Seminar: A first look at N-manifolds

February 13th, 2014 by ajb
Higgs event I will be giving a talk at the Geometry and Differential Equations Seminar at IMPAN (Warsaw) on Wednesday 26th February 2014. The title is “A first look at N-manifolds”.

Abstract
In this talk I will introduce the concept of an N-manifold as refinement of the notion of a supermanifold in which the structure sheaf carries an additional grading, called weight, that takes values in the natural numbers. I will provide several motivating examples which largely come for the theory of jets, before discussing some generalities.

Link
Geometry and Differential Equations Seminar

New exhibition at Jodrell Bank near Manchester

February 9th, 2014 by ajb
Higgs event Big Telescopes, Big Science is a brand new exhibition which will be unveiled in February at Jodrell Bank visitors centre. the exhibition will include hands-on activities showing how telescopes work and how it is possible to use many smaller telescopes to act as one large telescope.

There will also be running family science shows as part of the half term activities.

Follow the link below for more details.

Link
Big Telescopes, Big Science

Son of LHC!

February 7th, 2014 by ajb
Higgs event CERN is putting plans in place to build a successor to the Large Hadron Collider (LHC). Possible options for the next generation of colliders will be discussed at the University of Geneva next week.

There are plans for a massive circular collider – with a circumference of 80–100 km – that would accelerate protons to energies of about 100 TeV! The LHC has a 27 km circumference and can collide protons with energies up to about 7 TeV.

Link
CERN kicks off plans for LHC successor.

The Brighton Science Festival 2014

January 28th, 2014 by ajb

Brighton

The Brighton Science Festival 2014 is going to be held from the 6th February to the 2nd March at various locations throughout the city of Brighton, East Sussex. A list of the events can be found here.

If you are in Brighton or nearby over the above dates I am sure you could find something to interest you.

Link
Brighton Science Festival 2014

Scientists get the Hollywood treatment!

January 27th, 2014 by ajb
Higgs event Monday night on the Science Channel, the Breakthrough Prizes are having their USA television debut!

Preview and clips at http://tinyurl.com/peo97q3

Hosted by Kevin Spacey, the show hands out seven $3 million prizes, and honors physicists Joseph Polchinski, Andrew Strominger, Cumrun Vafa, John H. Schwarz and Michael B. Green, as well as leading medical researchers James P. Allison, Mahlon R. DeLong, Michael N. Hall, Robert Langer, Richard Lifton and Alexander Varshavsky.

Helping to celebrate the scientists are celebrities Conan O’Brien, Anna Kendrick, Glenn Close, Rob Lowe, Michael C. Hall, as well as tech leaders Mark Zuckerberg, Larry Page, Sergey Brin, Anne Wojcicki, Jimmy Wales and Yuri Milner.

The show is at 9:00ET/PT, 8:00CT, 7:00MT.

On curves and jets of curves and on supermanifolds

January 22nd, 2014 by ajb
I placed a preprint on the arXiv entitled “On curves and jets of curves and on supermanifolds” [1]. The abstract can be found below

Abstract
In this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.

Quick overview
The k-th order tangent bundle on a classical manifold can be defined as equivalence classes of curves at the points of the manifold that agree up to velocity, acceleration, rate of change of acceleration on so on up to k-th order. The tangent bundle which is much more widely known, is the “1st order tangent bundle”. These higher order tangent bundles have found applications in higher derivative Lagrangian mechanics for example and appear in the work of Grabowski & Rotkiewicz [2].

The main question posed and answered in this preprint is “can we define the k-th order tangent bundles of a supermanifold is such a kinematic way?”

The problems with simply generalising the classical notions directly are two fold:

  1. Supermanifolds do not consist solely of topological point, thus point-wise constructions need some careful handling.
  2. The naive definition of a curve as a morphism of supermanifolds $latex \mathbb{R} \rightarrow M $ totally misses the odd dimensions of the supermanifold $latex M$.

To resolve these issues we look at a “superised” version of curves defined via the internal Homs, a notion from category theory. In sort we consider curves that are paramaterised by all supermanifolds, however all the constructions should be functorial in this parametrisation. This allows us to define what I call S-curves that can “feel” both the even and odd dimensions of a supermanifold. Moreover, this allows us to think in terms of S-points and parallel the classical constructions of jets of curves rather closely.

This allow us to define the k-th order tangent bundle of a supermanifold at first as a generalised supermanifold (a functor from the opposite category of supermanifolds to sets ) and then we show that this is representable, that is a genuine supermanifold. Moreover, locally everything looks the same as the classical higher order tangent bundles.

References
[1] Andrew James Bruce, On curves and jets of curves and on supermanifolds, arXiv:1401.5267 [math-ph].
[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. geom. Phys. 62 (2012), 21-36.

My Thursday Colloquium talk

December 30th, 2013 by ajb
board On the 9th of January 2014 I will be giving a talk at the Algebra and Geometry of Modern Physics seminar here in Warsaw.

Title “What the functor is a superfield? ”

Abstract Physicists are usually quite happy to formally manipulate the mathematical objects that they encounter without really understanding the structures they are dealing with. It is then the job of the mathematician to try to make sense of the physicists manipulations and give proper meaning to the structures. (Confusing physicists is not the job of mathematicians, however mathematicians are good at it!) In this talk we will uncover the structure of Grassmann odd fields as used in physics. For example such fields appear in quasi-classical theories of fermions and in the BV-BRST quantisation of gauge theories. To understand the structures here we need to jump into the theory of supermanifolds. However we find that supermanifolds are not quite enough! We need to deploy some tools from category theory and end up thinking in terms of functors! :-O

I will try to post my notes here at some point after the talk…

My phone number is in pi!

December 22nd, 2013 by ajb
pi I found a website that allows you to search the number $latex \pi$ for stings of numbers. The Pi Searcher can search for any string of digits (up to length 120) in the first 200 million digits of $latex \pi $.

The number $latex \pi $ is transcendental meaning it cannot be written as any combination of rational numbers and their n-th roots.

Phone numbers here in Poland are 7 digits and it turns out that there is a  99.995% chance that the Pi Searcher can find it, Of course, if it can’t find it then it may well be still in $latex \pi $ somewhere,  which is related to the fact that we think it is a normal number.

Anyway my home phone number here in Poland appears 10 times in the first 200 million digits! My IOP number (6 digits) appears 186 times!

Try it for yourself at the link below.

Link
The Pi Search Page

Could Newton have “chosen” something else?

December 14th, 2013 by ajb

Introduction

We all know Newton’s second law “F=ma”. In words, we only have acceleration when an external force is applied. But could he have some how chosen not acceleration but say something higher order ?

Acceleration is the rate of change of velocity, which itself is the rate of change of position. Thus acceleration is second order in derivatives with respect to time. This is not just a feature of Newtonian mechanics but is rather general and found throughout the fundamental laws of nature.

There are some rather general results using the Hamiltonian formalism that tells us that theories with equations of motion that are higher order than two are unstable. In particular the energy is not bounded from below and this can lead to problems classically and quantum mechanically. This is the famous Ostrogradski instability of (non-degenerate) Lagrangian theories with higher order derivative terms.

Note that such theories are still of interest as effective theories, but they cannot be seen as fundamental.

I won’t say anything more here about this.

So the question is can we understand simply why Newton could not really have picked anything of higher order in “F=ma”?

Notation: I will use Newton’s dot notation for the first and second order derivative with respect to time. For the n-th order derivative (n>2) I will use $latex x^{(n)}(t)$.

Order three

Let us just pick a different form of Newton’s law as

$latex F = N x^{(3)} (t) $,

that is let us suppose the force is proportional to the third order derivative of the position. (That would be the rate of change of acceleration). Here N is some property of the particle analogous to mass.

Now let us think about the motion of a free particle. So set the force term equal to zero and see what happens. Solving our “higher order” Newton’s law with no force is simple. We have

$latex x(t) = c_{3} t^{2} + c_{2}t + c_{1}$.

The constants here are set by our initial conditions;

$latex x(0) = c_{1}$, $latex \dot{x}(0) = c_{2}$ and $latex \ddot{x}(0) = 2 c_{3}$.

So what do we notice? The velocity as a function of time is given by

$latex v(t) := \dot{x}(t) = \ddot{x}(0)t + \dot{x}(0)$.

This means that even if we set the initial velocity to zero the isolated free particle will speed up! Remember this is without any forces acting on the particle.

Newton’s first law says (in part) that “a particle at rest will remain at rest unless acted upon by an external force”. This higher order form of the second law is inconsistent with the first.

Worse than this, there are no forces here and so no potentials. The particle just speeds up all by itself and so clearly we lose conservation of energy. The particle can gain kinetic energy, defined as usual, but at no loss of potential energy! We would have to abandon our usual notion of conservation of energy in simple mechanics!

Higher order again
The same arguments work for higher order terms. The particle will just speed up by itself in violation of the first law and conservation of energy.

Modification of Newton’s law

Let us consider a slightly different situation in which “F = ma” becomes

$latex F = M \ddot{x}(t) + N x^{(3)}(t)$.

That is we will add a higher order term to Newton’s law. Again, let is consider the case with no force term. We want to solve the equations of motion

$latex \ddot{x}(t) + \frac{N}{M} x^{(3)}(t) =0$.

Here we assume that M is not equal to zero and is positive. For now we make no assumption at all about N.

One can directly solve the equations of motion

$latex x(t) = \left( \frac{N}{M}\right)^{2} c_{1} e ^{-(M/N) t} + c_{2}t + c_{3}$.

Again we notice that the velocity is not constant and so we do not have conservation of energy in this situation either. But let us have a look at the particles trajectory for different ratios M/N.

newton1

In the above we have set M/N = 2 and all the constants to 1. The purple line is what we expect from F=ma. Note that we have a quick decay to the classical case. This itself signifies we do not have conservation of energy as we have no mechanism for the loss in kinetic energy, it just happens!

newton2

In the above we have set M/N =1. This case is very similar to the previous case.

newton3

Now in the above we have set M/N = 0.5. Again this is very similar to the previous cases.

We have a decay so that after some period of time everything looks the same as the standard Newtonian case. However, we still have to violate the first law to achieve this.

Now what happens if we let N be negative?

newton4

Here we see we have a runaway situation in which the particle just keeps on speeding up! Even if initially the trajectory is very close to the standard one after some time it just blows-up. Again this is in violation of the first law and conservation of energy.

Lower order

What about lower order laws?

Well if we had $latex F \propto x$ then when there are no forces we simply have $latex x =0$. Everything not in motion would have to sit at x=0. Meaning we cannot have any extended objects that are not in motion. This cannot be consistent with our Universe.

What about $latex F \propto \dot{x}$? Again let us set the forces to zero and we see that the solution is just $latex x(t) = x_{0}$, some constant. However, this does not sit comfortably with our notion of relativity. Different inertial observers will not agree on the value of $latexx_{0}$. Thus if we don’t want to introduce absolute space we cannot allow this lower order form.

Conclusion
So as Newton wanted his first law to be true, have a good notion of statics and did not want to introduce absolute space he could have only have picked “F = ma”.

Physics World 2013 Breakthrough of the Year

December 13th, 2013 by ajb

icecube

The Physics World award for the 2013 Breakthrough of the Year goes to “the IceCube South Pole Neutrino Observatory for making the first observations of cosmic neutrinos”. Nine other achievements are highly commended and cover topics ranging from nuclear physics to nanotechnology

cube

Links
IceCube homepage

Physics world