Abstract
The quantum mechanics of bound states with discrete energy levels is well understood. The quantum mechanics of scattering processes is also well understood. However, the quantum mechanics of moving bound states is still debatable. When it is at rest, the space-like separation between the constituent particles is the primary variable. When the bound state moves, this space-like separation picks up the time-like separation. The time-separation is not a measurable variable in the present form of quantum mechanics. The only way to deal with this un-observable variable is to treat it statistically. This leads to rise of the statistical variables such entropy and temperature. Paul A. M. Dirac made efforts to construct bound-state wave functions in Einstein’s Lorentz-covariant world. In 1927, he noted that the c-number time-energy relation should be incorporated in the relativistic world. In 1945, he constructed four-dimensional oscillator wave functions with one time coordinate in addition to the three-dimensional space. In 1949, Dirac introduced the light-cone coordinate system for Lorentz transformations. It is then possible to integrate these contributions made by Dirac to construct the Lorentz-covariant harmonic oscillator wave functions. This oscillator system can explain the proton as a bound state of the quarks when it is at rest, and explain the Feynman’s parton picture when it moves with a speed close to that of light. While the un-measurable time-like separation becomes equal to the space-like separation at this speed, the statistical variables become prominent. The entropy and the temperature of this covariant harmonic oscillator are calculated. It is shown that they rise rapidly as the proton speed approaches that of light.
Highlights
The bound state in quantum mechanics with discrete energy levels is well understood. How would those energy levels appear to moving observers? What will happen to the size of the bound state? the Bohr-Einstein issue of the hydrogen atom leads to the problem of moving quantum bound states in Einstein’s Lorentz-covariant world
Since the language of special relativity is the Lorentz group, and harmonic oscillators provide a starting point for the present form of quantum mechanics, Dirac considered the possibility of using harmonic oscillator wave functions to construct representations of the Lorentz group [19]
This problem arises because the time separation between the constituent particles is not a measurable quantity in the present form of quantum mechanics
Summary
Bohr was worrying about the electron orbit of the hydrogen atom, while Albert Einstein was interested in how things appear to moving observers. The Bohr-Einstein issue of the hydrogen atom leads to the problem of moving quantum bound states in Einstein’s Lorentz-covariant world. We review first efforts made in the past to resolve this quark-parton issue [16] [17], using the Lorentz-covariant oscillator wave function. This wave function allows us to Lorentz-boost the hadron at rest to its speed very close to that of light. The question is whether the quarks and partons are two different ways of looking at the same entity We resolve this issue using the Lorentz-covariant oscillator wave functions constructed in Sections 3 and 4. It is possible to calculate them as functions of the hadronic speed using the density matrix
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