Category Archives: General Mathematics

Supersymmetry and mathematics

CERN 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.

Mathematical Physics IOP booklet

The Institute of Physics (IOP) has written a report that examines the relationship between mathematics and physics.

There is no completely agreed upon definition of mathematical physics, which is closer to mathematics that physics in most respects. The IOP report puts it like this;

Mathematical physics is best described as consisting of two parts: physical research that proceeds primarily through mathematical means and areas of mathematics that work to solve the problems posed by physics.

The Journal of Mathematical Physics define mathematical physics as;

…the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.

The IOP booklet looks at a wide range of topics in mathematical physics from quantum mechanics, gravity & black holes, random matrix theory, solitons and topological insulators.

Link
Mathematical Physics: What is it and why do we need it?

Ever wondered how to calculate a given number?

How you ever wondered what is special about your favourite number? As we all know, the answer to the ultimate question of life, the universe and everything is \(42 = 2100/50 \approx sin(\pi 3/26)-sin(\pi 11/39)\) and can also be approximated very well by many other expressions.

By using the The Inverse Symbolic Calculator (ISC) you can take numerology to another level by finding closed expressions that well approximate the number (any truncated decimal) you think has some special meaning. It used a mixture of lookup tables and integer relation algorithms. The tables were first compiled by S. Plouffe.

Whatever you do please have some fun with it.

Link
The Inverse Symbolic Calculator
Simon Plouffe homepage

Nuclear theory research in the UK to be exanded

Flag A new nuclear theory group is going to be set up at the University of York. The Science and Technology Facilities Council (STFC) will make a special funding award to set up the group and will provide funding to appoint a nuclear physics theory chair and PhD studentships. Furthermore, the university of York will fund a nuclear physics theory lectureship.

The need to expand the UK’s capability in theoretical nuclear physics was part of the Institute of Physics review in October 2012. For sure, although the UK has some good researchers in this field, the numbers of people in theoretical nuclear physics is small. One number that has been suggestion is that there are about seven permanent researchers in the UK working on theoretical nuclear physics.

The establishment of a new group must be welcome news for the UK nuclear physics community.

Link
Gap in nuclear physics research identified by IOP is to be filled.

Assume spherical sheep in vacuum…

sheep Scientists have now used GPS to uncover the rules that describe how sheepdogs are able to herd sheep.

Strömbom et al [1] have shown that there are surprisingly few rules here; in fact they suggest just two rules.

  1. The sheepdog learns how to make the sheep come together in a flock.
  2. Whenever the sheep are in a tightly knit group, the dog pushes them forwards.

The sheepdogs make the most of what is know as “selfish herd theory”, that is the tendency of a given sheep to want to be near the centre of the flock when under threat.

There is a Welsh connection here. One of the authors, Dr. A. King is based at Swansea University, which is where I studied for my undergraduate degree.

Now, anyone know any good jokes about Welsh people and sheep? Can’t say that I have herd many…

References
[1] Strömbom et al, Solving the shepherding problem: heuristics for herding autonomous, interacting agents, J. R. Soc. Interface 11(100) (2014).

Happiness is a long equation

As you can imagine as a mathematician, the bigger and harder the equations the happier I am. Not really, we look for pattens and elegance rather than just difficult equations, though of course difficult equations can be elegant and contain a lot of interesting structure.

Anyway, scientists now have an equation for happiness and here it is

happy
Taken from [1].

Now we just need to apply some calculus to find the maxima (local or global I’m not fussy) and find out just how happy a mathematician can be!

Reference
[1] Robb B. Rutledge, Nikolina Skandali, Peter Dayan, and Raymond J. Dolan, A computational and neural model of momentary subjective well-being, PNAS 2014 : 1407535111v1-201407535.

Link
Equation ‘can predict momentary happiness’ BBC News

More experiments with random walks

I have again been playing with some random walks, using the same method as here. This time I used 1000000 iterations and added some colour.

Below are random walks, on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.

walk

walk

walk

walk

Once again, these images are rather for artistic purposes than scientific purposes.

This is like so random…

Below are random walks on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.

I may play with these further, but they make some interesting pattens. We have approximate self-similarity and so these patterns have fractal-like properties. Anyway, enjoy….

random10

random1

random2

random5

These images were created for artistic rather than scientific reasons. That said, random walks are have been applied to many fields including ecology, economics, psychology, computer science, physics, chemistry, and biology.

Probably the most famous application of a random walk is to Brownian motion, which describes the trajectory of a tiny particle diffusing in a fluid. I have no idea if there is anything scientific in these images, but I would not be surprised if for small step sizes we have approximately Brownian motion. However, I would need to think a lot more about this before making concrete statements.