Ms –> PhD

Posted September 30, 2006 by Davi
Categories: Physics, Uncategorized

Since the last day 21 I’m a doctor in physics. My thesis defense was really entertaining. I was not worried with not being approved so I went to the defense keeping in mind what I could learn from the examiners. You can find my presentation here (PDF, Portuguese, my first presentation with Beamer). It’s huge, I had to skip some details, and others are not written.

My thesis is ready a long time ago and it has some peculiarities. Since it was clear to me that I knew everything that is written in the papers I am a co-author, instead of wasting my time detailing in the thesis those results, I just translated them to Portuguese, adding some comments to the older results, and used much of the time I had to explain the context from which those results belong, i.e., I did a review of noncommutative space-time, its connection with string theory, dualities, symplectic geometry, differential forms etc. These reviews were useful to myself and to the examiners, since not all of them were familiar with all these concepts. Some very small corrections to the thesis remain to be done, after that I will upload it here.

In the following day of the defense, 09/22, I just rested. In 09/23 I made some adjustments to my thesis presentation to turn it into the 15 minute presentation I did in the meeting ENFPC XXVII. This presentation is in English and only concernes the last work of mine, you can find it here (PDF, English).

The ENFPC XXVII was a very good meeting, perhaps my favourite of all the six ENFPC’s I’ve been present (all since 2001). The most significant problem for me was the distance from Rio de Janeiro to Águas de Lindoia. We needed 8 or 9 hours to go that city. The return we did in 1 hour less, however we arrived in Rio in a thursday after 17:00. Normally I need from 45 minutes to 1 hour to left UFRJ and arrive home (in Niteroi), in that day the same trip took 2 hours and 5 minutes!

Symplectic geometry, phase space and constraints: heuristics

Posted September 16, 2006 by Davi
Categories: Physics

In the last months I read some books on symplectic geometry. Most of them have in common this line of development: definitions & axioms –> main theorems –> “symplectic geometry is useful to physics!”. I do like the axiomatic approach, but I think an heuristic approach is also useful.

During my masters course I read V.I. Arnold’s Mathematical Methods of Classical Mechanics and found Chap. 8 (Symplectic Manifolds) a very good introduction to the subject. Only about 3 months ago I decided to read Arnold’s Chap. 9 (Canonical formalism), its title had not called my attention, so I jumped to others books. But when I started to read it, wow, its approach to symplectic geometry is very interesting and very different from the others approaches! Symplectic geometry is in that chapter ‘more or less’ constructed by physical assumptions. One may even first read Chap. 9 and only after that read the more standard approach of Chap. 8.

One of his main arguments goes like this (these words are mine, more details in Arnold’s book): the phase space is even dimensional (2n) and should not, a priori, have any preferential direction (some physical systems can have a phase space anisotropy, but this is not necessary). We may define an extented phase space which is 2n+1 dimensional, the extra dimension playing the role of the time. In this extended phase space, every physical system is expected to have a unique preferential direction, determining the phase space evolution unambiguously (1). Its easy to show that 2-forms in odd dimensional spaces are always degenerated (2). Let w be a nondegerated 2-form in the phase space (even dimensional), therefore any extension of it to the extended phase space is degenerated and has one, and exactly one, zero-mode (3). We associate this zero-mode v with the direction of the phase space evolution. (!!)

How this zero-mode is determined for each physical system? First lets suppose we are dealing with a very simple physical system: a one-dimensional free particle. We know that the evolution of q and p are given in the phase space by the usual Hamiltonian equations. On the other hand, the zero-mode v is the responsible for the phase space evolution, which means that

\dot q = v1 and \dot p = v2

Therefore, the first component of v is \partial H/ \partial p and the second is – \partial H / \partial q. Which 2-form we in the extended phase espace has this zero-mode v? The solution is we = dL, where L = p dq – p2/(2m) dt (i.e., L is the Lagrangian of the system expressed in first order in the velocities).

More general physical systems have a more general 1-form L. The most general 1-form L (with no explicit dependence on time) is given by

L (\xi) = a\alpha (\xi) d\xi\alpha – H (\xi) dt

In this more general case w_e is still given by dL and w=da\alpha \wedge d \xi\alpha. Is it necessary to w be exact? No, we can generalize this picture a little bit by requiring w to be only closed (not necessarily exact), but here I’ll not go that further, w being exact is sufficiently general to deal with both unconstrained and constrained systems through the usual procedures. Moreover, being w exact it’s easier to prove Liouville theorem (see Arnold’s Chap. 9 for a very elegant demonstration; Liouville theorem is lost if w is not at least closed).

So the most general case we’re going to deal with is we=dL. One should note that we = dL may in general have more than one zero mode, so there is is a problem to specify the evolution of the phase space. Indeed, this happens with constrained theories. One needs first to insert new variables (Lagrange multipliers) to embed the physical manifold (constraint surface) in a R2n+m phase space, then determine the nondegenerated closed 2-form w, which yields unambiguously the phase space evolution.

Nondegenerated closed 2-forms are called symplect structures. A manifold with a symplectic structure is a symplectic manifold. Occasionally degenerated closed 2-forms are called pre-simplectic structures.

=> I’ll stop here. No one can learn symplectic geometry or hamiltonian reduction through this message alone, but I hope I have written some useful remarks.

For more details in the symplectic approach for constrained systems see for instance L. Faddeev, R. Jackiw Hamiltonian reduction of unconstrained and constrained systems, Phys. Rev. Lett. 60 (1988) 1692; M. Henneaux, C. Teitelboim, Quantization of gauge systems, Princeton (1992). There are many references online on symplectic geometry, see A. Cannas homepage for instance (some references are in Portuguese).

(1) Gauge symmetry and constraints will be commented soon.
(2) Let w = 1/2 w_{\alpha \beta} dx^\alpha \wedge dx^\beta, with \alpha, \beta = 1,2,… D. We know that w_{\alpha \beta} = – w_{\beta \alpha}. Hence, det(w_{\alpha \beta}) = (-1)^D det (w_{\alpha \beta}). If D is odd, we have det(w_{\alpha \beta}) = 0.
(3) A zero-mode is an eigen-vector whose eigen-value is null. If w \not= 0 and w_{\alpha \beta} v^\beta = 0 then v is a zero-mode of w.

MikTex 2.5 and Hyperref

Posted September 2, 2006 by Davi
Categories: Uncategorized

Just now I upgraded my old 2.4 version of MikTeX to the 2.5 version. The details of the new version can be found in the last link. Those who already have version 2.4 installed may use the MikTex Update Wizard.

There was a change in the default option of hyperref, it was changed from hypertex to dvips. The line \usepackage{hyperref} is no longer compatible with the DVI building (but pdf documents are generated normally). To correct this problem one may either replace \usepackage{hyperref} with \usepackage[hypertex]{hyperref} in each tex file (*) or, which I prefer, return to the original default option. To do the latter, first locate the file hyperref.cfg (its standard folder is C:\texmf\tex\latex0miktex), then open it with an ascii editor (e.g., Notepad “Bloco de notas”). This file has only one line. Where it reads “dvips” replace with “hypertex”. Save the file “hyperref.cfg” and now everything will work fine. For both DVI and PDF outputs one may use \usepackage{hyperref}.

For more info, see TUG, this manual and Miktex 2.5.

(*) A PDF document can be built with that option, but the hyperlinks won’t work. For some reason, the modification in the hyperref.cfg file doesn’t have this drawback. Explanations?

Some news: thesis, previous E. Witten speech, D. Gross, D. Shostakovich etc.

Posted August 29, 2006 by Davi
Categories: Music, Physics, Uncategorized

Almost three weeks since the last uptade, I have a lot of things to comment.

1. Thesis. My thesis is ready since the last week. The defense will be in 21/09/06, now I’m working on its presentation (and listening my new CD’s).

2. An old order of mine finally arrived. Just to cite the titles (some I know for a long time, others are new to me): Frank Zappa – Yellow Shark/ Ensemble Modern; Frank Zappa – Freak Out; Caspar Brötzman Massaker – Home; Charles Mingus – At Antibes; Charles Mingus – Jazz Portraits; Igor Stravisky – Stravinsky in America / M. Tilson Thomas (it includes Agon besides some other pieces from Strvinsky’s final years); Igor Stravinsky – Le Sacre du Printemps & Petrushka / Boulez (my third version of Le Sacre and second of Petrushka); Mr. Bungle – Mr. Bungle; Mr. Bungle – Disco Volante; John Coltrane – Love Supreme [remaster] (my first non-free-jazz from J. Coltrane); Henry Threadgill – Carry the Day.

3. Pointless news. Now the About page is much more complete. More general improvements soon.

4. E. Witten. I went to the Witten’s speech cited in the last post. He talked about universe expansion, string theory, multiverses and many other topics due to the participants questions (the questions section was huge). This speech of him was very clear and intended to an audience of nonspecialists, as previously announced. Neverthess, physicists would not loose an opportunity to do many questions to him. While talking about multiverses and the string landscape he first pointed that he find those ideas interesting and that they answer some of his old questions; however, he hopes that they are wrong, since he still wish that string theory could answer all his original questions including the deduction of many parameters which, from the multiverse viewpoint, are regarded as devoid of a fundamental significance [they are essential parameters to the existence of our lives, but their values could be others, other universes may exist with different parameters; likewise the distance of the Sun from the Earth is essential to the existence of the forms of life we know, but it’s only an accident without a fundamental physical significance (there are many planets with different orbits)]. Among many other discussions and papers, see, for instance, Bee’s Thoughts on the Anthropic Principle. Moreover, Witten questioned the scientific value of the multiverse hypothesis on the grounds of physical predictions and experimental verification, without entering in the details. When questioned about Loop Quantum Gravity he just said that he doesn’t find this proposal intelectually attractive for him.

5. D. Gross. David Gross will give a speech on “The future of physics” in the next thursday, 08/31, for details see here. A more technical speech of him will be given in the XXVII ENFPC.

6. Shostakovich. In the same day and local, just after the D. Gross speech, there will be a performance of some works of D. Shostakovich (see here). Shostakovich has some great works, like his Symphony N° 13 (“Babi Yar”), I still don’t know the works which will be presented I’ll look for them.

ENFPC XXVII, SW map, dualities and E. Witten tomorrow.

Posted August 9, 2006 by Davi
Categories: Physics

I completed my registration on XXVII ENFPC (National meeting on particle and fields physics), a traditional Brazilian meeting already announced in Christine’s blog. This year’s meeting will be held in Águas de Lindóia, SP, a near city for the physicists from São Paulo, we from Rio have to travel more than the usual (I hope this time the bus’ air-conditioned will work and the trip will take less than 10 hours)!

I wish I already have exposed here more details of mine thesis and research (including a tentative explanation for my friends who are not physicists), I will again postpone that. In this link (PDF) you find an “extended abstract” of what I pretend to present in that meeting. The title is 3D and 4D Noncommutative electromagnetic dualities and the role of the slowly varying fields limit. Comments?

Tomorrow Edward Witten will present a speech on the expansion of the universe (details here or here). I’ll try to be there. Unfortunately I lost Lisa Randall‘s presentation yesterday and I could not attend the Young Researchers Symposium (see also Sum Over Histories). Moreover I almost don’t have time to write in this blog, I’ll stop now. Thesis, thesis, thesis…

The boring part of writing a thesis

Posted July 27, 2006 by Davi
Categories: Uncategorized

Now I’m translating from English to Portuguese some papers which were mostly written by my self to include them in my thesis. This really sounds to me as a waste of time. I wish I could simply bundle the papers in English with the introductory part and conclusions written in Portuguese.

Offtopic: Magma (the French group) fans should see this video . I think the father of that little girl should listen less Magma. :-)

Thoughts on Ockham’s razor and modern physics

Posted July 23, 2006 by Davi
Categories: Philosophy, Physics

razor_75.jpg vs. kite_string.jpg ?

While reading [1] Bertrand Russell’s History of Western Philosophy the above picture occured to me and I decided to write here some comments on Ockham’s razor and its possible usefulness to modern physics. Bellow you’ll find some lines which you might enjoy (explicitly, I just mention modern physics issues a couple of times, perhaps in the future I’ll write more). Paraphrasing a Siegel’s parody blog, if this post were really serious, it would not be here, it would be published elsewhere. :-)

The first time I heard about Ockham’s razor (aka, Occam’s razor), about 10 years ago, I thought it was a strong principle with a leading role in science. Curiously for me by then, “serious” philosophy of science books seldom mention Ockham’s razor, it appears it’s frequently cited and used only in “soft” books on philosophy of science or science divulgation. Some years later and until recently, it became clear to me that Ockham’s razor was completely useless. However, as now I understand, Ockham’s original statements were relevant to the establishment of modern science and, in a certain sense, they’re still important to science. As a criterion to decide what is the best explanation or theory, nonetheless, I still think it’s useless. I think Ockham’s razor is blunt, although his original statements are not.

Some history: William of Ockham. William of Ockham (~1290 – 1349 a.D.) was a Franciscan born in Ockham, a village in Surrey, England. Ockham had a different view on the role of faith and reason than the advocated by the Catholic Church of his time [which, as far as I understand, was inclined to St. Thomas Aquinas‘s philosophy, who brought Aristotle’s metaphysics to the Catholic Church and had demonstrated the existence of God through 5 different ways(1)]; consequently Ockham was excommunicated in 1328 by pope John XXII. In accordance with [1], Ockham proposed that “The reality is inherent to individual things, and only these can be object of experience, leading to a direct and correct knowledge. It means that, for an explanation of the being, the complex features of Aristotle’s metaphysics was completely superfluous. It is in this sense that we should interpret Ockham’s statement that “it is useless to do with more what can be done with less”“.

From Ockham’s viewpoint, it was nonsense to use reason to justify what concerns faith and God. For him, faith should not be restricted by logical judgments, and nor philosophy nor science are restricted by faith. If I understand his point, for him faith concerns the ultimate truth, while reason the apparent, the “effective”, truth in the secular world given by our ordinary perceptions. This puts a sharp difference between the matters of faith and the matters of reason, difference which was, and still is, influent to the comprehension of science and religion.

The Razor. Ockham’s principle of exclusion of the necessity of metaphysics or teology in order to study Nature has lead to many similar, but stronger, expressions. These versions are commonly grouped under the unique name of “Ockham’s Razor” and have in common a certain quest for simplicity. Some of these Ockham’s Razor expressions have become quite popular in the context of science divulgation, e.g.,

i) Entities should not be multiplied beyond necessity;
ii) The simplest answer is usually the correct answer;
iii) Of two equivalent theories or explanations, all other things being equal, the simpler one is to be preferred.

The third one, I think, is the most “sharp” and commonly found formulation of Ockham’s razor. At least a priori, it appears to be a good principle to select one scientific theory among many others which are in practice equivalent.

How to cut with the razor? Well, in principle is very simple: given two different theories which describes some Nature’s aspect equally well (from the perspective of possible physical experiments), if one can be regarded as simpler than the other, then the more complex is eliminated, cut, in favor of simplicity. Classical examples of the application of this razor in science include the victory of the heliocentric model over the geocentric one and the establishment of special relativity with the subsequent abandon of the aether hypothesis.

I think this principle casts 3 major difficulties: i) what precisely means “two different theories or explanations” which are equivalent? ii) can something be absolutely simple? At least in principle, nothing is simple by itself, something is simple in regard to a certain criterion, so what is the criterion? iii) what’s the meaning of “cutting” the more complex explanation? Are we getting closer to the truth? Or is it just a pragmatical move, that is, is it done just to simplify conceptual or technical manipulation? Why the necessity of cutting, can’t we use both theories?

I will not try to answer above questions, I would need much more time. In particular, the last question of item iii) and the item i) may seem very easy to answer (e.g., i) different models which foresees the same experimental output; iii-last) yes, can, but since they’re equivalent it’s waste of time), but depending on the interpretation of the Ockham’s razor, I think these questions can be very problematic. For instance, in modern physics the concept of duality plays a major role (it’s essential to string theory, while it’s very useful to quantum field theory/particle physics) and duality does exactly this: connects two different “mathematical” descriptions which have the same physical content. Depending on the criterion adopted to establish the scale of simplicity, one of these dual theories may be seen as simpler than the other. Nevertheless, in physics, while dealing with duality issues, we don’t discard one description in favor of the other, we use both! Both have the same physical content, but, depending on what is being studied, one is more suitable (i.e., easy to manipulate) than the other.

Two classifications.. The Wikipedia’s article on the Ockham’s razor uses a distinction very interesting between two kind of applications: one is similar to William of Ockham’s original statements, being called ontological application, while the other is the inter-theoretic competition, which is closer to the Ockham’s Razor definition present above and in “soft” books. The first application is frequently used: one should not care about a metaphysical statements while doing science (2), science is about what we can perceive using our ordinary senses (directly or not), moreover science is capable of achieving its own truths. See the B. Russell’s citation in the begining of this post. On the other hand, the last application, concerning theory competition, is very problematic, as above vaguely pointed. If one is concerned with theory competition or development of science, I think one should look for the “hard” books on philosophy of science and forget about Ockham’s razor. In this sense, I think, Ockham’s razor is not a good vague principle related to science, it’s indeed useless.

Well, these were mine loose thoughts on Ockham’s razor. Wikipedia has a lengthy text on the subject with many external links. I need to get back to my thesis’ writing now. Comments are welcome. :-)

(1) I don’t know any site about these demonstrations, but one of them is really short and funn…, I mean, interesting. It starts with the assumption that in our world we can find things with different degrees of perfection [a rather dubious start], therefore there MUST be something completelly perfect, which is God [clearly, St. Thomas Aquinas was a not a great mathematician].
(2) To avoid an ingenuous positivism some subtleties need to be clarified, but I’ll not enter in the details here, perhaps in the future.