Witten on supermanifolds and integration on them

Edward Witten placed some notes on the arXiv on 11th September entitled “Notes On Supermanifolds and Integration” [1]. In the notes he outlines the the theory of supermanifolds and integration on them. His motivation is to present what is needed for the RNS formalism in string theory.


Edward Witten

As ever Witten has produced very readable and concise notes without over complicating things. I recommend that anyone interested in how to integrate objects on supermanifolds start here.


[1] Edward Witten. Notes On Supermanifolds and Integration. arXiv:1209.2199 [hep-th]

Four UK universities in the world top ten.

The QS World University Rankings have been released and four UK universities come in the top ten. There are 18 UK universities in the top 100, showing that for quite a small Island we have have a good reputation. The warning is however, that due to changes in the funding UK universities may struggle in the ranking in the future.

The top 10 world universities are listed below:

    • Massachusetts Institute of Technology
    • University of Cambridge
    • Harvard University
    • UCL
    • University of Oxford
    • Imperial College London
    • Yale University
    • University of Chicago
    • Princeton University
    • California Institute of Technology

Heisenberg’s Measurement-Disturbance


Werner Heisenberg, one of the founding fathers of quantum mechanics is famous for his uncertainty principle. Initially he came to his result based on a rather heuristic argument that measurement of a system disturbs the system and this leads to an inherent uncertainty in all measurements. We have the “Heisenberg’s microscope” in which he imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.

Later on the uncertainly principle was formulated correctly in terms of quantum operators that do not commute. This is of course the true deep reason why we have quantum uncertainty.

It has now been shown that Heisenberg’s original argument is wrong. Aephraim Steinberg and other researchers the Centre for Quantum Information & Quantum Control and Institute for Optical Sciences at the University of Toronto have published work disproving Heisenberg [1].

I will stress the claim is not that the uncertainty principle is wrong, it is not, the claim is that Heisenberg’s original argument is wrong.


[1] Lee A. Rozema, Ardavan Darabi, Dylan H. Mahler, Alex Hayat, Yasaman Soudagar, and Aephraim M. Steinberg. Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements. Phys. Rev. Lett. 109, 100404 (2012)

Fractal from Binomial Coefficients


Above is a discrete fractal generated by creating a table of zeros and ones by deciding if the binomial coefficients are even or odd. The “key” here is paint black if odd, otherwise leave light blue.

The pattern is closely related to Pascal’s triangle.

The pattern clearly shows self-similarity as all fractals do.

As far as I know, this pattern was first noticed in [1]. Also note that we have a structure very similar to the Sierpinski Sieve. In the limit of infinite rows we recover the Sierpinski Sieve, up to a shift in the positions of the zeros and ones.

A slight variant


Just for fun I used the same algorithm to study the pattern associated with modified binomial coefficients of the form

\(\left( \begin{array}{c} (-1)^{k}n\\ k \end{array} \right)\)

Again the pattern shows lots of self-similarity.


[1] S. Wolfram: American Mathematical Monthly, 91 (November 1984) 566-571

A Random Walk


Random walks can be found throughout nature in many different contexts. For example they been used in ecology, economics, psychology, computer science, physics, chemistry, and biology. Above is an example of a (simple) random walk I created. There is 8 directions to this walk and 1000000 points.

The random walk above is an example of a Markov process, that means that the next step only depends on the present step. Such processes have “no memory”.

Random walks are closely related to Brownian motion, which is the physical phenomenon of minute particles diffusing in a fluid.

Random walks are examples of discrete fractals. They show self-similarity on large scales (such as in the picture above), but on the smaller scales the discrete nature of the grid becomes apparent. See the picture below.


Here we have another random walk generated in exactly the same way as the previous one, but not just 1000 points. One can consider this as a “zoom in” on the random walk with a million points. The finite step size is apparent and the resemblance to genuine fractals is far less clear.




More fun with Julia sets

I have been rather creative and explored an interesting Julia set. I will say that I picked it for the way it looks, rather than anything scientific. I make no claims that this Julia set is of any real mathematical interest, nor that it is related to any interesting dynamical system or anything like that.


Here is part of the Julia set for \(F_{c}=(1 + \sin(z) ) \log(|z|)\) and with \(c = – 0.5 i + 2 \). I have included grid lines to help us navigate.

So, let us have a closer look.








The self-similarity in this Julia set is quite striking. the generic features here are also quite generic; the branching off and swirls.

Science cash hike 'a wise gamble', say Brain Cox


“The UK science budget is about £5.5bn each year… on a government spend of over 600 billion. That’s for everything – medical research, Cern, engineering, arts and humanities; the whole thing. It’s below most global averages, the OECD’s for example,”

Prof. Brian Cox, University of Manchester

Prof. Brian Cox was speaking at the British Science Festival in Aberdeen. The Manchester University researcher was speaking before his guest lecture at the festival.

He also pointed out that public engagement with science was vital to the future sustainability and growth of British science.


BBC News

British Science Festival

Brian Cox’s homepage

Splitting the electron


In Horizon: How Small is the Universe? broadcast on BBC Two at 21:00 BST on Monday 3 September, the narrator talked about “splitting the electron” and that this had been achieved.

The scientist speaking, Jeroen van den Brink was a little more careful. The properties of the electron were split between quasi-particles.


Jeroen van den Brink

There are three quasi-particles here; holons, spinons and orbitons [2]. Before I say something about these, you should note that these are not really fundamental particles, but rather they arise from the collective behavior of the system. Quasi-particles are emergent phenomena that occur when a microscopically complicated system behaves as if it contained different weakly interacting particles in free space.

A good example here is the electron effective mass as it travels through a semiconductor. The system behaves like a system of weakly interacting electrons, but with a different mass. These “wrong mass” electrons are quasi-particles and arise due to the collective interactions.

Another example from semiconductor physics are holes. The absence of an electron in the valence band of a semiconductor behaves like the presence of a positive charge carrier.

Holons, spinons and orbitons

The electron, in certain circumstances can be considered as a bound state of the the three quasi-particles; holons, spinons and orbitons. The holons carries the charge, the spinons carry the spin and the orbitons the orbital location. That is these three quasi-particles describe the fundamental properties of the electron in some material.

Amazingly, there are situations in which these quasi-particles become deconfined. That is they are no longer strongly bound, but exist as free particles. This is what Horizon was really talking about.


In 1996 Kim et.al. split an electron into a holon and spinon [1].

The third part, the orbiton was far more elusive. Only this year has it been reported that “spin–orbital” separation of the electron has been observed [3].

I won’t comment further here as experimental physics is outside my expertise. I suggest those that are interested read the original papers I cite.

Has the electron really been split?

No, but the properties, the quantum numbers have been shared between quasi-particles and these quasi-particle have experienced deconfinement. This is an amazing result that is deeply rooted in quantum mechanics and collective phenomena, but does not signal that we should think of free isolated electrons as not being fundamental.

Today there is no compelling evidence that the electron is not fundamental, that is has some internal structure. It may well do, string theory for example suggests that the electron is just one vibrational mode of a fundamental string, but right now the evidence is not there.


[1]C. Kim, et.al. Observation of Spin-Charge Separation in One-Dimensional SrCuO2. Phys. Rev. Lett. 77, 4054–4057 (1996)

[2]K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231 (1982)

[3]J. Schlappa et.al. Spin–orbital separation in the quasi-one-dimensional Mott insulator Sr2CuO3. Nature 485, 82–85 (03 May 2012)