20 Incompleteness Theorem

20 Incompleteness TheoremWhat is the Universe?】Dialogue with the Universe

The proposition which cannot judge correction exists.
・・・Gödel (Kurt Gödel, Austria 1906-78) proved in 1930.
・・・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.
Reference: 「世にも美しい数学入門」Masahiko FujiwaraYouko Ogawa(p.147151)

If there is a proposition which cannot confirm that verification is impossible and it is unverifiable, it can say, "Search is eternal."
 ・・・Although nothing may remain even if it continues search throughout life・・・

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 CardanoItaly 15011576
17th century  Minus is accepted in Europe.
・・・Descartes draws a number line.   René DescartesFrance 15961650
The first half of the 19th century  An imaginary number is denoted by an axis of coordinates.complex plane  Carolus Fridericus GaussGermany 17771855
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.

20 不完全性定理

20 不完全性定理【宇宙とは】宇宙との対話

・・・1930年にゲーデル(オーストリア 190678)が証明した。



19 Uncertainty Principle

19 Uncertainty Principle【What is the Universe?】Dialogue with the Universe

19-1 Uncertainty principle
Introduced first in 1927, by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. 
You can answer only according to probability on the level of quantum.

○The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp
:No matter how improve the accuracy of the measurement, the product of the standard deviation of the position and momentum is greater than h.
:One is uncertain when you try to confirm the other.
・h:Planck constant(6.626×10^-34 J・s)

○In the case of energy and time
:Beating only a little law of conservation of energy, it may be borrowed energy. 
※However it must return as fast as borrow a lot.
・E:energy、t: time

⇒Energy is possible to materialize.⇒Pairs of matter and anti-matter particles are constantly being created and annihilated.
=Quantum fluctuations 
⇒Also in vacuum, pairs of matter and anti-matter particles are constantly being created and annihilated.: Vacuum fluctuations
⇒The vacuum has a vastly complex structure.
・Vacuum fluctuations causes a measurable force (Casimir effect) in infinitesimal.
※Vacuum fluctuations [3]

19-2 Casimir effect
○The typical example is of the two uncharged conductive plates in a vacuum, placed a few nanometers apart. The Casimir effect produces the negative pressure. It is due to quantum vacuum fluctuations of the electromagnetic field.
○As the two mirrors move closer to each other, the longer waves will no longer fit--the result being that the total amount of energy in the vacuum between the plates will be a bit less than the amount elsewhere in the vacuum. Thus, the mirrors will attract each other, just as two objects held together by a stretched spring will move together as the energy stored in the spring decreases. [4]
○The dynamical Casimir effect is the production of particles and energy from an accelerated moving mirror. [5] Further, there may help development of quantum computers.


19-3 Phase Transitions of Vacuum /Inflation Cosmology/Quantum Tunneling
○Vacuum is to shift to more of the state of low energy:phase transition.
○In physical cosmology, cosmic inflation, cosmological inflation, or just inflation is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from 10 minus 36 square seconds until after 10 minus 34 square seconds.
(The size of the universe (10^-27m) is much smaller than atom (10^-10m).
⇒3 millimeters (10^-3m))
○The driving force behind this inflation has been said that the energy that was released during the phase transition of the vacuum.
○In early inflationary models, the phase transition of the vacuum was considered that it is generated by the Quantum tunnelling or tunnel effect. However, in a model named new inflation or slow-roll inflation, instead of tunneling out of a false vacuum state, inflation is considered that occurred by rolling down a potential energy hill.

※Quantum tunnelling or tunnel effect to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classicallycould not surmount. In the quantum theory it is explained by the uncertainty principle.
It has important applications to modern devices such as the tunnel diode, quantum computing, and the scanning tunnelling microscope. [3]

1.Wikipedia: Uncertainty principle
2. ハイゼンベルクの不確定性原理 - nifty  Uncertainty principle- nifty
3.RevoScience”Young researcher proposes new explaination for unsolved problems in physics…”
4. “What is the Casimir effect?”SCUENTIFIC AMERICAN 1998
5. Wikipedia:Casimir effect

19 不確定性原理

19 不確定性原理【宇宙とは】宇宙との対話

19-1 1927年 ハイゼンベルク




図:真空のゆらぎ [3]

19-2 カシミール効果(Casimir effect) 
○また、二枚の金属板を振動させると光子やエネルギーが生じる。これを動的カシミール効果という。 [4] 

19-3 真空の相転移/インフレーション宇宙/トンネル効果 [6]


1. Wikipwdia:不確定性原理
2. ハイゼンベルクの不確定性原理 - nifty
3. 島根大学集中講義 真空の性質2009
4. Wikipedia:カシミール効果
5. SJN news” チャルマース工科大、真空から光子を生成。「動的カシミール効果」を実験で確認”(2011)
6. Wikipwdia:偽の真空
7. Wikipwdia:トンネル効果

18 Quantum Mechanics

18 Quantum Mechanics【What is the Universe?】Dialogue with the Universe

18-1 What is Quantum mechanics?
○Energy, length, speed, and time, those physical quantities have the smallest unit(quanta) that can not be divided any more. 

○About the particles like the size of electron (: 2.8 × 10 ^ -15m), you can see the state (the value of the energy), if you get the wave function by solving the wave equation (Schrodinger equation: partial differential equation 1926) , or if you solve the equation of motion of the Heisenberg (the determinant 1925) .
※The wave equation and the equation of motion of Heisenberg is mathematically equivalent.

○The Wave equation
• The Wave equation is a Fundamental equation of quantum mechanics. It is contrasted with Newton's equation of motion of classical mechanics.
・We can calculate the state of the matter wave stochastically by the equation.
・The Wave equation cannot be obtained strictly by using classical mechanics or mathematics. It is the equation that successfully describes the physical phenomena.
※Relativity is classical mechanics.

: reduced Planck constant
m: mass of particle
Ψ(Psi):wave function: Complex quantity, which was introduced to represent the wave
V: potential energy

○The squared modulus of the wave function is a real number interpreted as the probability density of measuring a particle's being detected at a given place. 
・The figure below shows the graph of the probability density and the energy state of matter. [1]

※The red line is the energy state(n). The probability density and n is changed at intervals.

☆The universe had particles that have exactly the same properties without limit, at energy following the mathematical rules!

○The universe was born from a quantum fluctuation.

○Wave function is a function of the complex because the wave equation contains complex number. [2]
・・・The universe is represented by imaginary!

18-2 Material Wave(de Broglie wave)
○On a submicroscopic level, there is no boundary strict between particles and wave.
→When existence is regarded as a wave motion, in physics, it is called a "place." For example, light is wave motion of an electric field and magnetic field.

※ The electron is wave.
※ And the length of orbit around the nucleus is an integer multiple of the wavelength (10 minus 10 square m).
※ It is impossible to determine the size of the electron.
※ It is under a very microscopic situation like an electron beam that the character as a wave is actually observed.

○The image of the wave of complex number [1]

※Whole wave represents one of the particles.
※Height and depth of the wave corresponds to the size of the wave function.
:de Broglie wave=the wave function

○The Wave of particles:A wave that moves the space represented by the complex number. [3]

○Applications [4]
・Description of the behaviors of the subatomic particles
・Computational chemistry
・The presence description of antimatter and negative energy
・Basic theory of semiconductor engineering → Electronic devices design
(the laser, the transistor (and thus the microchip), the electron microscope, and magnetic resonance imaging (MRI) , the light emitting diode, USB drives)
・Basic theory of superconductivity and superfluidity
・Quantum cryptography
・Quantum computing
・Quantum teleportation
・Parallel universes

○Unsolved problem
・Quantum gravity theory is incomplete.

18-3 Uncertainty Principle → No.19

1. 広江克彦”EMANの量子力学” Katsuhiko Hioe” EMAN’s Quantum mechanics”(Japanese)
2.Lecture 3: The Wave Function - MIT OpenCourseWare
3. EMAMの物理学・量子力学/調和振動子 EMAN’s Physics ・Quantum mechanics/ harmonic oscillator (Japanese)
4. Wikipedia:Quantum mechanics

18  量子力学

18  量子力学【宇宙とは】宇宙との対話

18-1 量子力学とは? [1]

○電子(:2.8×10^-15m)程度の大きさの粒子(物質)は波動方程式(シュレディンガー方程式:偏微分方程式 1926)を解いて波動関数を得るか、ハイゼンベルクの運動方程式(行列式 1925)を解くとその状態(エネルギーの値)が分かる。


Ψ(プサイ):波動関数(wave function):波動を表すために導入した複素量





・・・宇宙は虚数によって表現される! [3]

18-2 物質波(ド・ブロイ波(de Broglie wave))の理論


○複素数の波のイメージ [2]


○粒子の波:複素数で表される空間を動く波 [4]

○量子力学の成果 [5]
・半導体工学の基礎理論 → 電子機器の設計


18-3 不確定性原理 → No.19

【参 照】
2.広江 克彦”EMANの量子力学”
3.虚数は私たちの世界観を変えてしまった。 - とね日記
4. EMAMの物理学・量子力学/調和振動子


17  Theory of Relativity(Albert Einstein)

17  Theory of Relativity(Albert Einstein)【What is the Universe?】Dialogue with the Universe

○The universe is a four-dimensional space-time consisting of three-dimensional space and time, and the space-time can shrink or elongate. (Special relativity [1])
○ The speed of light(electromagnetic wave) traveling space-time is constant. (Special relativity)
○ You can write the relationship of energy and mass and speed of light with following equation. (Special relativity)
  ※c^2:raise c to the 4 power

   (E:energy(J)、m:mass(g)、c:speed of light(m/s))
○If the mass of the object which exercises reaches speed of light, it will become infinite. (Special relativity)

○The space-time is distorted. (General relativity[2])
○ The universe is expanding. (General relativity)
○ Black hole exists. (General relativity)

※ In accordance with Special relativity and General relativity, it is called Theory of relativity.
※ According to theory of relativity, the phenomenon in which an error becomes large if Newtonian mechanics describes (movement near the speed of light and movement in a big gravitational field) , can be described correctly.

1. Special relativity(1905)[Electromagnetic force ]
:The theory which dealt with the inertia system in the state where there is no gravity.
○The universe is the 4-dimensional world of time and space. The "space-time" seen from the watcher is a relative thing which is rectified by Lorentz transformation, and is extended or is shrunk.
○The length of a spacecraft is shrunk by a cross direction at a rate of.
(V: Motion speed, C:Light speed)
○At the system which exercises, compared with an external system, only is delayed in time. 

2. General relativity [Electromagnetic force + Gravity]
:Theory of gravity and space-time. (1915 - 1916)
• This theory described that gravity is generated by distortion of spacetime by mass.
・It is the extended equation of the Newton's law of universal gravitation, so that it can be applied to the strong gravitational field . By substituting the material and energy density such as a star on the right side, it is possible to estimate the curvature of the space-time around the star.
・Genera relativity can properly describe phenomenon(ex. exercise close to light-speed, or movement in a large gravitational field of such a black hole) more than Newtonian mechanics.
・Generally relativity predicts that black hole is formed where large mass is concentrated in a limited space.

○ Einstein’s equation: If you assign the mass of the material such as a star or energy in the right-hand side, it is possible to calculate the curvature of space-time around the star.


※ Left-hand side: Curvature of space-time (Geometrical quantity representing the bending of space-time):Total energy of the galaxy
※ Right-hand side: Density of matter and energy (causing a gravity) that exist in the universe 
⇒In the flat universe of curvature zero, the total energy for the entire galaxy becomes zero.
=In the flat universe, the sum of the negative energy due to gravity (repulsion) and the positive energy(kinetic energy) is zero.

・Gμν:Einstein tensor(Tensor represents the distortion)
・Λgμν : The cosmological constant
・κ :Einstein's constant of gravitation
(κ=8πG/c^4 (π:pi、G:Gravitational constant、c :Speed of light))
  ※c^4:raise c to the 4 power
・Tμν:energy-momentum tensor
・μ, ν: Identify the coordinates of space-time.