３１ Mathematics

３１　Mathematics【What is the Universe？】Dialogue with the Universe

Mathematics is the alphabet with which God has written the Universe. -Galileo Galilei （1564 - 1642）

○Key word：Symmetry [1]

※Forecast of mathematics：Ｔhose that do not can still prove.

（It is well in the world of mathematics.）

【Mathematical history】

About 7000 years ago 　Number was born. (Mankind began to group living.)

16th century    Imaginary number is accepted.

・・・"Forgetting the mental anguish received with imaginary number, and merely introducing this." Gerolamo Cardano（Italy 1501～1576）

17th century　　Minus is accepted in Europe.

・・・Descartes draws a number line. 　　René Descartes（France 1596～1650）

The first half of the 19th century　　An imaginary number is denoted by an axis of coordinates.：complex plane　　Carolus Fridericus Gauss（Germany 1777～1855）

1935　　Gödel's Incompleteness Theorem

・・・Then, Turing(Aan Mathison Turing, UK 1912-1954) proved that there is no unific method of confirming beforehand whether a certain proposition is a proposition which cannot judge truth.

1964　　Murray Gell-Mann (1929-)predicted the existence of quarks by the theory　of　pure mathematics(the group:SU (3)).

【References】

[1] Edward Vladimirovich Frenkel(1968-)

http://www.nhk.or.jp/hakunetsu/math/

３１ 数学

３１　数学【宇宙とは】宇宙との対話

数学は神が宇宙を書いたアルファベットだ。－ガリレオ・ガリレイ

○キーワード：対称性 [1]

※予想：まだ証明できていないもの。（数学の世界ではよくある。）

【数学の歴史】

約7000年前　数字が誕生した。（人類が集団生活を始めた。）

１６世紀　　虚数を認める

・・・「虚数によって受ける精神的苦痛は忘れ、ただこれを導入せよ」カルダノ（イタリア　1501～76）

１７世紀　　ヨーロッパでマイナスが認められる・・・デカルトが数直線を描く

　　　　　　　　　　　　　　　　　　　デカルト（仏 1596～1650）

１９世紀前半　　座標軸で虚数を現す：複素平面　ガウス（独1777～1855）

1931年　　　不完全性定理　ゲーデル（オーストリア　1906－78）

・・・その後、チューリングが、ある命題が真偽を判定できない命題であるかどうかを、あらかじめチェックする統一的な方法がないということを証明した。

1964年　　マレー・ゲルマン(1929-)は純粋数学の理論（群：ＳＵ（３））を使ってクォークの存在を予言した。

【参　考】

[1]  NHK 数学ミステリー白熱教室  エドワード・フレンケル(1968-)

 http://www.nhk.or.jp/hakunetsu/math/

３０ What is the Universe？＋The History of Cosmology

３０　What is the Universe？＋The History of Cosmology【What is the Universe？】Dialogue with the Universe

Q:Why does the universe exist in spite of trouble？

Q:Can a unified theory of the universe explain the universe accurately？

Q:Or we need the Creator？

Q:If so, the Creator gives the effect of something else in the universe？

Q:And who created the Creator？

Source: "Brief History of Time" Stephen William Hawking (issued April 15, 1995)

【The history of cosmology】

1895　　William James(1842-1910) coined the term Multi-Universe(Multiverse).

1900  　Max Karl Ernst Ludwig Planck(1858 -1947) advocated that the light, X-rays and other electromagnetic waves were released as a constant mass that he called quantum.

1905　　Special relativity[1]・・・Theory of electromagnetism that Albert Einstein（1879-1955） announced.

1913　　 Bohr's model(Niels Henrik David Bohr(1885-1962))・・・A pioneer of quantum mechanics (Old quantum theory).

・Electron is a wave.

・The length of the electrons orbit around the nucleus is an integer multiple of the electron wavelength (10 minus 10 square m).

(Bohr,  de Broglie)

1915-1916   General theory of relativity（General relativity）(Einstein) [1] [Electromagnetic force + gravity]

・Einstein explained that the gravity caused by the mass distort the space-time.

• When written in Newtonian mechanics[2] the phenomenon of error increases (movement in the near to the speed of light or large gravitational field) can be correctly described.

· In general relativity, spacetime is expected never to stay inflated or deflated steady or the black hole[3] is expected to forme when a large mass to concentrate on limited space.

・Einstein predicted the existence of gravitational waves[4] in 1916 in his general theory of relativity. 

1920s　 Quantum mechanics

・Atomic, electronic and electromagnetic waves have the characteristics of the particles and waves.

1922　Friedman found the expanding universe solutions in general relativity equations.

1925 　Pauli exclusion principle

1926 　Werner Heisenberg was put together the uncertainty principle[5].

※Uncertainty principle：It is a characteristic of the system, such as the wave. In knowing pair of physical properties (eg position and momentum) at the same time, there is a fundamental limit of accuracy.

1928　Dirac theory

　・Either both consistent：theory of quantum mechanics and special relativity

・There is positron(antimatter) that is the partner of electron.

1929　 Discovery of cosmic expansion(the redshift)（Edwin Powell Hubble）

1932   Discovery of positron(antimatter)

1934　 Fritz Zwicky coined the term dark matter. [6]

1948　－Big Bang model [7]

　　　－Quantum electrodynamics, QED [Electromagnetic force ]

· Electromagnetic interaction acting between charged particles, including electrons is caused by the transfer of particles called photons.

1957　Bryce Seligman DeWitt (1923–2004)  coined the term many-worlds interpretation; MWI.

1964

・Peter Ware Higgs proposed the theory of generation of mass. [8]

・Cosmic microwave background (radiation)（; CMB、CMBR）[7]discovered.

1968     Weinberg-Salam theory[9] [Electromagnetic force + Weak interaction]

1970　　Penrose and Hawkins （co-authored paper）

・On the basis of the general relativity theory, they were finally prove that should there was a big bang singularity in the very early universe.

• In the singularity, the effect of small scale of quantum mechanics dealing can not be ignored.

· However, if you take into account the quantum effect, singularities disappear. ...?

1970s

・Only one force was present in the beginning of the universe, then, the force was divided into electromagnetic force, weak interaction, strong interaction and gravity, so physicists  are trying to  expresse these four forces in one form.

　－Standard Model, SM [Electromagnetic force + Weak interaction（+ Strong interaction）] [9]

－Grand unification theory or Grand unified theory、GUT[Electromagnetic force + Weak interaction+ Strong interaction] [10]

The late 1970s　 －Quantum chromodynamics [9]

1980s　　Cosmology based on theory of elementary particles[10]

1981　　Cosmic inflation Theory [11]

1984　　Superstring Theory [12] [Electromagnetic force + Weak interaction＋Strong interaction ＋ Gravity]

1986~　　Loop Quantum Gravity [13]

1995　　 M-theory[14] [Electromagnetic force + Weak interaction＋Strong interaction ＋ Gravity]　

1998　　－Michael S. Turner coined the term dark energy. [6]

　　　　－Observed the accelerated expansion of the universe（Brian P. Schmidt）

2016 　　Studio Gooda！ coined structure of dark matter. [7]

【References】

1. Ｎｏ． １６　Theory of Relativity

2. Ｎｏ． １５　Newtonian Mechanics

3. Ｎｏ． ８１　Black Hole

4. Ｎｏ．２９　Gravitational Wave

5. Ｎｏ．１８　Uncertainty principle

6. Ｎｏ．１１　Dark Matter and Dark Energy

7. Ｎｏ．２０　The Big Bang model

8. Ｎｏ． ２４　Higgs Boson

9. Ｎｏ．２１　Standard theory(Standard model)

10. Ｎｏ．２３　Cosmology based on theory of elementary particles

11. Ｎｏ．２３－２　Inflation cosmology

12. Ｎｏ．２５　Superstring Theory

13. Ｎｏ．２６　Loop Quantum Gravity

14. Ｎｏ．２７　M-theory

15. Ｎｏ．８８－４－２　A Structure of Dark Matter

３０ 宇宙とは？＋宇宙論の歴史

３０　宇宙とは？＋宇宙論の歴史【宇宙とは】宇宙との対話

Ｑ：宇宙はなぜ、存在するという面倒なことをするのか？

Ｑ：統一理論には自分自身の存在をもたらすほど大きな強制力があるのか？

Ｑ：それとも創造主が必要なのか？

Ｑ：もしそうだとすれば、創造主は宇宙に何か他の影響も与えるのではなかろうか？

Ｑ：そして、創造主を創造したのはだれなのか？

出典：「ホーキング宇宙を語る」スティーブン・Ｗ・ホーキング（1995年4月15日発行）

 　(p.239)

【宇宙論の歴史】

１８９５年　ウィリアム・ジェームズ　多元宇宙という言葉を作る。

１９００年　マックス・プランク(1858 -1947) は、光、Ｘ線その他の電磁波は、量子と彼

が呼んだ一定のかたまりとして放出されると唱えた。

１９０５年　特殊相対性理論 [1]：アルベルト・アインシュタイン（1879-1955）が発表した電磁気学の理論。[電磁気力]

１９１３年　ボーアの原子模型（ニールス・ボーア(1885-1962)）：電子は波で、原子核を回る軌道の長さは波長（１０のマイナス１０乗ｍ）の整数倍（ボーア、ド・ブロイ）

１９１５－１９１６年　一般相対性理論 [1] [電磁気力＋重力]

・質量が時空間を歪ませることによって、重力が生じることを説明した。

・ニュートン力学[2]で記述すると誤差が大きくなる現象（光速度に近い運動や、大きな重力場における運動）を正しく記述できる。

・時空は膨張または収縮し、定常にとどまることがないこと、限られた空間に大きな質量が集中すると、光さえ脱出できないブラックホール[3]が形成されることを予測した。

・重力波[4]の存在を予測。

１９２０年代　量子力学

・原子・電子・電磁波は粒子としての特徴をもつと同時に波としての特徴をもつ。

１９２２ 年　フリードマンが一般相対論方程式の中に膨張宇宙解を見つけた。

１９２５年　パウリの排他原理

１９２６年　ヴェルナー・ハイゼンベルグが、不確定性原理[5]をまとめあげた。

※不確定性原理：波のような系に備わっている特性。一対の物理的性質（例えば位置と運動量）を同時に知ることには、精度の根本的限界がある。

１９２８年　ディラック理論・・・量子力学と特殊相対性理論のどちらとも整合性のある理論

→電子にはパートナーとなる陽電子（反物質）がある。

１９２９年　宇宙膨張（銀河の赤方遷移）の発見（エドウィン・パウエル・ハッブル）

１９３２年　陽電子（反物質）の発見

１９３４年　フリッツ・ツビッキーが暗黒物質を提唱 [6]

１９４８年－ビッグバン・モデル（ガモフ）[7]

－量子電磁力学(Quantum electrodynamics, QED）[電磁気力]

・電子を始めとする荷電粒子間に働く電磁相互作用を光子という粒子の受け渡しにより説明した。

１９５７年　　ブライス・デウィットが多世界解釈を提唱。

１９６４年－ピーター・ヒッグスが質量の生成に関する理論を提唱した。[8]

－宇宙マイクロ波背景放射[7]の発見

１９６８年　ワインバーグ＝サラム理論[9] [電磁気力＋弱い力]

１９７０年　ペンローズとホーキングの共著論文

：一般相対論が正しく、かつ宇宙が、われわれが現に観測しているのと同じ程度の物質を含んでさえいれば、宇宙のごく初期にはビッグバン特異点があったはずだということを最終的に証明した。

・特異点では、量子力学が扱う小さな尺度の効果が無視できなくなる。

・しかし、量子効果を考慮に入れると特異点は消え去る。・・・？

１９７０年代　

・宇宙の始まりに存在したのは唯1つの力だけで、その後、電磁気力、弱い力、強い力、重力の四つに分かれたという考え方から、これら四つの力を一つの形で表して統一しようとしている。

○標準理論( Standard Model, SM) [電磁気力＋弱い力（＋強い力）] [9]

○大統一理論(Grand unification theory or Grand unified theory、GUT）[電磁気力＋弱い力＋強い力]　 [10]

１９７０年代後半　量子色力学 [9]

１９８０年代　素粒子論的宇宙論 [10]

１９８１年　インフレーション理論 [11]

１９８４年　超弦理論(Superstring theory　1984) [12] [電磁気力＋弱い力＋強い力＋重力]

１９８６年～　　ループ量子重力理論 [13]

１９９５年　Ｍ理論[電磁気力＋弱い力＋強い力＋重力] [14]

１９９８年　－マイケル・ターナー　暗黒エネルギーという言葉を作る。[6]

　　　　　　－宇宙の加速膨張を観測（ブライアン・シュミット）

２０１６年　スタジオ・グーダ！　暗黒物質の構造を提案 [15]

【参　照】

1. Ｎｏ．１６　相対性理論

2. Ｎｏ．１５　ニュートン力学

3. Ｎｏ．８１　ブラックホール

4. Ｎｏ．２９　重力波

5. Ｎｏ．１８　不確定性原理

6. Ｎｏ．１１　暗黒物質と暗黒エネルギー

7. Ｎｏ．２０　ビッグバン・モデル

8. Ｎｏ．２４　ヒッグス粒子

9. Ｎｏ．２１　標準理論（標準モデル）

10. Ｎｏ．２３　素粒子論的宇宙論

11. Ｎｏ．２３－２　インフレーション理論

12. Ｎｏ．２５　超弦理論

13. Ｎｏ．２６　ループ量子重力理論

14. Ｎｏ．２７　Ｍ理論

15. Ｎｏ．８８－４－２　暗黒物質の構造

２９ Gravitational Wave

２９　Gravitational Wave【What is the Universe？】Dialogue with the Universe

２９－１　What are Gravitational Waves？[1] [2]
○ Gravitational waves are ripples of spacetime that propagate at the speed of light.
○Albert Einstein predicted the existence of gravitational waves in 1916 in his general theory of relativity. 
○On February 11, 2016, the LIGO Scientific Collaboration and Virgo Collaboration teams announced that they had made the first observation of gravitational waves, originating from a pair of merging black holes using the Advanced LIGO detectors. On June 15, 2016, a second detection of gravitational waves from coalescing black holes was announced. [3] [4]
○"Gravitational waves go through everything. They are hardly affected by what they pass through, and that means that they are perfect messengers," said Prof Bernard Schutz, from Cardiff University, UK. [3]
○In the framework of quantum field theory, the graviton is the name given to a hypothetical elementary particle speculated to be the force carrier that mediates gravity. However the graviton is not yet proven to exist and no reconciliation yet exists between general relativity which describes gravity, and the Standard Model which describes all other fundamental forces.

２９－２　What can we know and what is possible by the gravitational wave？
１）Direct observation of black holes
２）Construction of quantum gravity theory, the integration of general relativity and quantum theory (construction of theory to integrate the four forces of the universe)
３）Direct observation of the big ban and the earlier of the universe
４）Communication in the universe or between the parallel universe 

２９－２－１　Direct observation of black holes 
○Nothing can escape from a black hole, so it is impossible to see one directly. (Quantum field theory does predict that black holes give off an extremely tiny amount of thermal radiation, but it's so little that it we can't detect it from Earth.)
○Measurement of the waveform and amplitude of the gravitational waves from a black hole merger event makes possible accurate determination of its distance. [5]
○The accumulation of black hole merger data from cosmologically distant events may help to create more precise models of the history of the expansion of the universe and the nature of the dark energy that influences it. [5]

２９－２－２　Construction of quantum gravity theory, the integration of general relativity and quantum theory
○One of the difficulties of quantum gravity is that quantum gravitational effects are only expected to become apparent near the Planck scale, a scale far smaller in distance (equivalently, far larger in energy) than what is currently accessible at high energy particle accelerators. [6]
○A quantum gravity theory is required in order to understand problems involving the combination of very high energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe. [6]

２９－２－３　Direct observation of the big ban and the earlier of the universe
○What is currently observable is only the surface of the big bang.
○If we observe primordial gravitational waves, the inside of the big bang can be seen. Their detection exclude different models of inflation. Their detection exclude different models of inflation.

※The universe expanded approximately 10 ^ 30 times in the inflation era. In addition, since the transition to Friedman space-time, The universe inflated about 10 ^ 60 times together. Therefore, information of the initial Planck scale has not been disturbed by the Big Bang which come after inflation.

２９－２－４　Communication in the universe or between the parallel universe
○Because graviton travels through the extra dimensions, communication of between parallel universes of using the gravitational waves can be expected. [7] [8]

２９－３　Observation of gravitational waves
２９－３－１　LIGO（Laser Interferometer Gravitational-Wave Observatory）
○LIGO is one side of the 4km michelson laser interferometer.
○LIGO succeeded in direct observation of gravitational waves in 2015. [3] [4] Gravitational waves are a prediction of the Theory of General Relativity.

※LIGO [9]

○LIGO’s passive damping system holds the all-important mirrors perfectly still through a 4-stage pendulum called a "quad". [10]



２９－３－２　eLISA（Evolved Laser Interferometer Space Antenna）[11]
○The eLISA concept has a constellation of three spacecraft, arranged in an equilateral triangle with 1 million-kilometre arms as a ESA(European Space Agency) mission.
○LISA is good at gravitational wave detection at low frequencies than LIGO and VIRGO.
○A tentative launch date is 2034.

【Referendes】
1. Wikipedia：Gravitational wave
2. LIGO”What are Gravitational Waves?”
https://www.ligo.caltech.edu/page/what-are-gw
3. Pallab Ghosh Science correspondent, BBC News “Einstein's gravitational waves 'seen' from black holes”
http://www.bbc.com/news/science-environment-35524440
4. Maddie Stone”Scientists Have Detected Gravitational Waves Again” 2016
http://gizmodo.com/scientists-have-detected-gravitational-waves-again-1782024611
5. Wikipedia：First observation of gravitational waves
6. Wikipedia：Quantum gravity (QG)
7. Ｎｏ．２８　The Parallel Universe
8. Ｎｏ．９６　Communication with the alien
9. Robert Naeye”Gravitational Wave Detection Heralds New Era”at SKY ＆ TELESCOPE 2016
http://www.skyandtelescope.com/astronomy-news/gravitational-wave-detection-heralds-new-era-of-science-0211201644/
10. LIGO”LIGO Technology”
https://www.ligo.caltech.edu/page/ligo-technology
11. Wikipedia：Evolved Laser Interferometer Space Antenna