# Botzmann, Einstein, and the Physics of Ignorance

 Great thinkers at the main quadrangle of the University of Vienna.

 S = k.log W from his grave stone in Vienna.
Until 1950, Vienna served as the music capital of the world, and we still regard Vienna as a music city.

• Physicists are music lovers, and the physics conferences usually include music programs. Let us start this page with a music story.

1. Click here for my Vienna page with some music stories. These days, there are many other music cities .

• In addition to those music composers, Vienna produced many great thinkers who changed this world. Let us go to the main campus of the University of Vienna, and got to the main quadrangle. You will notice busts of great thinkers who changed this world. There are so many whose names you cannot recognize. However, you should recognize three of them. I had photos with them.

1. Christian Doppler. Doppler formulated his formula for sound waves originating from a moving object. This concept is applicable to light waves. Michelson-Morely experiment!

2. Erwin Schrödinger. If you are a physicist, you live with him everyday. Click here for a webpage dedicated to him.

3. Ludwig Boltzmann is known to us for his formula

• S = k.log W

relating the entropy to the thermodynamic probability. He is known to us for

4. Where does the entropy stands in Thermodynamics? We know how to measure temperature, pressure, and volume. We also know about the internal energy, Helmholtz function, Gibb's function, as well as the enthalpy.

The Maxwell relations tell us the role of this unmeasurable quantity, but the list of formulas is not enough. John A. Wheeler attempted to organize these quantities in a two-by-two matrix form. Here is the table I refurbished for the purpose of this webpage.

## Entropy in Quantum Mechanics

 von Neumann in his Princeton house.
The purpose of this webpage is to talk about what role his entropy formula plays in Einstein's Lorentz-covariant world.

• In quantum mechanics, the concept of probability plays the central role. There are also measurement problems. There are variables which can be measured as well as those that cannot be measured. For the variables that cannot be measured, the best we can do is to take their average values. With these ideas in mind, von Neumann introduced the density matrix.

• In his 1972 book on statistical mechanics, Feynman explains why we need the density matrix. He says

When we solve a quantum-mechanical problem, what we really do is divide the universe into two parts - the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the entire universe. To motivate the use of density matrices, let us see what happens when we include the part of the universe outside the system.

Feynman then used the density matrices and Wigner functions to illustrate his rest of the universe. However, he used one harmonic oscillator to illustrate what he said about the rest of the universe. Yes! The harmonic oscillator is the basic tool to illustrate the Wigner function. But how could he explain two different worlds with one oscillator?

In order to understand fully his rest of the universe, we have to use at least two coupled oscillators where one of them serves as the observable universe while the other is for the rest of the universe.
Click here for a paper published for the American Journal of Physics. As you probably know I wrote many papers on two-oscillator systems, and I love coupled oscillators.

 Wigner was a no-nonsense man, so was Feynman.

• Eugene Wigner was intensely interested in the density matrix. There for good reasons.

1. Wigner in 1932 introduced the phase-space distribution function known as the Wigner function whose purpose is very similar to that of the density function.

2. Wigner and von Neumann came from the same high school in Hungary. Click here for their high school in Budapest.

3. During the period 1942-45, they both worked in the Manhattan Project for developing the first nuclear bomb, based on a Hungarian science fiction talking about super high explosive material wrapped around by the gunpowder.

4. They both had their houses in Princeton, and the spoke Hungarian whey met.

• During the period 1985-1990, my job was to tell Wigner the stories he wanted to hear. I was able to do this job because I knew what he was interested in. I knew he was interested in the information content of the density matrix while I was a graduate student (1958-61). There was a visitor named Matsuo Yanase from Japan working with Wigner. Yanase told me about Wigner's interest in von Neumann's density matrix and measurement problems. Wigner and Yanase published a paper on this subject in the proceedings of the National Academy of Sciences, Vol. 49, pages 910-918 (1963). While talking with Yanase, I realized the entropy is a measure of ignorance.

As for Wigner's papers, I was primarily interested in Wigner's 1939 paper on internal space-time symmetries of relativistic particles, and my research team was able to construct his representation based on the Lorentz-covariant oscillator wave functions, based on the mathematics of the two coupled oscillators. Click here for the paper. Eventually, with Marilyn Noz, I was able to write a book on this subject. Click here for a review of this book. With this book, I was able to approach Wigner.

I knew from one of his papers that Wigner was also interested in the time-energy uncertainty relation. He talked about Dirac's time-energy uncertainty relation in his article in the volume entitled "Aspects of Quantum Theory" dedicated to P. A. A. Dirac to commemorate his seventieth birth day and his contribution to quantum mechanics, edited by Abdus Salam and E. P. Wigner (Cambridge University Press, 1972).

 If there is a space-like separation, there must also be a time-like separation, according to Einstein. Yet, this variable was thoroughly hidden behind Bohr and Einstein. It is not enough to blame them. We have to figure out what is going on.

• The time-separation variable between the two constituent particles plays an essential role in the Lorentz-covariant world. This variable was mentioned by Feynman and his students in their 1971 paper. However, they said they would drop this variable because they did not know how to handle it. This is not what we expect from Feynman's papers, but it is not enough to blame Feynman.

Indeed, this blame should be directed to Bohr and Einstein. Einstein was worrying about how things appear to moving observers, while Bohr was interested in the hydrogen atom. Bohr is responsible for the minimum limit on the separation between the proton and the electron. This spatial separation is called the Bohr radius. Click here for a detailed story. They should have addressed this issue, but they did not.

• Do you know how to deal with this problem?. Feynman did not know, but it is clear that this variable belongs to the rest of the universe Feynman discusses in his book of 1972 based on his lectures delivered in earlier years. Indeed, the density matrix tells us how to take care of the variable we are not able to observe, and our ignorance over this variable appears as the entropy in the obserbable world.

We can study this problem using harmonic oscillator wave functions. If the wave function takes the Gaussian form

where z and t are the space and time separations respectively. This wave function is separable in the these two variables, and the z component of the density matrix is not affected by our ignorance over this time-separation variable. However, the story is different when the system is Lorentz boosted, and the result is

The z and t variables become entangled. We can translate these formulas into the following cartoon.

• I told this story to Wigner and he liked it, since he was the first one to worry about non-separable (entangled) variables. We then calculated the density matrix and published a paper entitled

What is missing in that paper is the entropy plotted against the Lorentz-boost parameter. Here is the graph. I did not know how to use Wolfram's Mathematica. Did it exist at that time?

• At the same time, I was working with my younger colleagues and published a paper about the entropy leading to the temperature rise. Click here for detailed calculations. The result of this paper can also be summarized as

## Gaussian Entanglement

• These days, there are many papers based on the Gaussian entanglement. There there is another observer in Feynman's rest of the universe. What happens the first observer when the second oberver makes his/her observations.

Indeed, this Gaussian picture of the entanglement, as well as Feynman's rest of the universe is based on the followong mathematical formula.

• Do you know who derived this formula? He was an undergraduate student doing a summer research work with me in 1976. His name is Seog Oh, and is now a professor of physics at Duke University.

This result was included the appendix of the paper by Kim, Noz, Oh which was published in J. Math. Phys. However, the referee insisted that the paper be shortened, and we had to delete the appendix. On other hand, the editor thought this formula was important enough to be archived. He asked us to submit it to Physics Auxiliary Publication Service. The editor at that time was Morton Hammermesh.

Since this result is a product of an undergraduate research program, we thought it is appropriate to publish in the American Journal of Physics. Click here for this AJP paper. Since then, I used this formula very frequently in my papers with Marilyn Noz. This formula is also contained in my paper with Wigner mentioned above.

Y. S. Kim (March 2016)

copyright@2016 by Y. S. Kim, unless otherwise specified. The photo of Bohr and Einstein is from the E. Segre Visual Archives of the American Institute of Physics.