain a Lie algebroid, in complete analogy with the Lie groups and Lie algebras. The ‘integration’ is a little more complicated and not all Lie algebroids can be globally integrated to a Lie groupoid. Recall that for Lie algebroids we can always integrate them to a Lie group.

Previously we defined the notion of a weighted Lie algebroids (and applied this to mechanics) as a Lie algebroid with a compatible grading. A little more technically we have Lie algebroids in the category of graded bundles. The question of what such things integrate to is addressed in our latest paper [1].

Lie groupoids in the category of graded bundles
The question we looked at was not quite the integration of weighted Lie algebroids as Lie algebroids, but rather what extra structure do the associated Lie groupoids inherit?

We show that a very natural definition of a weighted Lie groupoid follows as a Lie groupoid with a compatible homogeneity structure, that is a smooth action of the multiplicative monoid of reals. Via the work of Grabowski and Rotkiewicz [2] we know that any homogeneity structure leads to a N-gradation of the manifold in question; and so what they call a graded bundle.

The only question was what should this compatibility condition between the groupoid structure and the homogeneity structure be? It turns out that, rather naturally, that the condition is that the action of the homogeneity structure for a given real number be a morphism of Lie groupoids. Thus, we can think of a weighted Lie groupoid as a Lie groupoid in the category of graded bundles.

I will remark that weighted Lie groupoids are a nice higher order generalisation of VB-groupoids, which are Lie groupoids in the category of vector bundles. These objects have been the subject of recent papers exploring the Lie theory and application to the theory of Lie groupoid representations. I direct the interested reader to the references listed in the preprint for details.

Some further structures
Following our intuition here that weighted versions of our favourite geometric objects are just those objects with a compatible homogeneity structure in [1] we also studied weighted Poisson-Lie groupoids, weighted Lie bi-algebras and weighted Courant algebroids. The classical theory here seems to be pushed through to the weighted case with relative ease.

Contact and Jacobi groupoids
The notion of a weighted symplectic groupoid is clear: it is just a weighted Poisson groupoid with an invertible Poisson, thus symplectic, structure. By replacing the homogeneity structure, i.e. an action of the monoid of multiplicative reals, with a smooth action of its subgroup of real numbers without zero one obtains a principal $\mathbb{R}^{\times}$-bundle in the category of symplectic (in general Poisson) groupoids. Following the ideas of [3] this will give us the ‘proper’ definition of a contact (Jacobi) groupoid. We will shortly be presenting details of this, so watch this space.

References
[1] Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, Graded bundles in the category of Lie groupoids, arXiv:1502.06092

[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36

[3] Janusz Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013), 27–58.

Supersymmetry and mathematics

February 15th, 2015 by ajb
 Prof Beate Heinemann, from the Atlas experiment at CERN had said that they may detect supersymmetric particles as early as this summer. But what if they don’t?

What if nature does not realise supersymmetry? Has my interest in supermathematics been a waste of time?

Superysmmetry

We hope that we’re just now at this threshold that we’re finding another world, like antimatter for instance. We found antimatter in the beginning of the last century. Maybe we’ll find now supersymmetric matter

Prof Beate Heinemann [1]

In nature there are two families of particles. The bosons, like the photon and the fermions, like the electron. Bosons are ‘friendly’ particles and they are quite happy to share the same quantum state. Fermions are the complete opposite, they are more like hermits and just won’t share the same quantum state. In the standard model of particle physics the force carriers are bosons and matter particles are fermions. The example here is the photon which is related to the electromagnetic force. On the other side we have the quarks that make up the neutron & proton and the electron, all these are fermions and together they form atoms.

Supersymmetry is an amazing non-classical symmetry that relates bosons and fermions. That is there are situations for which bosons and fermions can be treated equally. Again note the very different ‘lifestyle’ of these two families. If supersymmetry is realised in nature then every boson will have a fermionic partner and vice versa. In one swoop the known fundamental particles of nature are (at least) doubled! Moreover, the distinction between matter and forces becomes blurred!

A little mathematics
Without details, the theory of bosons requires the so called Canonical Commutation Relation or CCR. Basically it is given by

$[\hat{x},\hat{p}] = \hat{x} \hat{p} – \hat{p} \hat{x} = i \hbar$.

Here x ‘hat’ is interpreted as the position operator and p ‘hat’ the momentum. The right hand side of this equation is a physical constant called Planck’s constant (multiplied by the complex unit, but this is inessential). The above equation really is the basis of all quantum mechanics.

The classical limit is understood as setting the right hand side to zero. Doing so we ‘remove the hat’ and get

$xp- px =0$.

Thus, the classical theory of bosons does not require anything beyond (maybe complex) numbers. Importantly, the order of the multiplication does not matter here at all, just think of standard multiplication of real numbers.

The situation for fermions is a little more interesting. Here we have the so called Canonical Anticommutation Relations or CAR,

$\{\hat{\psi}, \hat{\pi} \} = \hat{\psi} \hat{\pi} + \hat{\pi} \hat{\psi} = i \hbar$.

Again these operators have an interpretation as position and momentum, in a more generalised setting. Note the difference in the sign here, this is vital. Again we can take a classical limit resulting in

$\psi \pi + \pi \psi =0$.

But hang on. This means that we cannot interpret this classical limit in terms of standard numbers. Well, unless we just set everything to zero. Really we have taken a quasi-classical limit and realise that the description of fermions in this limit require us to consider ‘numbers’ that anticommute; that is ab = -ba. Note this means that aa= -aa =0. Thus we have nilpotent ‘numbers’, that is non-zero ‘numbers’ that square to zero. This is odd indeed.

Supermathematics and supergeometry
In short, supermathematics is all about the algebra, calculus and geometry one can do when including these anticommuting ‘numbers’. The history of such things can be traced back to Grassmann in 1844, pre-dating the applications in physics. Grassmann’s interests were in linear algebra. These odd ‘numbers’ (really the generators of) are usually referred to as Grassmann variables and the algebra they form a Grassmann algebra.

One of my interests is in doing geometry with such odd variables, this is well established and a respectable area of research, if not very well represented. Loosely, think about simple coordinate geometry in high school, but now we include these odd numbers in our description. I will only reference the original paper here [2], noting that many other works evolved from this including some very readable books.

What if no supersymmetry in nature?
This would not mean the end of research into supermathematics and its applications in both physics & mathematics.

From a physics perspective supersymmetry is a powerful symmetry that can vastly simplify many calculations. There is an industry here that works on using supersymmertic results and applying them to the non-supersymmetric case. This I cannot see simply ending if supersymmetry is not realised in nature, it could be viewed as a powerful mathematical trick. In fact, similar tricks are already mainstream in physics in the context of quantising classical gauge theories, like the Yang-Mills theory that describes the strong force. These methods come under the title of BRST-BV (after the guys who first discovered it). Maybe I can say more about this another time.

From a mathematics point of view supergeometry pushes what we know as geometry. It gives us a workable stepping stone into the world of noncommutative geometry, which is a whole collections of works devoted to understanding general (usually associative) algebras as the algebra of functions on ‘generalised spaces’. The motivation here also comes from physics by applying quantum theory to space-time and gravity.

Supergeometry has also shed light on classical constructions. For example, the theory of differential forms can be cast neatly in the framework of supermanifolds. Related to this are Lie algebroids and their generalisations, all of which are neatly described in terms of supergeometry [3].

A very famous result here is Witten’s 1982 proof of the Morse inequalities using supersymmetric quantum mechanics [4]. This result started the interest in applying physics to questions in topology, which is now a very popular topic.

In conclusion
Supermathematics has proved to be a useful concept in mathematics with applications in physics beyond just ‘supersymmetry’. The geometry here pushes our classical understanding, provides insight and answers to questions that would not be so readily available in the purely classical setting. Supergeometry, although initially motivated by supersymmetry goes much further than just supersymmetric theories and this is independent of CERN showing us supersymmetry in nature or not.

References
[1] Jonathan Amos, Collider hopes for a ‘super’ restart, BBC NEWS.

[2] F. A. Berezin and D. A. Leites, Supermanifolds, Soviet Math. Dokl. 6 (1976), 1218-1222.

[3] A Yu Vaintrob, Lie algebroids and homological vector fields, 1997 Russ. Math. Surv. 52 428.

[4] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. Volume 17, Number 4 (1982), 661-692.

Plaque for Sir William Grove

January 22nd, 2015 by ajb
 A plaque will be placed in Swansea’s Grove Place to commemorate the 19th Century scientist Sir William Grove.

Sir William, was the founder of the Swansea Literary and Philosophical Society, and managed to combine a legal career with several important scientific achievements. In particular he anticipated the conservation of energy and was a pioneer of fuel cell technology. He was the first to produce electrical energy by combining hydrogen and oxygen in 1842, a technology that went on to supply water and electricity for space missions.

Just another example of a Welsh person having done great things for the world.

William Robert Grove, Wikipedia.

Higher order mechanics on graded bundles

December 9th, 2014 by ajb
 In collaboration with K. Grabowska and J. Grabowski, we have applied the recently discovered notion of a weighted algebroid to mechanics on graded bundles[1].

In our preprint “Higher order mechanics on graded bundles” We present the corresponding Tulczyjew triple for this situation and derive the phase equations from an arbitrary (maybe singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. This is all done essentially in the first order set-up of mechanics on a Lie algebroid subject to vakonomic constraints. The amazing this is that the underlying graded bundle structure gives this whole picture the flavour of higher derivative mechanics. Within this framework we recover classical higher order mechanics, but we can study some more exotic situations.

For example, we geometrically derive the (reduced) higher order Euler-Lagrange equations for invariant higher order Lagrangians on Lie groupoids. To our knowledge, not much work has been done in understanding such systems [2,3]. We hope that the example on Lie groupoids turns out to be useful, maybe in say control theory.

References
[1] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).

[2] L. Colombo & D.M. de Diego, Lagrangian submanifolds generating second-order Lagrangian mechanics on Lie algebroids, XV winter meeting of geometry, mechanics and control, Universidad de Zaragoza, (2013). http://andres.unizar.es/ ei/2013/Contribuciones/LeoColombo.pdf

[3] M. Jozwikowski & M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus, arXiv:1306.3379v2 [math.DG] (2014).

Breakthrough Prize in Fundamental Physics Symposium Videos

December 5th, 2014 by ajb
 The videos from the Breakthrough Prize in Fundamental Physics Symposium are now available to watch, follow the link below. The symposium was held on the 10th November at Stanford University and co-hosted by UC-San Francisco and UC-Berkeley.

There was a panel discussion with Adam Riess, Brian Schmidt, Saul Perlmutter and Yuri Milner, and individual 20-minute talks from Nima Arkani-Hamed, Juan Maldacena, Andrei Linde, Stephen Shenker, Alexei Kitaev, Patrick Hayden, John Preskill, Nathan Seiberg, Joe Polchinski and Uros Seljak.

My Birthday

November 23rd, 2014 by ajb
 I did not realise this until today, but I share my birthday with John Wallis (23 November 1616 – 28 October 1703). Wallis made contributions to infinitesimal calculus, analytic geometry, algebra and the theory of colliding bodies. He is best known for the infinity symbol $\infty$.

Wallis was also a code breaker and used his mathematical skills during the Civil war to decode Royalist messages for the Parliamentarians.

John Wallis Wikipedia.

Engineers should embrace their artistic side.

November 22nd, 2014 by ajb
 More warnings about a shortage of STEM graduates. Where are these messages coming from? There is no shortage, many graduates in engineering for example are forced to work in jobs that are nothing to do with their degrees, or even worse they are in non-graduate jobs.

Engineering needs to emphasise its creative side to encourage more young people to take it up as a career.

Sir John O’Reilly

This is fine. The fields of science and engineering do require some creative thinking. There is an element are art to this.

But I am worried about the continual message of a lack of STEM graduates where in reality many are unemployed. I really don’t understand the B.S. here.

Poor maths skills in Welsh Schools

November 13th, 2014 by ajb
 Schools watchdog Estyn says too many pupils struggle with the basics of mathematics and maths skills were “at best, average” in more than half of schools inspected in Wales last year.

The mathematics skills and the teaching methodologies in Welsh schools seems to be a continuing source of worry.

The attitude that it is okay to not be good at basic mathematics needs to change. There is some social acceptability in being poor in mathematics. I am sure that when English teachers, lets say at a party, explain that they are English teachers are not met with a funny look and the response “I hated English at school and I cannot read or write”…

Why there is no equivalence principle for electromagnetic theory

November 10th, 2014 by ajb
 Quite regularly one will come across a website, blog or some thread on a forum that says the gravity is just electromagnetism. For sure they are not the same. However, I am not sure what would constitute the ‘nail in the coffin’ for all these clams.

With this in mind, I am not going to try to debunk every such claim. However, I think the reason for this seeming equivalence comes from the static limit. In this limit it is true that there are many formal similarities between Newtonian gravity and electrostatics. In fact there are many formal similarities in the full classical theories, both are very geometrical in nature, but I won’t go into details here.

I just wanted to point out one very clear difference between gravity and electromagnetism that can be seen in this static limit. That is the lack of a generalisation of the equivalence principal for electromagnetic theory. This principal in gravity is very important and one that I will comment on in due course.

The static limit
I am guessing that we have all seen Coulombs law for electrostatics and Newton’s law for gravity. Let me just write them down

$F = k \frac{qQ}{r^{2}}$,

where $k$ is Coulombs constant, it is a measure of the strength of the electrostatic force and $q$ and $Q$ are the electric charges of two point particles. The above expression is the electrostatic force between two such charged particles.

Similarly we have Newton’s law of gravity

$F = G \frac{mM}{r^{2}}$,

where $G$ is Newton’s constant which measures the strength of the gravitational force and $m$ and $M$ are the masses of two point particles. The above expression measures the gravitational force between these particles.

These expressions for the forces should be seen as the static non-relativistic limit. I just mean that as long as the particles are moving slow enough then the change in the fields can be viewed as instantaneous. This is okay for many applications, but it is not the full picture. However, it is the one you see at high school.

The formal similarities at this level are clear. You just need to swap constants and interchange charge and mass. But this does not mean they are the same, and there is a subtle issue here. Before that we need Newton’s law of motion

Newton’s second law
Newton’s second law tells us that the force exerted on a particle is proportional to the acceleration of that particle. Moreover, the constant of proportionality is the (inertial) mass.

$F = m a$.

That is all we will need.

The gravitational equivalence principal
Let us think of the particle of mass $m$ as a test particle. That is we will think of how it is moving in the gravitational field generated by the particle $M$ and that it does not generate a gravitational field of its own. This approximation is good for small objects moving in the gravitational field of big objects; say planets around a star or satellites in orbit around the Earth.

Now we can examine how the small mass is influenced by the big mass. We should just equate the two expressions due to Newton

$ma = G \frac{mM}{r^{2}}$,

for which we can solve for the acceleration

$a = G \frac{M}{r^{2}}$.

We notice a very amazing thing. The small mass cancels from both sides of the equation. (We assume that gravitational mass and inertial mass are the same. This seems to be case in nature.)

This means that the motion of the test particle does not care about any of the intrinsic properties of that particle. The only things it does care about is the initial position and velocity. This is one form of the equivalence principal which has it’s roots in the experimental work of Galileo- acceleration of a test particle due to gravity is independent of the mass being accelerated.

Thus, really all the information about the test particles motion is encoded in the gravitational field alone. All test particles whatever their mass will behave the same. This is the clue that gravity can be formulated very intrinsically in terms of space-time geometry alone; this leads to general relativity which is not the subject for today.

The electromagnetic version
Now let us play the same game with electrostatics…

$ma = k \frac{q Q}{r^{2}}$,

where we think of the test particle $(m, q)$ moving in the electric field generated by the particle $(M,Q)$. Now solving for the acceleration gives us

$a = k \left(\frac{q}{m}\right) \frac{Q}{r^{2}}$.

Now we see the difference. The motion of the test particle does depend on the intrinsic properties of that particle, namely the charge-mass ratio. There is no similar statement like for gravitational physics; there is no equivalence principle.

Closing remarks
Everything above is done in a particular limit of the full classical theories. The same picture is true when we examine the motion of test particles in general relativity and the Lorentz force in electromagnetic theory. I have chosen these limits because I think this is clear and also the source of the instance that gravity is just electromagnetic theory. I have pointed out one clear and explicit difference.

One can do the same with Coulomb’s law for the magnetic force. Although magnetism is a bit more complicated we can examine the situation for point-like poles. This is okay for small enough poles that are well separated. You will reach the same conclusion that there is no equivalence principal in this situation. Thus, gravity is not magnetism either.

Talk for Polish physics students

November 5th, 2014 by ajb
 On Friday 7th November I will be giving a short talk to physics students entitled ‘Fermions in Physics: from anticommuting variables to supermanifolds’, as part of the Ogólnopolska Sesja Kół Naukowych Fizyków.

This translates as ‘Scientific Session of the Nationwide Circle of Physicists’. This year the session is in the Tricity area.

I hope to tell them a little about the passage from the canonical anticommutation relations to Grassmann algebras and then supermanifolds. I have 45mins to get them interested in this geometric side of mathematical physics.

I will post the slides here after the event, but they may not make much sense without me telling this you this story.