Do we need new quantum mechanics?

One of my friends told me that Murray Gell-Mann wants to change quantum mechanics, but he does not want to change it. It appears that Gell-Mann knows what he is doing. If you wish to change quantum mechanics, you should first understand thoroughly the present form of quantum mechanics.

Do you understand the hydrogen atom? This atom consists of the wave function of the electron circling around the proton. We know how to solve the Schroedinger equation to get the Rydberg levels. We also know how to make fine-structure corrections while taking into account relativistic effects on the electron motion. In addition, we know how to calculate the Lamb shift. Thus, you know everything about this fundamental atom. NO!

Quantum electrodynamics is a covariant theory. Do you know how to get the localization of the wave function which leads to the Rydberg levels within the framework of QED? The answer to this question is NO. The Bohr radius a measure of space-like separation between the electron and proton. For an observer in a different Lorentz frame, the time-separation becomes an essential variable, but we never talk about it.

The existence of non-measurable variables and their effects on measurable variables have been an outstanding issue in quantum mechanics since the publication of von Neumann's paper on density matrices [1]. The main idea of von Neumann was brilliantly summarized by Feynman in his book on statistical mechanics [2]. 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."

Then, is there a concrete physical example which illustrates Feynman's rest of the universe? It is possible to use a pair of coupled harmonic oscillators. One of the oscillators is the system in which we are interested, and the other is the rest of the universe [3]. If we do not measure the rest of the universe, the result is an increase in entropy in the system in which we are interested. Yes, this oscillator system is only a toy model. Is there a serious physical problem where this becomes an issue?

When we Lorentz-boost a system, the longitudinal and time-like coordinates become coupled exactly like the system of two coupled oscillators. Let us consider a hadron bound together by two quarks. There is space-like separation between the quarks as in the case of the Bohr radius. However, the time separation variable comes in when we boost the hadron. On the other hand, we do not measure the time-separation variable. As the hadronic speed approaches the speed of light, the quark model becomes the parton model [4]. Our failure to measure the time-separation variable results in an increase in entropy [5].

Feynman's parton picture presents also a coherence problem. When we deal with quarks, we add amplitudes, while we add cross sections when we use partons. Does this mean that the Lorentz boost destroys coherence? Interesting question indeed!

There are many problems in modern physics which do not appear to obey the rules of the existing form of quantum mechanics. However, we should examine those problems thoroughly before making attempts to modify quantum mechanics. Otherwise, we have to go through an unproductive process of talk-talk-talk- .. -talk.

1. J. von Neumann, Die mathematische Grundlagen der Quanten- mechanik (Springer, Berlin, 1932). See also J. von Neumann, {\it Mathematical Foundation of Quantum Mechanics} (Princeton University, Princeton, 1955).
2. R. P. Feynman, Statistical Mechanics (Benjamin/Cummings, Reading, MA, 1972).
3. D. Han, Y. S. Kim, and M. E. Noz, Am. J. Phys. 67 , 61 (1999).
4. R. P. Feynman, The Behavior of Hadron Collisions at Extreme Energies , in High Energy Collisions , Proceedings of the Third International Conference, Stony Brook, New York, edited by C. N. Yang et al , Pages 237-249 (Gordon and Breach, New York, 1969).
5. Y. S. Kim and E. P. Wigner, Phys. Lett. A 147 , 343 (1990).