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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns.

2u+1d=1 1u+2d=0

$$a_1d+b_1u=c_1$$ $$a_2d+b_2u=c_2$$ $$u=\frac {c_1b_2-c_2b_1}{a_1b_2-a_2b_1}$$ $$d=\frac {a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$$

u=+2/3 d=-1/3

there ain't no experiment that could be done, nor is there any observation that could be made, that would say, "You guys are wrong." The theory is safe, permanently safe. Is that a theory of physics or a philosophy? I ask you.

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closed as not a real question by Qmechanic, genneth, Luboš Motl, David Z Sep 24 '12 at 3:03

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This whole question is based on a false premise. –  dmckee Sep 22 '12 at 14:39

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up vote 3 down vote accepted

Deep inelastic scattering experiments at SLAC in the 1970's confirmed that quarks exist and that they have fractional charges. See this from which the following quote is taken:

These properties were so odd that for a number of years it was not clear whether quarks actually existed or were simply a useful mathematical fiction. For example, quarks must have charges of + 2/3e or - 1/3e, which should be very easy to spot in certain kinds of detectors; but intensive searches, both in cosmic rays and using particle accelerators, have never revealed any convincing evidence for fractional charge of this kind. By the mid-1970s, however, 10 years after quarks were first proposed, scientists had compiled a mass of evidence that showed that quarks do exist but are locked within the individual hadrons in such a way that they can never escape as single entities.

This evidence resulted from experiments in which beams of electrons, muons, or neutrinos were fired at the protons and neutrons in such target materials as hydrogen (protons only), deuterium, carbon, and aluminum. The incident particles used were all leptons, particles that do not feel the strong binding force and that were known, even then, to be much smaller than the nuclei they were probing. The scattering of the beam particles caused by interactions within the target clearly demonstrated that protons and neutrons are complex structures that contain structureless, pointlike objects, which were named partons because they are parts of the larger particles. The experiments also showed that the partons can indeed have fractional charges of + 2/3e or - 1/3e and thus confirmed one of the more surprising predictions of the quark model.

While it is true that we cannot seperate a single quark from a proton due to the color confinement property of the strong color force, it turns out that another property of the strong color force, asymptotic freedom, allows very high energy deep inelastic scattering to probe the properties of "free" quarks. These deep inelastic scattering experiments of electrons on protons established that there really were point like constituents of protons that had fractional electric charge and thus validated the quark model of hadrons.

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Evidence for the quarks comes, not from some trivial argument that it would get the charges of two particles right, but from a host of sources.

Looking just at the "get the numbers to add up" approach we need a mechanism to simultaneously explain

  • Mass spectrum of the baryons
  • The spins of the baryons
  • The charges of the baryons
  • The parities of the baryons

and it was found that the so-called "constituent quark" models could do exactly that.

Indeed the constituent quark model predicted the mass, charge, and spin of the $\Omega^-$ baryon which was subsequently found right where it was expected.

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