Saturday, April 30, 2016

The signature problem in the obsolete supergravity

I looked up the old textbook of ``Supersymmetry and Supergravity" (Princeton, 2nd edition, 1992 - Japanese translation and footnotes from Maruzen press, 2011) of Julius Wess (1934-2007) and Jonathan Bagger at my nearby library for some reasons. The English edition in my bookshelf says the old super-gravity theory (believed to be renormalizable at the old days of 1980s?) of Physics Letters B (1978) and Nuclear Physics B (1976) was (at page 145, Chapter XVIII The Supergravity Multiplet) \begin{align*} \delta \bar{\psi}_{m \dot{\alpha}} = -2 \mathcal{D}_m \bar{\zeta}_{\dot{\alpha}} - ie_m^{\quad c}\\ \times \left\{ \frac13 M^{\ast} (\zeta \sigma_c)_{\dot{\alpha}} + b_c \bar{\zeta}_{\dot{\alpha}} - \frac13 b^d (\bar{\sigma}_c \sigma_d \bar{\zeta})_{\dot{\alpha}} \right\}. \end{align*} In the Japanese edition, some corrections were done under the guideline of the posthumous writings of Prof. Wess (according to the translater Kazunari Shima). [page 141, Chapter 18] \begin{align*} \delta \bar{\psi}_{m \dot{\alpha}} = -2 \mathcal{D}_m \bar{\zeta}_{\dot{\alpha}} - ie_m^{\quad c}\\ \times \left\{ \frac13 M^{\ast} (\zeta \sigma_c)_{\dot{\alpha}} + b_c \bar{\zeta}_{\dot{\alpha}} + \frac13 b^d (\bar{\sigma}_c \sigma_d \bar{\zeta})_{\dot{\alpha}} \right\}. \end{align*} The theoretical physics and mathematical physics students (graduate students) had to handwrite a tremendous amount of dirty typesetting and calculation at the old days before the superstring theory and M / F theory. [Maybe this is an official press release of the Princeton Press (at least the Maruzen Press). So, this is not my original contribution to the theoretical physics of this historic literature.]

Tuesday, March 22, 2016

Spring = Revival.

I was thinking about the future of mathematics. Why was it divorced from the mainstream physics? Where did the `natural' definition of mathematics come from? Has it passed away? Still in development?

No answer. But we want to know.

Wednesday, August 26, 2015

The Theoretical Minimum (not Susskind but Lev Davidovich Landau): at least (orbital / spin) angular momentum

From right to left: "A Shorter Course of Theoretical Physics (Japanese) [=Краткий курс теоретической физики. В двух томах (Russian)]" Volume 1 and 2 (Mechanics and Electrodynamics / Quantum Mechanics respectively), Mechanics (Volume 1, Japanese), The Classical Theory of Fields (Volume 2, Japanese), Quantum Mechanics: Non-Relativistic Theory (Volume 3, English), Quantum Mechanics: Non-Relativistic Theory 1/2 & 2/2 (Volume 3, Japanese), Quantum Electrodynamics 1/2 & 2/2 (Volume 4, Japanese), Quantum Electrodynamics (Volume 4, Russian),
Statistical Physics, Part1, 1/2 & 2/2 (Volume 5, Japanese),
Fluid Mechanics 1/2 & 2/2 (Volume 6, Japanese),
Theory of Elasticity (Volume 7, Japanese),
Electrodynamics of Continuous Media 1/2 & 2/2 (Volume 8, Japanese),
Statistical Physics, Part 2: Theory of the Condensed State (Volume 9, Russian),
Physical Kinetics (Volume 10, Russian).

Although there is no "volume 9, Statistical Physics Part 2" (and no "volume 4, Quantum Electrodynamics, Part 2" & no "volume 10, Physical Kinetics") on the web, the well-known Course of Theoretical Physics was a mandatory (before the entrance exam of graduate schools of those days) series of textbooks written by students of L.D.Landau (Nobel Laureate in Physics [the prize for the theory of liquid helium's superfluidity], 1962 and his Nobel speech), which is now available for free here (U.S. archives) and here (an old version of the Russian original). However, editions of English translation are not up-to-date and such editions do not have a good TeX typesetting; there are some things like the followings: \begin{align*} <M \mid L_{+} \mid M-1 >=<M-1\mid L_{-}\mid M>\\ =\sqrt{ }[(L+M)(L-M+1)]. (27.12) \end{align*} [from the Quantum Mechanics [Volume 3, English ed.]] The Japanese edition is TeXnically correct.
\begin{align*} <M \mid L_{+} \mid M-1 >=<M-1\mid L_{-}\mid M>\\ =\sqrt{(L+M)(L-M+1)} (27.12) \end{align*}(a period or a comma should be inserted.) Likewise, the semi-classical approximation is
\begin{align*} f_{12} \sim \exp \left\{ -\frac{1}{\hbar} \text{im} \Big[ \int^{x_0} \sqrt{ } [2m (E_2-U)] dx\\ - \int^{x_0} \sqrt{ } [2m (E_1 - U)] dx \Big] \right\} (51.6) \end{align*} in the English edition. It should be \begin{align*} f_{12} \sim \exp \Bigl( -\frac{1}{\hbar} \text{Im} \Big[ \int^{x_0} \sqrt{2m (E_2-U)} dx\\ - \int^{x_0} \sqrt{2m (E_1 - U)} dx \Big] \Bigl) (51.6) \end{align*} as is written in the Japanese edition. (Russian originals of volumes 4, 9, 10 seem OK for the square roots including fractions.)

Today, I added three up-to-date editions (volume 4 [year 2006, 4th ed.] "Квантовая электродинамика"=Quantum Electrodynamics [2nd ed.] (or Relativistic Quantum Theory [1st ed.]), 9 [year 2004, 4th ed.] "Статистическая физика. Часть 2. Теория конденсированного состояния"=Statistical Physics Part 2: Condensed Matter Theory (or Statistical Physics, Part 2: Theory of the Condensed State), 10 [year 2007, 2nd ed.] "Физическая кинетика"=Physical Kinetics) of the original Russian to my bookshelf -- since the new / re-print editions of Japanese translation are no longer available for a long time. This situation is the same for the undergraduate students at the University of Tokyo of 17 years ago, and we had to share the old sombre fragile archives of the [physics / liberal arts / city] library (a stack room for books including Russian), or physics-oriented students can purchase some of the easier-to-obtain editions from secondhand booksellers at the Kanda (Jimbo-cho -- where I went and buy the three books above) city in Tokyo. [The exceptionally well-sold editions of volume 1, 2, 3 1/2 from the Tokyo-Tosho press and the volume 5 1/2, 5 2/2 of Statistical Physics, Part 1 from the Iwanami press are not out-of-print].

I was lucky in my undergraduate days that I could obtain (by a reasonable price) the volume 6 1/2 & 2/2 (=3rd ed. of original Russian) of Fluid Mechanics and the volume 7 (=4th ed. of original Russian) of Theory of Elasticity in the Japanese translation from the CO-OP (student union) of the University of Tokyo in my undergraduate days (1998 April-2002 March). I did not buy the volume 9 of Statistical Physics, Part 2 from the Iwanami press in my undergraduate days.

More nostalgically speaking, I was reading the volume 1 of Mechanics and the volume 5 of Statistical Physics when I was a freshman of the University of Tokyo of the year 1998-1999. This was only the prologue of my professional work of theoretical & mathematical physics in addition to pure algebraic (& arithmetic) geometry and algebraic analysis. While I was thinking about becoming a professional mathematician, it turned out my pursuit of learning both modern math [including number theory and arithmetic geometry] and theoretical physics [including elementary particle theory] was impossible at the governance system of the University of Tokyo at those days -- there was no Kavli-IPMU institute, there was no double major Ph.D., and there was a strong bashing / adversity against pure science and pure math. There was no communication between the physics department of UT's Hongo campus and the math department of UT's Komaba campus.

In my undergraduate days, some of the applied physicists and literature / social science students accused my particular interest was "inside the philosophy" or "religion-like" -- which was (as if) the same words "This is not mathematics; this is theology." [1890s] as the critics (of Paul Gordon) to Hilbert when Hilbert tried to defend the set theory [its ultimate initial plan of so-called Hilbert's program was not achieved -- but I don't write about this misleading popular science in this post. I just draw your attention to the fact that the Hilbert program was something that the earlier-20th-century mathematical physicists (including von Neumann) were sharing but its historical meaning has no consensus between arithmetic geometers and mathematical physicists.] of Cantor at the beginning of the 20th century (from Kronecker -- whose argument with Cantor was reconciled at the very end of Kronecker's life [but it is not confirmed in the literature]). The initial goal to establish the set theory was not the topology or the real / complex number, but the uniqueness of the Fourier transform / inverse transform as the trigonometric series. This concrete goal was not achieved as the unification of number theory and physical mathematics, but its idea (or, Cantor's dream) is still alive in another form of the category theory of homotopy algebras in arithmetic geometry and higher topos theory (and the elementary topos theory [not always Grothendieck topos] of sheaves in logic and the type theory).

At the days of Cantor, there was no delta function, no Bourbaki, no Grothendieck, no Landau-Lifshitz, no CERN LHC for Higgs bosons, and no superstring theory / M-theory. However, the pure mind of Cantor is still alive everywhere in the garden of modern mathematics.

Tuesday, January 20, 2015

André Weil on Gauss

I'm not trying to recall the history of the 20th century by Bourbaki before the late Grothendieck. I'm not a historian and not a nostalgic mathematician. Rather, I would like to reminisce the phrase by one of the greatest mathematicians of the 20th century, André Weil, who was admired by Japanese mathematicians including Prof. Kunihiko Kodaira of complex algebraic geometry.

Other words by A. Weil (not his sister Simone Weil) were usually cited in French or in English translations like Wikiquote, but I found out that the very famous phrase is not well-known to non-Japanese mathematicians when I googled on the web. The followings are my quotation of his words from Japanese reference.

(night of 10-Oct-1955, when Weil came to Japan, Fuji hotel.)
"Well, start with your own idea.
Gauss did like that.
You start like Gauss as well.
Then, soon you will realize you are not Gauss.
It is OK.
Anyhow, start like Gauss."
(by Weil, a feeble translation by Makoto Sakurai)

Some mathematical physicists misunderstood these words were by Atiyah or Serre, but it is not true. In the original bibliography of the complete posthumous works of Yutaka Taniyama (1927-1958), it is written at the first page of the article "In touch with A. Weil" (page 199-208 of the second edition, 1994).

The words were so famous that they were cited elsewhere in different Japanese expressions, but the original precise Japanese did not appear at the Sugaku seminar or Japanese popular literature.

In the above, I did not mean to reflect the past mathematics, but I meant young mathematicians and mathematical physicists were extremely inclined to copy-and-paste the works of (Fields / Nobel or any other kinds) medalists. The grant agency is very crucial in the career of young scientists, so that the competition is harsh.

However, it is a sad situation that some wrong translations of physics works by "pure" mathematicians are prevalently overwhelming the original works by either pure mathematicians or (let me say) native physicists. I do not accuse any individual case that I regret in this post, but let me recall this episode of Weil as a warning against such prize-winners.

As I wrote above, I do not like to cite eminent people's articles (including Weil). Nevertheless, the copy-and-paste machines on the web are so harmful that their plagiarism will eradicate the sincere efforts of young talented mathematicians and physicists.

It is like Japanese classic musicians; the music students are said to learn the music score very well, but there is no joy of discovery. As I'm not a musician, let me say about mathematics that the abroad conferences or symposiums are in any case very rejoicing.

On the other hand, the Japanese people are not so involved with the speakers when it comes to the cases of Japanese symposiums or international conferences in English. The organizers are trying to ask questions, but the communications among participants and speakers are not so active. Well, there are language problems as well, but I said to the Chicago people, "my English is not so good, but it is better than ordinary Japanese English." (The Chicago person said that my English was very good.) A French mathematician said language was not the problem, the problem was always mathematics.

Saturday, September 27, 2014

Weighted Arithmetic-Geometric means and logarithm

This was what I wrote yesterday after the April visit for the University of Chicago. This short article is written as a test of MathJax and the complement of the freshman math lecture at the year 2011. There is an extraction of my 9 page PDF file. \begin{eqnarray} \sum_{k=1}^N w_k a_k & = & w_1 a_1 + w_2 a_2 + \cdots + w_N a_N \nonumber\\ & \ge & a_1^{w_1} a_2^{w_2} \cdots a_N^{w_N} \\ & = & \prod_{k=1}^N a_k^{w_k} \nonumber\\ & := & \prod_{k=1}^N \exp \left[ w_k \ln a_k \right]. \end{eqnarray} The auxiliary conditions of antilogarithm and weight are as follows. \begin{eqnarray} 0 < a_1, a_2, \cdots, a_N \\ \Longleftrightarrow & a_k \in \mathbb{R}_{>0}, \\ 0 < w_1, w_1, \cdots, w_N (< 1)\ s.t.\\ w_1 + w_2 + \cdots + w_N = & \sum_{k=1}^N w_k = 1,\end{eqnarray}, where the equality holds if and only if a1 = a2 = ... = aN.

For the plan to avoid the misuse of ill-definition of general logarithmic function (of complex variables or matrix elements, or even some other completion of non-Archimedean / quantum dilogarithm), I showed the inequality without using Jensen's inequality (convex inequality). [I used the elementary calculus of polynomials in one variable for N->N+1. Or, I simplified the forward-backward induction by a one-step backward induction.]

See the PDF resume of mine, later.

Tuesday, January 01, 2013

Gauss Way

Japan has already leaped the end of year 2012, but California is still before the beginning of year 2013.

The MSRI (Mathematiical Sciences Research Institute) at the Gauss Way, California, is planning a series of workshops: Noncommutative Algebraic Geometry and Representation Theory including
, but I cannot visit California in January 2013 for the following reasons.

1.  I have to teach my students as the academic calendar of Japan is written.
2.  I cannot afford to visit California without the financial support of Japan and/or the USA, which is on the verge of serious budget cut (fiscal cliff (財政の崖) or fiscal wall (財政の壁)).

About the no.2, I have more things to append. Although European people are still thinking Japanese people are "economic animals," it is not the case. Since the so-called financial bubbles, IT-bubbles, and biotechnology bubbles are gone within this/these 2 decades, Japanese people have been struggling to overcome simultaneous crises in the tax rate, the employment rate,  the welfare, and some civilian control of nuclear plants after the aftershocks of Tohoku earthquakes and Tsunami at 14:46, 11-March-2011 (JST).

On the contrary, before the Diet (Japanese Parliament or the Lower House) campaign at the December of 2012, I have done some electric annual ballot for the AMS (American Mathematical Society) vice-president, Boards, and Councils. The financial circumstance of AMS is not optimistic; as an AMS member of 8 years, I was asked for signature against the budget cut:

Tell Congress to Avoid January 2013 Sequestration

Although I am not a resident of the U.S.A., I am affiliated with the American Mathematical Society for a long term, including some eminent immigrant mathematicians from the overseas to American universities, but I do not cite individual researchers' names in this post.

OK. Let me go back to my personal financial problem. Even though I registered in the participants' list of the MSRI, they say they cannot afford to financially support my visit for the travel and living expenses while the vice deputy sent me an encouragement message about attending one or more workshops of the MSRI at the last year. I really interpreted this letter in the literal sense, but the U.S. mathematicians are now planning the January (9-12) 2013's San Diego, CA, Joint Meetings with the SIAM, the MAA(Mathematical Association of America) and so on, which encourage graduate students' NSF grant proposals with a little chance of travel expenses.

Needless to say, I am not a graduate student any more, but Japanese policy on the science development is not promising for post-doctoral members or part-time lecturers. Thus I was much interested in the events of January-2013, but neither Japanese researchers nor American institutes let me join in such important meetings since I have not obtained the (three or more) strong recommendation letters from eminent mathematicians in the world-wide. All I can do is write my original preprint from my original motivation, but the number of papers is not large.

As Carl Friedrich Gauss -- the prince of mathematics in the 19th century -- said, a genuine mathematician does not always have plenty numbers of recommendation letters; whereas he said "Few, but ripe" for his remarkable achievements in pure mathematics and mathematical physics by himself while writing his diaries.

Tuesday, October 09, 2012

Higgs Hunter

Last month, I visited Kyushu University, which lies in one of the southern islands of Japan, for the oral presentation at the Mathematical Society of Japan. Before the presentation at the morning session (at 21-Sep-2012) of algebraic geometry, I submitted my update of Ph.D. thesis with the elimination of some of my typos and obsolete references. I also noticed my change of permanent receivable e-mail address of the University of Tokyo. See Mixed anomalies of chiral algebras compactified to smooth quasi-projective surfaces and the slides uploaded at the researchmap database. The conference hall was at the "2nd Centre Zone" of the university, which was a little far from the Fukuoka (福岡) airport; we need to connect two railways of subway and local trains, and the transportation of school (semi-shuttle) bus was necessary from the nearby train station to the "Big Orange Station" at the entrance of Ito Campus.

My presentation was about the "Geometric Quantization of Wess-Zumino-Witten model" and not "Geometric Quantization of Chern-Simons theory". In addition, my presentation was a revisit of Spontaneous Symmetry Breaking by Nambu-Goldstone bosons and soliton's micro-local calculus (Dynamical Symmetry Breaking) of co-tangent bundle of beta-gamma conformal field theory without explicitly assuming the motivic integration of Kontsevich-Soibelman or Feynman's oscillator integrals. See the English abstract for the conference at the MSJ official website. The latter half of my presentation title was on the "virtual localization formula", but it is not the WKB (semi-classical) approximation of pre-quantization and polarization after the Oxford's school of differential geometers. My trial was to formulate the (topologically half-twisted) Super-Conformal Field Theories as an extension of Wess-Zumino-Witten term of level 1, which was claimed by N.Nerasov[2005] but not uploaded to the arXiv unlike proposed.

Last but not least, let me mention my travel airline briefly. The Skymark airline, which is a recently hot topic of an LCC (Low-Cost-Carrier) airline agency, was used from 06:20am to 08:15am (Japan's Standard Time) at 20-Sep-2012, from Haneda (羽田) airport in Tokyo to Fukuoka in order to attend the domestic meeting of the Algebra Session.

However, the returning way back from Fukuoka to Haneda was terribly delayed. I first waited more than half an hour at the Big Orange Station, which made me thirsty and sunburnt. I arrived at the Fukuoka airport one hour before the schedule while I left the conference hall at 14:00. I had to wait for the boarding at the gate 2 since the aircraft of LCC was delayed to arrive for the schedule of departure time 17:55, which was postponed to be after 18:15 and I had patiently awaited since otherwise I would have to wait for the next LCC flight of more than 2 hours later at night.

Unlike supposed, my oral presentation was not performed regularly because of the irregularly worked projector from my EEEPC901 net-book with Fedora 17, which I used for several times and demonstrated twice within the conference dates. The Algebra Session was tight in schedule, so that I used my OHP backup for the optical camera with the use of my green laser pointer. I hope my above-mentioned slide file will be useful for someone who is interested in the relation between the Higgs mechanism (of physicist's sense) and the chiral algebra (Hamiltonian method on the worldsheet Riemann surface) theory with the base change of target space Kaehler manifolds (Lagrangian method of factorization algebra).

The chairperson's question was about my next research plan and I said "I have worked more than 5 years on this topic. From the beginning around the AMS talk of Okounkov, an extension to non-toric cases is a long-standing problem. I also would like to apply the DGA method to quantum invariants." It was because Japanese mathematics is under a serious view from the outside society and Japanese mathematicians have to explain the value to the general public; since Japanese economy faces 20 years of serious decline and the universities are under severe budget cuts.

In the good-old-days of Greek geometry, mathematicians can naively and Platonically quote "Evidently Euclid did not stress the practical aspects of his subject, for there is a tale told of him that when one of his students asked of what use was the study of geometry, Euclid asked his slave to give the student threepence, "since he must make gain of what he learns.""

Nevertheless, nowadays world economics suggests a serious reconsideration of our position of academics in the real-world of general public. So I just started my oral explanation of presentation by saying "If you have to talk to a person in an area other than mathematics, you can appeal algebraic geometry has useful applications (such as blow-ups) to natural science, and not just calculus alone".

I hope I can keep my trial on the recovery of academic value in the calculation-based financial society.