Thursday, February 16, 2017

Shortening the Theoretical Minimum: We can learn. We can choose.

Since my Russian is terrible, I can only understand some translations of the Russian literature in math and physics. I recaptured the "theoretical minimum" of the good old Landau school. These textbooks were translated to many languages including English and Japanese.

I zoomed the first easier chapters of the classical mechanics, the classical field theory, and the non-relativistic quantum mechanics. These materials are everywhere including the scattering theory (the so-called S-matrix theory and the related algebraic analysis [D-modules] of the RIMS, Kyoto).

However, the materials treated in the Landau textbooks are obsolete in a sense. I mean, the quantum electro-dynamics (QED) is not well-described in the Landau course -- the Feynman diagram does not have a standard space-time axis and the [single / double] arrows of the modern theoretical physics after 't Hooft. Landau abandoned the quantum field theory at his later years.

Apart from the advanced materials of (quantum) statistical physics / stochastic calculus [not always the condensed matter physics for the applied engineering and high-temperature superconductivity], we can see some of better modern textbooks. It is like Bourbaki. Bourbaki was a classic and it was written at the highest level at those ages, but now we have better textbooks and we can understand the curriculum more easily.

Although the fluid mechanics of the 19th century (well, I don't want to talk much about the plasma physics) and the elastic theory are not well inherited, we have the books of "[the] Ginzburg-Landau Equations and Stability Analysis" [Iwanami Press, still no English translation available] by Shuichi Jimbo and Yoshihisa Morita [ISBN=978-4000075619] and "An Introduction to Quantum Field Theory" by Michael E. Peskin and  Daniel V. Schroeder. [ISBN=978-0201503975]

I read the Peskin-Schroeder at the junior and senior [the 3rd and 4th year of the UTokyo, the academic year 2000-2001]. I performed the Peskin-Takeuchi's "Physics beyond the standard model" [phenomenology of elementary particle theory including muons [leptons: the anomalous magnetic moment g-2] & (Yukawa's) pions [pi mesons of quarks decayed to multiple photons: chiral abelian anomaly and the Riemann-Roch-like theorem of Fujikawa's method by chiral fermions] but without any sound mathematical foundation] at the 1st year summer seminar of my master's period.

Peskin seminar was after the seminar of "Renormalization: An Introduction" by Manfred Salmhofer [Springer, ISBN=978-3642084300]  during the winter holidays [a little before 2002-March] with the theoretical physics students working on the condensed matter physics. I didn't sufficiently understand the Fermi surface problem [still a mystery in the AdS/CFT correspondence conjecture] at those days, which is not treated in the renormalization group theory in the quantum field theory of the elementary particle theory.

To sum up, the Feynman rules [including the statistical factors of second quantization of Bose-Einstein / Fermi-Dirac statistics of identical particles] for the perturbative [asymptotic series of] Feynman integrals [as well as symmetry breakings after the late Yoichiro Nambu, the phase transition theory, and some of the so-called standard model] are treated more shortly in the textbook of Peskin-Schroeder. Non-perturbative effects of solitons, supersymmeties, and D-branes / M-branes [of the so-called superstring theory, which I still doubt in the effectiveness to the real-world problems] are not treated in this pedagogical textbook.

I still think Peskin is worthy of attention for undergraduate physics students and determined mathematical physicists [including mirror symmetrists and twistor theorists among other geometric representation theorists at the math department]. Of course, I mentioned no classic literature of quantum field theories including Zinn-Justin and Bjorken-Drell. Ryder [ISBN=978-0521478144] might be a second choise and available on the web.

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 (Mathematical 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.