Feynman's Parton Picture

Before Einstein, the energy-momentum relation for massive particles and massless particles were E = p2/2m and E = cp respectively. Einstein combined these two separate formulas into one. As we know well, Einstein's work was not trivial.

In 1964, Murray Gell-Mann invented the quark model in which hadrons are quantum bound states of more fundamental particles called quarks. Five years later, in 1969, Richard Feynman observed that a hadron, when it moves with a speed close to that of light, appears as a collection of infinite number of partons with a wide-spread momentum distribution. Partons interact incoherently with external signals.

It is only natural to regard the parton picture as a Lorentz-boosted quark model. However, since the partons are so different from the quarks, we have been wondering whether they are really Lorentz-boosted quarks.

  1. Further Contents of E = mc2
  2. Covariant Model
  3. Lorentz Boost
  4. Feynman's Decoherence
  5. Infinite Number of Partons
  6. Boiling Quarks and Phase Transition


Further Contents of Einstein's Energy-momentum Relation


According to Feynman, the adventure of our science of physics is a perpetual attempt to recognize that the different aspects of nature are really different aspects of the same thing.


How can we construct a covariant model for quarks and partons?

Let us start with a hadron consisting of two quarks bound together by a harmonic oscillator potential. The simplest wave function for this two-particle system is is the harmonic oscillator wave function. Can this wave function be Lorentz-boosted?


Can you boost a quark-model wave function to get a parton distribution?

    Y. S. Kim, Phys. Rev. Lett. 63 , 348 (1989).
    This figure explains how a Lorentz-squeezed hadron become a collection of partons.

    1. As the hadron moves fast, the quark distribution becomes concentrated along one of the light-cone axes. The amplitude of the oscillation becomes larger, indicating that the spring constant appears to become weaker. The particles become free!

    2. The momentum distribution becomes also widespread. This is what we see in the world through Feynman's parton model.

    3. The number of partons is infinite because free particles have continuous momentum distribution as in the case of black-body radiation.

    4. The major axis of the ellipse measures the period of oscillation. As Feynman observed, the interaction time between the quarks is dilated.

    P. Hussar, Phys. Rev. 23, 2781
    (1981).


    We are of course interested to know whether the parton distribution calculated from the covariant oscillator formalism is in agreement with the distribution observed in the real world. This graph will indicate that the answer is YES.






Feynman's Decoherence


Infinite Number of Partons


Boiling Quarks

In his 1972 book on statistical mechanics, said

    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.

    The Bohr radius is an important quantity in quantum mechanics. It is a space-like separation between two particles. However, if the system is boosted, the time-like separation becomes prominent. This problem goes back to Bohr and Einstein. They met often, but they never discussed this issue. Click here for a story.

    Since there are no theoretical tools to deal with this problem, this variable is in the Feynman's rest of the universe.

    On the other hand, von Neumann's approach to entropy tells us how to deal with the variable we do not measure. If they are not measured, they appear as an entropy increase. Let us look at this figure.

    Thus, we are led to the phase transition of the form



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