We have gone past the first week of Ramadhan. I decided not to write anything before this, to keep the sanctity of Ramadhan, However, my first week has been challenging due to my bad back to the extent that I had to take an injection last Tuesday to get some relief of the pain.
Now, I break my pause in writing just to document my youngest's birthday three days ago. Here are the pics.
During Ramadhan, it is the best time for us to reflect upon (many) things that we have done. Going down a self-pity lane, I was comparing myself with others. First as a theoretical physicist; when I came back to Malaysia from abroad, there were not many active local theorists. So what I did, was to compare myself with theorists abroad. With the greats, there was simply a huge gap and my concern has always been on how to reduce this gap. Despite that the internet gave us free flow of information, there is still a need for a sizable local community in which one can always have regular discussions, feedback on ideas, giving a fertile resesrch ecosystem in which one can grow. I took plenty of students to help build this community and at a time we had regular meets (like journal clubs). Many times, I tried to instill Gelfand spirit into this meets (see here and here), but sometimes the technical level is not enough when we limit the meets within our own circle and the next thing I did, was to invite experts from abroad to help us (e.g. our EQuaLS acquaintances). Indeed many of us know that PhD training is not enough and it takes years to build sufficient technical level and maturity that we see in experts abroad. Abroad, this would be realised with years of postdoctoral study. Just a side note: Dr. Tay did a year of postdoc with me and if I was not wrong, was probably the first postdoc in UPM since at the time, the research management centre seemed unsure of the procedures involved for the appointment.
I still remember the challenge posed by Prof. David Fairlie who said that if you want see how successful one's student is, observe his/her paper or research program independent of his/her supervisor. My PhD work was partly driven by my supervisor's paper on anomalies in conservation laws. The idea is to generalize this to field theories. Anomalies are essentially failure of classical symmetries when brought over to its quantum version. In order to investigate this, one needs a quantization scheme that uses symmetries as an underlying ingredient but yet sensitive to global features (to address domain problems). So, I was introduced to Isham's canonical group quantization programme (for a more pedagogical introduction, it is best to refer to his Les Houches lecture notes; the voluminous book is pretty hard to get now but it was in our UPM library). This paper was my attempt to frame his work within the canonical group quantization scheme. The generalization to field theory is nontrivial; one needs a field-theoretic version of the external magnetic field. This can be given by a topological term like the Wess-Zumino term. Thus, this led me to do this work, but in a way, the paper also shows something unexpected where generalization of loops in field configurations might not quite give the desired ingredient.
Later, my supervisor gave me the paper of J.Stuart Dowker (the father of Fay Dowker) to study. The gist of the paper is that the usual homology group associated to the Wess-Zumino term used in sigma-models is only a subgroup (spherical homology) of the homology group. The review that I did of the paper was presented in a local (PERFIK) conference. At the time of my PhD study, I did not realise the significance of this paper to my own PhD work and was not included in my PhD thesis. It was only much later then I realised that the possible connection would be the domain problem mentioned earlier. Still, I thought it was too technical for me to look into (probably would require a mastery of functional analysis to even formulate the problem of field configurations with the domains concerned). I proceeded nevertheless, to formulate canonical group quantization of strings with an external field (realized by the Wess-Zumino term) with a surprising result of the winding number operators are no longer Abelian. This part of the work, I consider it as incomplete since I did not study the representation theoretical parts and was only published as a conference paper (in conjunction with the two-decade anniversary of UPM).
Back to the Fairlie challenge earlier; what was the research I initiated after my PhD? It was a similar problem of what I did for my PhD but using a different configuration space, namely the two-sphere nstead of the two-torus. With an external transverse magnetic field, there is a similar change of canonical group, though it is more subtle for the case of the sphere (requires explicit construction of obstruction of the original canonical group without the magnetic field). Recently, there was interest in this work (see here and our work was cited - reference 21 with the note that unavailable conference proceeding/unpublished references in the cited work). Our interest went beyond the sphere, we started to investigate quantum systems on punctured surfaces realized by hyperbolic geometry on upper half plane. Realizing that the Lie-algebraic level of treating quantum systems on such hyperbolic surfaces will be insensitive to the presence of punctures (physically this means that a particle on such surfaces will almost always scatter to infinity), the next phase was to consider explicit numerical solutions to Schroedinger-like equations on punctured surfaces, representing bound states on such surfaces (given by Maass cusp forms) with an automorphy condition for the solutions. Since I have been computer-phobic since my undergraduate days (recalling the hours of work in the computer lab to program a scientific calculator using Pascal language), I had to have external help. Indeed during my PhD study, my supervisor asked whether I would like to work with mathematical structures or work with computer calculations (e.g. soliton solutions), I answered that I prefer the former (and my contemporary colleague Robert Leese work in the latter). Embarking on this numerical phase, I sought help from Holger Then and Fredrik Stromberg and invited them over to UPM. I actually even wrote to Dennis Hejhal but he replied saying that he was no longer interested in the topic. At the time, I had just opened Theoretical Studies Laboratory (TSL) at ITMA (Institute of Advanced Technology) and had purchased computers and softwares and the numerical research phase help partially justify the purchases (for the lab). In fact, I was trying to convince colleagues to start adopting Mathematica (now Wolfram Language) for research but since those with engineering-like research preferred Matlab and on the symbolic side, the mathematicians preferred Maple, I failed.
In research, I tend to follow (established) researchers of topics of interest. Having started with canonical group quantization, Chris Isham is certainly one I do follow closely.
He is described by Lee Smolin as the theorists' theorist (see Three Roads to Quantum Gravity).
Some of Isham's later work was on the use of topos for Kochen-Specker theorem. The KS theorem is a relatively lesser known theorem in comparison with Bell's theorem in quantum foundations, but it essentially arises from the mathematical structure of quantum theory itself, particularly the separation between the observables and states. Before one could even begin to understand the topos approach, one need to overcome two hurdles: i) the conventional proofs of the KS theorem; and ii) the sophisticated machineries within category theory. The latter is the larger hurdle as it is often not within the arsenal of mathematical tools of theoretical physics, but it has gained familiarity when one frames it as bird-eye's view of maps occuring in physics partcularly the modern coordinate independent view of doing geometry. Category theory is also being used to reframe quantum mechanics within this picture of maps as expounded by Bob Coecke and collaborators. We have followed this route to familiarize ourselves with ths new approach. This was done right before my official retirement with two of my MSc students. For KS theorem in conventional setting itself, I have managed to convince my philosophically inclined student to consider KS theorem as a research topic, who later worked out several different KS proofs (after the PhD work) including one proof with only using 17 rays. I was also considering the approach made by Michel Planat (connecting with our other reserach directions), but I have not made too much progress in finding a niche (the mathematics is quite sophisticated - see here for a brief introduction).
Quantum information research was also brewing over the years but again, our problem is to find a niche that we could contribute meaningfully. Given our attention on geometrical and topological approaches, the natural route would be to look at geometry of quantum states. We had invited Karol Zyczkwski over to UPM to help us identify open problems.
One particular direction that we are trying to pursue is to use tools of symplectic topology in elucidating geometry of quantum states. I mention this in a conference in Thailand and recently we had this write-up being published here.
One initiated research direction while I was at the Maths Institute is complex networks because I thought the wide applicability of the subject matter. But the paper that got me started with this direction is the paper of Tomaso Aste entitled "Complex Networks on Hyperbolic Surfaces". Again, what I was trying to do is to tie up the different research strands that we were pursuing. I have also gotten very fascinated with hyperbolic geometry all the while, partly because there are 'more' hyperbolic spaces than other spaces (flat or positive curvature) and that they may capture the complexity of the real world with the hope of seeing applications - see this article. Finally, there ara other research topics that were suggested by (international) students, namely cosmology and thermal fied theory, but I insisted these were taken up with an external supervisor who is in the area.
As one can see, all these seem to be diverse though I would like to think that most of them have flavors of geometry and topology in them. I guess all these must have meant something (refering back to the Fairlie challenge). I am now out of academia though I wish I could do more. My other half was telling me that I have not accepted the reality that I am already retired. Perhaps.
I have more in my mind that I wanted to write on (e.g. the E*stein-related matters that seem to fill my news and video feeds) but this has gotten too long. In the midst of this writing, a war has erupted in the middle east. May Allah help us.
