# Feynman's Rest of the Universe

In his book on statistical mechanics, Feynman 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.

Feynamn then used the density matrices and Wigner functions to illustrate his rest of the universe. However, he used only one 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?

• It appears that the best way to understand what Feynman said above is to use two coupled harmonic oscillators. One of them is in the world in which we do physics, and the other is for the rest of the universe. It is straight-forward to construct the Hamiltonian for the system of two coupled oscillators. It is not difficult to construct wave funtions, density matrices, as well as Wigner functions for this two-oscillator system. Then there is a well-established procedure for constructing the density matrix and Wigner function when one of the two oscillators is not obseved: to sum over or integrate out the un-observed variable. The net result is the increase in entropy through the density matrix, and an addition of statistical uncertainty through the Wigner function.

• However, it was difficult for me and my co-authors to publish the paper saying this. The reason is very simple. The referees kept saying the paper is too trivial, but the truth was that they did not like my research group becoming more famous than they are by talking about Feynman. This is the Herod complex shared by all physicists.

It took us eight years to publish, and it appeared finally in the American Journal of Physics in 1999. The final referee said the paper is right and appropriate for the journal, but he could not recommend publication. He did not explain why. The editor over-ruled the referee's Herod complex after our appeal.

• Click here for the article using two oscillators to illustrate Feynman's rest of the universe. One oscillator for the world in which we do physics, and the other for the rest of the universe.

## Two-mode Squeezed States

P. A. M. Dirac was quite fond of harmonic oscillators and the Lorentz group. He attempted to construct a representation of using two oscillators, and ended up with the O(3,2) deSitter group applicable to five dimensional space with three space and two time coordinates. Here is his paper.

In this paper, he starts with the step-up and step-down operators for two oscillators attempted to construct a Lie algebra with a closed set of commutators. He then ended up with ten generators given in this page.

• This big O(3,2) group has many small O(2,1) subgroups. Every O(2,1) subgroup is the Lorentz group applicable to two space and one time dimensions. One of them later became the basic language for two-photon coherent states commonly called "two-mode squeezed states." While both photons can be observed in laboratories, we can pretend not to see it. In this way, we can see clearly what Feynman was talking about when he mentioned the rest of the universe.

• With this preparation, we can become ambitious enough to tackle the problem of hidden variables in quantum mechanics. The question is whether they are hidden in Feynman's rest of the universe.

## Time Separation hidden in Feynman's Rest of the Universe.

 They could have talked the Lorentz-boosted Bohr radius, but they did not. The time separation was a hidden variable to them.
Niels Bohr (1885-1962) was born six years after Einstein. Whenever Bohr mentions the word "space" in his philosophical writings, he adds "time." This is how much Bohr respected Einstein. Einstein also had a great respect for Bohr. At the same time, Einstein had his objections to the Copenhagen interpretation of quantum mechanics. This aspect is well known and is still actively debated by the present generation of physicists.

• Niels Bohr is also well known for his Bohr radius. To him, the proton was sitting at the center of the universe, even though he knew the existence of the time variable associated with the space. Einstein perhaps thought about the hydrogen atom in a moving frame, but he never said anything about it. His Lorentz transformations are strictly for point particles.

Before saying there was something wrong with them, let us look at Thomas Jefferson. He said "All men are created equal." But he forgot to mention "women" there, and he was not able to practice his concept of equality with some of the the people who served him at his Virginia mansion called Monticello. Jefferson's freedom remained as a hidden variable to at least one half the world population for many years. It took many years of evolutionary processes to see the full extent of this variable. It is quite possible that this variable is still not completely discovered.

Let us go back to Bohr and Einstein. Physics is an experimental science. It is still unthinkable for the hydrogen atom to move with a relativistic speed, but the proton can be accelerated to a speed very close to that of light. The question then is whether the proton has it own quantum mechanics like the hydrogen atom. Yes, the quark model formulated in 1964 could tell the story. We can therefore illustrate the evolution of the hydrogen atom has gone through with this figure.

Needless to say, this evolution was made possible thanks to the advances in the accelerator physics. At the proton proton speed of 0.9999999999c. At magnitude of time separation is essentially the same as that of the spacial separation.

In order to study this hidden variable, I spent many years with Marilyn Noz. We published many papers, but we also had many problems with the referees, because the time-separation variable is thoroughly hidden to them.

• The first person who recognized the existence of this variable was Richard Feynman. Based on his talk given at the 1970 APS spring meeting, he published his paper with his students in Phys. Rev. D 3 , 2706 (1971)]. In this paper, Feynman et al. state the existence of the time-separation variable. They said they could not do anything about this variable because they do not anything about it. I like very much their honest statement.

• As for the quantum mechanics of the time variable, Paul A. M. Dirac worried about the time-energy uncertainty relation in his paper

• P. A. M. Dirac, Proc. Roy. Soc. (London) A114, 243 (1927).

He noted in this paper that the time-energy uncertainty relation is a c-number relation, and that this lack of time-like excitations makes it difficult to combine quantum mechanics with special relativity. In his 1945 paper:

• P. A. M. Dirac, Proc. Roy. Soc. (London) A183 , 284 (1945).

Dirac uses the four-dimensional harmonic oscillator to understand the Lorentz-covariant world. The fourth variable is the time variable. I assume he had in mind the time-separation variable. If we combine Dirac's 1927 paper with his 1954 paper, we end up with this figure with a circle.

In 1949, Einstein became 70 years old, and the Reviews of Modern physics published a special issue. This issue contains Dirac's paper:

1. P. A. M. Dirac, Rev. Mod. Phys. 21 , 392 (1949).

It is then not difficult to combine the circle and the rectangle to construct the ellipse. Does this sound like a science fiction to you? We observe this effect whenever we turn on a particle accelerator. This is a picture of the proton released from the accelerator. Click here for a detailed story.

Paul A. M. Dirac was of course a great physicist. I seem to work strictly within his framework. I could do my own research because he left some soft spots in his papers.

1. Dirac's papers are poems, but have you seen figures there. I seem to enjoy translating his poems into cartoons.

2. Dirac wrote many history making papers. However, Dirac seldom takes advantage of his own papers published earlier. It is therefore profitable to link up the papers Dirac published at different times.

3. Dirac had a "famous-brother-in-law" named Eugene Wigner who wrote a paper on representations of the Poincare group in 1939. In his 1949 paper, Dirac said the task of constructing relativistic quantum mechanics is the same as that of constructing a suitable representation of the Poincare group.

When I talked with Dirac in 1978, he was still uncomfortable about the lack of time-like excitations in the Lorentz-covariant world. He used to call this space-time asymmetry. He could have learned about Wigner's O(3)-like little group to clarify the question of space-time asymmetry.

• Time Separation as a Hidden Variable: In spite of all I said above, the time separation remains as a hidden variable to the absolute majority of physicists even these days, especially to the orthodox Copenhageners. How do you define the word "hidden" in physics? This concept came from the uncertainty principle which does not allow a point in the phase space of position and momentum.

The best way to approach this problem is to use the Wigner phase space, where position and momentum is confined in an area not smaller than Planck's constant. The area of uncertainty can increase if we fail to measure other variables. In that case, the region of uncertainty expands. In order to clarify this, we can use Feynman's rest of the universe.

The mathematics of the coupled oscillator system is exactly the same as the covariant harmonic oscillator which leads to Dirac's form of relativistic quantum mechanics, where one of the variable is the time-separation variable. It is not necessaary to write down lengthy formulas. The best way is to use a cartoon. It is even better if the cartoon is the one you have seen before. The net result is

The space-time portion of this paper is from the paper I published with Wigner in 1990. The title of the paper is Entropy and Lorentz Transformations. If we do not measure the time-separation variable assuming it is hidden in Feynman's rest of the Universe, the net result is an increase in entropy and an expansion of the uncertainty region in the Wigner phase space. Click here for the phase space expansion.

This figure can be used for the two-photon coherent system or two-mode squeezed state. The labels "first photon" and "second photon" were added in my paper with Noz entitled "Time separation as a hidden variable in the Copenhagen school of quantum mechanics," in Advances in Quantum Theory, American Institute of Physics Conference Series No.1327, pp 138-147 (2001). ArXiv. As far as I can see, this story can be traced to Dirac's 1963 paper.

• The organization of this webpage can be improved. please come again. Making a webpage is like writing a paper. You know how difficult it is to write papers.

copyright@2009 by Y. S. Kim, unless otherwise specified.
The photos of Bohr and Einstein and of Dirac and Wigner are from the E. Segre Visual Archives of the American Physical Society.