Day Three

Lecture on Physics by Dr Yukawa

Day Three

Contents

　　1.量子“論”と量子“力学” Quantum "Theory" and Quantum "Mechanics"
　　2.波動ということ――エーテルから場へ Wave - From Ether to Field
　　3.不確定性関係を導く二つの方式 Two methods leading to Uncertainty relation
　　4.物理学における認識 Recognition in Physics
　　5.電子の拡散 Diffusion of electrons
　　6.古典的因果律からの転換 Shift from Classical Causality
　　7.シュレーディンガーの猫 Schlodinger's Cat
　　8.量子力学の完成――場の量子論 Completion of Quantum Mechanics - Quantum Theory of Field
　　9.量子力学と特殊相対論 Quantum Mechanics and Theory of Special Relativity
　　10.孤高の理論・一般相対論――一般共変性をめぐってLone Theory - Theory of General Relativity - About General Covariance
　　11.物理量と幾何学的量とのアイデンティフィケーション Identification of Physics Quantity and Geometric Quantity
　　12.入れ物（時空）と中身（物質） Container (Space-Time) and Contents (Matters)
　　13.一般相対論はミクロの世界と無関係か？ Does Theory of General Relativity have no relation with Micro World?
　　14.素粒子論――局所場と非局所場 Theory of Elementary Particles -Local Field and Non-local Field
　　15.差分的な考え方による可能性 Possibility of Finite Difference method
　　16.余話――外界認識の連続性と不連続性 Appendix - Continuity and discontinuity of recognition of the world

 

2.波動ということ――エーテルから場へ Wave - From Ether to Field


11.物理量と幾何学的量とのアイデンティフィケーション Identification of Physics Quantity and Geometric Quantity

I think that the creative activity is to reach an identification in a form of very high degree, through a remarkably increasing upward process of identification. This is a very big creation. For instance , although there many other examples, Boltzmann wrote

by using Boltzmann constant k, where S = Entropy, W= the number of micro states belonging to the macro states - Probability. He found this relationship. Thermodynamics explains that Entropy is the ratio of In and Out of Heat devided by the absolute temperature (T) when there is In and Out of Heat due to the application of energy. This is the macro quantity. It is an very outstanding thinking to connect the macro quantity with the micro quantity, which leads to the concepts of Probability and then Information Theory. This equation is not putting the same thing at each side of the equal sign (=). People tend to think that the both sides are the same. But it is only after Boltzmann put them this way that people think so. Boltzmann put the different things at the each side of equal (=).







Boltzmann's most important scientific contributions were in kinetic theory, including the Maxwell–Boltzmann distribution for molecular speeds in a gas. In addition, Maxwell–Boltzmann statistics and the Boltzmann distribution over energies remain the foundations of classical statistical mechanics. They are applicable to the many phenomena that do not require quantum statistics and provide a remarkable insight into the meaning of temperature.

Boltzmann’s 1898 I₂ molecule diagram showing atomic “sensitive region” (α, β) overlap.

Much of the physics establishment did not share his belief in the reality of atoms and molecules — a belief shared, however, by Maxwell in Scotland and Gibbs in the United States; and by most chemists since the discoveries of John Dalton in 1808. He had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient theoretical constructs. Only a couple of years after Boltzmann's death, Perrin's studies of colloidal suspensions (1908–1909), based on Einstein's theoretical studies of 1905, confirmed the values of Avogadro's number and Boltzmann's constant, and convinced the world that the tiny particles really exist.
To quote Planck, "The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases".^[7] This famous formula for entropy S is^[8][9]

where k = 1.3806505(24) × 10⁻²³ J K⁻¹ is Boltzmann's constant, and the logarithm is taken to the natural base e. W is the Wahrscheinlichkeit, the frequency of occurrence of a macrostate^[10] or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — number of (unobservable) "ways" in the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann’s paradigm was an ideal gas of N identical particles, of which N_i are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

where i ranges over all possible molecular conditions. ( denotes factorial.) The "correction" in the denominator is because identical particles in the same condition are indistinguishable.
Boltzmann was also one of the founders of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete.
The equation for S is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof — his second grave.

 ----

Information theory

Main articles: Entropy (information theory), Entropy in thermodynamics and information theory, and Entropic uncertainty

In information theory, entropy is the measure of the amount of information that is missing before reception and is sometimes referred to as Shannon entropy.^[65] Shannon entropy is a broad and general concept which finds applications in information theory as well as thermodynamics. It was originally devised by Claude Shannon in 1948 to study the amount of information in a transmitted message. The definition of the information entropy is, however, quite general, and is expressed in terms of a discrete set of probabilities :

In the case of transmitted messages, these probabilities were the probabilities that a particular message was actually transmitted, and the entropy of the message system was a measure of the average amount of information in a message. For the case of equal probabilities (i.e. each message is equally probable), the Shannon entropy (in bits) is just the number of yes/no questions needed to determine the content of the message.^[22]
The question of the link between information entropy and thermodynamic entropy is a debated topic. While most authors argue that there is a link between the two,^[66][67][68] a few argue that they have nothing to do with each other.^[22][69]
The expressions for the two entropies are similar. The information entropy H for equal probabilities is

where k is a constant which determines the units of entropy. For example, if the units are bits, then k = 1/ln(2). The thermodynamic entropy S, from a statistical mechanical point of view, was first expressed by Boltzmann:

where p is the probability of a system's being in a particular microstate, given that it is in a particular macrostate, and is Boltzmann's constant. It can be seen that one may think of the thermodynamic entropy as Boltzmann's constant, divided by log(2), times the number of yes/no questions that must be asked in order to determine the microstate of the system, given that we know the macrostate. The link between thermodynamic and information entropy was developed in a series of papers by Edwin Jaynes beginning in 1957.^[70]
There are many ways of demonstrating the equivalence of "information entropy" and "physics entropy", that is, the equivalence of "Shannon entropy" and "Boltzmann entropy". Nevertheless, some authors argue for dropping the word entropy for the H function of information theory and using Shannon's other term "uncertainty" instead.^[71]

Mathematics

Day Tow

Lecture on Physics by Dr Yukawa

Day Two

Contents
 
　　1.科学者分類学――孤立型　対話型　集団型 Classification of scientists - Lone type, Dialog type, Group type
　　2.会議の効用 Effectiveness of meeting
　　3.ニュートン力学における空間 Space of Newton Mechanics
　　4.ベクトルの歴史 History of Vector
　　5.空間の点に名前を付ける To name a point in space
　　6.見かけの力と本当の力 Factitious Force and Real Force
　　7.マッハの解釈 Mach's interpretation
　　8.ニュートンの偉大さ Greatness of Newton
　　9.絶対空間をめぐってAbout Absolute Space
　　10.“場”とは何か What  is "Field"?
　　11.相対論における場 Field in Theory of Relativity
　　12.特殊相対論による場の制約 Limitation of Field by Theory of Special Relativity
　　13.ニュートン力学的因果律――ラプラスの魔  Causality of Newton - Laplace's Demon
　　14.余話――ラプラスとその時代 Appendix - Laplace and his time
　　15.特殊相対論の因果律 Causality of Theory of Special Relativity


2.会議の効用 Effectiveness of meeting

In contrast to this (Lone type) there is another interest thing called "meeting". This is neither a class room meeting nor business meeting but precisely speaking called "debate".
One very important debate on physics is Slovay Conference. The first Conference was held probably in 1911 when the Theory of Relativity Theory had already been available in public, and Quantum Theory, Plancks' Quantum Theory was just a very popularly debating issue. Planck was originally a classic physicist and comparatively lone type. But at that time he was leading Quantum Theory. Besides Planck there were prominent classic physicists like Lorentz and Ehrenfest. Rutherford may have also attended. Those kind of people attended. At the conference they discussed whether the new and strange Quantum Theory proclaimed by Planck and Einstein was correct or not. The great physicists like Lorentz and Ehrenfest attended and (the new theory claimers) debated with them. What was the outcome? For instance Lorentz was a great physicist but a supporter of the classic physics so no matter how he tried to change the methods he could find only Rayleigh-Jeans Radiation Law, which was natural. But he must accept Planck's Radiation Law (Note 25)  as this is correct. However, it is very difficult to change the way of thoughts. This was what happened at the first Slovay Conference and is a very famous story. I do not think Niels Bohr attended this conference.

Note 25) is very long. "Classic Radiation Theory and Planck" introduces

1. Stefan–Boltzmann Law
2. Wien's Displacement law
3. Vien's Radiation Law 
4. Rayleigh–Jeans Law
5. Planck's Radiation Law

Since the Notes were not made by Dy Yukawa but added by the editor to the book version to help the readers. I do not translate them (for now) but these notes are helpful to understand what Dr Yukawa said in the lecture.


(This section will continue)

3. ニュートン力学における空間 Space of Newton Mechanics

Up to where did I lectured yesterday?  I was wandering about Newton's Mechanics and started talking about space. We tend to think that we know the space well. When we actually talk about the space, we are dealing with the space of the mechanics. Or we could say that it is about the ordinary experience world. We use some kind of ways to understand the ordinary experience world. In physics the matter is dissolved to a point mass or rigid body. Material objects are here and there and we regard that they exist in space. This is a very natural view. Nothing to say about it. This has been so since Democritus. According to Democritus the world is made of atoms and void so matter can move. It was not  possible to prove in the ancient time whether the atom theory was correct or not. But this was one of the easiest ways of thinking about the world. But space is the collection of points after all. In mathematics points cannot extend to 2D or 3D even they get together as many as possible. But do not worry much about this. If it matters to you, think of dividing the space into very small pieces and then divide each of them again and again. The limit will be a point. Did we talk about "point matter or rigid body ?" yesterday, which is relates with what I just talked. Space and matter are different. As of matter we have a problem - what angular momentum will become when making a rigid body very small.

The angular momentum (L) remains limited. Even when making r to very small r will be able to become zero as far as ω is limited. Think like this. I talked about this yesterday. But unless we have the condition of ω being limited, when making ω very large r may be limited and takes a certain value. This may relate with spin when going down to Quantum Mechanics (Note 26). But there is a gap between Classical Mechanics and Quantum Mechanics. It may not have any useful meaning to try to bridge Classical Mechanics with Quantum Mechanics or it may have. I cannot say for sure about this.

Meanwhile we do not have this kind of problem when we treat the point. The point is the thing which cannot be divided any more. This is the characteristics of the point. The point has something more than what Euclid defined. The point here means that the space is only one, or "unique" (only one) in another word. Someone may say it not so important, does not matter. But it is important and it matters.

As I explained yesterday usually matter is point mass. We think of the motion of matter as being minimized to the limit. This is a question where the matter is located in terms of time. To determine the position of the matter we introduce orthogonal (rectangular) coordinate system or Cartesian coordinate system. For instance we take the direction and the length of a radial vector (Note 27). Most of the teachers of mechanics use vectors nowadays and I think it natural as mechanics was originally made by introducing time to the Euclid geometry and further more by combining the laws of motion and force with it. Therefore it is quite natural to think of the things like vectors in the Euclid geometry.


Note 26)

In Quantum mechanics,

The extended value of electron is regarded as "r" (radius), ie   = r, which is the Compton wavelength, λ, of a particle is given by

where h is the Planck constant (energy x time, unit: Joule ・second), m is the particle's rest mass, and c is the speed of light.


The angular momentum of spin is

where

To replace ω by v/r, and put "r" = "λ"

                      h
 L = mvr = m  --  v
                      mc

When v = c (speed of light), L = h.

Note 27) Radial vector

To fix a certain point as the origin. To draw a straight line from the origin to an object. A vector which determines the location of the object by using this line having the length and direction is called a radial vector.


 .
4.ベクトルの歴史 History of Vector

However, the actual history of physics differs very much. Most science historians study and state that physics has been developed by following the rule of causality.There are many things derailing this rule. Very generally speaking we can see the basic flow of developments which seems following the rule of causality and could say to fairly reasonable extent that at a certain time a certain thing had come out as it should have. But there have been many exceptions.

One remarkable exception is vector. We think that vectors had come out as they should have and they have been used already since a very long time ago. But it was not so. As I have already mentioned several times in the books of Maxwell we can find ordinary components - ie Maxwell's equations which expressed by Cartesian components, each multiplied by a derivative and no or a very few (if any) vectors involved. This was a story in late 19th century when the symbol of vector was not found.
What existed before vector was quaternion. It is really strange. We know Hamilton. Those who study Quantum Mechanics study Hamilton and Lagrange. Hamilton lived in 19th century. His canonical formalism is studied even now. Both Hamilton and Lagrange were great physicists. Hamilton, also mathematician, created quaternion. What is Quaternion? This is the forth dimension number. Quaternions are not real numbers while the ordinary vectors are real. I do not know why a strange thing like quaternion was used but quaternion was popular back in the 19th century.  Although I had no particular interest in it I investigated it and found that an English (Scottish) physician Tait wrote a large volume of book titled "Quaternion". Tait worked with Lord Kevin.

A little later, Willard Gibbs, whom you may know well in the other field, came in. He is an American scholar. The Americans say that Gibbs is the greatest physicist in USA, which is fairly correct though I do not have any intention to rank the physicist. They say, Gibbs is a little Newton. This is because that Gibbs lived almost his life as a teacher at Yale university in New Haven just as living in an ivory tower and what he had done was out of mistakes. I think what he had done has no mistakes. In terms of statistical mechanics there are major two - one is Boltzmann and the other is Gibbs. Most of you may prefer Gibbs' statistical mechanics. Like Newton did what Gibbs did had no mistakes. One the other hand Boltzmann, whom I like, was struggling in his statistical mechanics. Gibbs looked at what Boltzmann was doing and took a part of what Boltzmann did and made a simply organized statistical mechanics. 

I do not intend to talk about statistical mechanics here. The Gibbs lecture notebooks in which he wrote for his lecture at at Yale university in 1880's  remain. In the notebooks he used scalar product and vector product although the currently using bold letter showing vectors are not seen. And also the symbols showing the equivalent to the current curl, gradient, nabra, etc  are also seen. In short he lectured physics by using vectors. This is supposed to be the first of using vectors in physics teaching and found by a brief research made by Mr Kawabe.

However in his writings we have found the fact that his lectures by using vectors were not welcome. Tait mentioned above criticized Gibbs' using vectors. This is a strange thing. We now may not be able to understand why strange quaternions should be used. In quantum mechanics some quantities which are very difficult to grasp by our intuitions are often important. Also difficult-to-grasp matrix showed up later. But these are in 20th century. The classic physics was so made that it did not required these strange things. Using vectors was natural but Tait was very against it. Gibbs responded to Tait's criticism. The contents of these arguments are not very important. Using vectors was all right but criticized, and then making remarks against the criticism.

It was again strange that after Gibbs the use of vectors became widespread. Even the text books published in Britain started to use vectors. In our student period these text books used more vectors than the textbooks from other nations. Quaternions disappeared. History is very interesting. Hamilton was no doubt a great man but the story seems reversed in this case.

 First someone conceived and proposed a strange idea. And then it became to be regarded as normal. And during dong the normal things a new strange thing was discovered. Without discovery of strange things no development of physics is made. But in case of vectors unnecessarily strange thing was proposed and then returned to the more normal vectors. However in 20th century onward quaternions has been considered to have a similar characteristics like spins. This is also very interesting. I could say that Hamilton had seen far future although this is my supposition. But this is a story of what happened.
 
 
5.空間の点に名前を付ける To name a point in space