Particle-Wave duality and Hamilton-Jacobi equation

[Quantum River]

I was fascinated by the quaternion ring in my senior middle school, but I dropped it when I find it is even impossible to define simple analysis concepts such as limit, derivative, integral in the quaternion ring. In my opinion, our/mathematicians' understanding of such noncommutative things such as quaternion, matrix, operators is quite limited. Even the simple group of 2*2 matrix is not quite clear. Our limited understanding may be rooted in our naive understanding of more fundamental problems.

It is difficult to imagine a physical theory without derivative and integral, so I think a physical theory based on quaternion may not be so bright. But I still admit quaternion is fascinating.

 

[Anonym]

I find it is even impossible to define simple analysis concepts such as limit, derivative, integral in the quaternion ring.

Notice, that the quaternions are 4-dim quadratic normal division algebra with the signature {1,-1,-1,-1}. Enter Google, type quaternion analysis and you will get list of 330,000 publications. However, I have bad experience with the mathematicians. Almost always I find what I don’t need and don’t find what I need. Usually, I adopt the approach: do it by yourself. It is common practice in our room. Otherwise, why you need study that volume of math.? Indeed, you may be much more sophisticated and bring close to you Weyl and Minkowski. You should find your very specific point. Try to make it as simple as possible. Then apply all what you know. Don’t forget that “raffinert ist der Herr Gott, aber boshaft ist Er nicht".

 

[Anonym]

Quantum River, I would like to add something. We mentioned algebra and analysis as background. It is obvious, but I would like to emphasize it explicitly: you should speak fluently the theory of continuous groups.

I understand that you are a student. Most important: you should choose for you the teacher. Check your environment, if there is somebody in it that able to answer intelligently your questions. If not, go to Utrecht at Europe or Stony Brook,N.Y.

Dany.

 

[Quantum River]

Quantum River, I would like to add something. We mentioned algebra and analysis as background. It is obvious, but I would like to emphasize it explicitly: you should speak fluently the theory of continuous groups.

I understand that you are a student. Most important: you should choose for you the teacher. Check your environment, if there is somebody in it that able to answer intelligently your questions. If not, go to Utrecht at Europe or Stony Brook,N.Y.

Dany.

Anonym, thanks!
Now I am reading some papers on string theory and loop quantum gravity. It is more interesting and there is a lot of creative work to do.

I think duality is at the heart of physics. Besides the most important particle-wave duality, there are electromagnetic duality, T-duality, S-duality, Maldacena duality and much more. From a philosophical point, there are always two sides of a coin. There could be more than two sides, but the case of two sides is the most important one. Today particle-wave duality expresses the duality most clearly. But we may find a more fundamental duality which should encompass the particle-wave one as a special case. In mathematics, there is Langlands program. Fermat's last theorem is just a very small part of the program. I will not say more. Above duality, there must be the oneness, which is my personal faith. I have read William Hamilton's original paper On a general method of expressing the paths of light, and the planets, by the coefficients of a characteristic function. I think Hamilton still viewed the dynamics of the light as a path/orbit. So Hamilton didn't have a sense of duality. I'm not absolutely sure about this. Sometimes looking at a problem from a historical perspective is really interesting.

I am going to apply to a USA graduate school next year. Besides, why should I go to Utrecht?

Quantum River

 

[Anonym]

I am going to apply to a USA graduate school next year. Besides, why should I go to Utrecht?

QR, I did not say that you should. I only mentioned that it is a good place not only to study physics but also to study how excerpt the relevant point and not worry too much whether the suitable math tools already available.

 

[Anonym]

[QUOTE =Quantum River;1223047]Gross Nobel lecture on QFT and QCD.
http://nobelprize.org/nobel_prizes/p...ss-lecture.pdf [/QUOTE]

QR, thank you. It is amazing document and I enjoyed reading it. Using this example we may discuss better what are the interconnections between physics, mathematics and education.

1.Elementary Particles Physics ( Particle Physcis in the 1960’s):

“There was hardly any theory to speak of… Symmetries were all the rage… The biggest advance of the early 1960’s was the discovery of SU(3) by Gell-Mann and Y.Neeman… The most dramatic example of this is Gell-Mann and George Zweig’s hypothesis of quarks… Quarks clearly did not exist as real particles; therefore they were fictitious devices (see Gell-Mann above)… {interesting interpretation; notice, that M.Gell-Mann introduced quarks}… One was not allowed to believe in this reality… Weinberg was emphatic that this was of no interest since he did not believe anything about quarks {who is The Weinberg? just cloned Bohr?}.

To summarize:absence of the physical intuition, absence of understanding of the elementary particle physics, absence of understanding of group theory and Cartan classification.

2.Quantum Field Theory

“In my first course on QFT at Berkley in 1965, I was taught that Field Theory= Feynman Rules… My Ph.D. thesis was written under the supervision of Geoff Chew, the main guru of the bootstrap, on multi-body N/D equations… Nonetheless, until 1973 it was not thought proper to use field theory without apologies… Yang-Mills theory, which had appeared in the mid 1950’s was not taken seriously… I decided, quite deliberately, to prove that local field theory could not explain the experimental fact of scaling and thus was not an appropriate framework for the description of the strong interactions.

The Yang-Mills theory explain the origin of minimal coupling in Maxwell electrodynamics and covariant derivative in GR.

To summarize:absence of the physical intuition, absence of understanding of the differential geometry.

3.Asymptotic freedom

I would show that there existed no asymptotically free field theory… I knew that if the fields themselves had canonical dimensions, then for many theories this implied that the theory was trivial, i.e. free…The second part of the argument was to show that there were no asymptotically free theories at all.

To summarize:absence of the physical intuition, absence of understanding of the functional analysis.

Obviously, his ideas were interpretation dependent. However, he was successful in his “proves” and finally the coin falled inside his brain.

To summarize, on Hebrew we say: Lo mevin klum (American Education System outcome). By the way, during one of his lectures he said that he consider himself more clever then A. Einstein (Dubrovnik, June 1983).
For the nobel prize motivated kids it is the experimental confirmation that they have a chance.

Dany.

P.S. I did not read F.Wilczek lecture. However, “If two separated bodies, each by itself known maximally, enter a situation in which they influence each other, and separate again, then there occurs regularly that which I have just called entanglement of our knowledge of the two bodies.” (E. Schrödinger, “THE PRESENT SITUATION IN QUANTUM MECHANICS”, Paragraph10, Theory of Measurements,Part Two).