Botzmann, Einstein, and the Physics of Ignorance
- Physicists are music lovers, and the physics conferences
usually include music programs. Let us start this page with a music story.
- Click here for my Vienna page with some music
stories. These days, there are many other music cities .
- Click here for other music cities around the world.
- 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.
- Christian Doppler. Doppler formulated
his formula for sound waves originating from a moving object. This concept
is applicable to light waves.
- Erwin Schrödinger. If you are a
physicist, you live with him everyday. Click here
for a webpage dedicated to him.
- Ludwig Boltzmann is known to us for
- S = k.log W
relating the entropy to the thermodynamic probability. He is known to us for
Click here for a large image.
- 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 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.
- Christian Doppler. Doppler formulated his formula for sound waves originating from a moving object. This concept is applicable to light waves. Michelson-Morely experiment!
Entropy in Quantum Mechanics
von Neumann in his Princeton house.
- 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
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
- Wigner in 1932 introduced the phase-space distribution function known as
the Wigner function whose purpose is very similar to that of the density
- Wigner and von Neumann came from the same high school in Hungary.
Click here for their high school in
- 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
- They both had their houses in Princeton, and the spoke Hungarian whey met.
- 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.
- 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.
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
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
- 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)