The proton decays because positive frequencies slowly turn to negative

Our blog post about charge nonconservation predicts proton decay. A quark will eventually turn into an antiquark, because its positive frequencies slowly turn to negative. There are 3 quarks in a proton plus a number of gluons, and the interaction is not strictly between two particles. There are random accelerations, which necessarily produce negative frequencies.

The radius of the proton is 9 * 10^-16 m.

Quark masses are only 2 or 5 MeV, while the mass of the proton is 938 MeV.

The Compton wavelength of a quark is around 2 * 10^-13 m. Most of the proton mass comes from gluons. If we assign the proton mass evenly to the three quarks, the Compton wavelength of a quark is 4 * 10^-15 m.

Since a quark is contained within 2 * 10^-15 m, its wavelength inside proton potential well has to be around 2 * 10^-15 m or less.

A quark will bump into the potential wall surrounding the proton at most some 10^24 times per second if it moves at the speed of light.

If the gluon cloud inside the proton is essentially independent of the position of an individual quark, then the quark will move in an essentially static potential and will not get much negative frequencies per bump.

Can we somehow estimate the production rate of negative frequencies?

Let us first calculate the ballpark figure what magnitude of acceleration "significantly" deforms a single wavelength of a reflected wave. Then there should be a significant negative frequency component in the reflection.

The acceleration must be such that it will move the mirror almost a wavelength during one cycle of the wave.

                    1/2 a t^2 = λ,

where t = λ / c. We get

                    a = 2 c^2 / λ.

Conversely, we get λ =  2 c^2 / a. Let us calculate the black body temperature associated with acceleration a. Wien's displacement law gives

                     b / T = 2 c^2 / a

                     T = 1/2 b a / c^2 = 1.5 * 10^-20 a

kelvins. The Unruh temperature is 0.4 *10^-21 a kelvins. The coefficient is in the same ballpark as our 1.5 * 10^-20.

What is the acceleration when a quark bounces back? There is asymptotic freedom inside the proton, and the bounce happens from a steep potential wall. The process will take around 10^-24 seconds and the speed change is 6 * 10^8 m/s. The acceleration is 6 * 10^32 m/s^2. The Unruh temperature is T = 2 * 10^11 K, which corresponds to a wavelength of 10^-14 m.

Unruh calculated that the (nonexistent) Unruh radiation has approximately the black body spectrum. Let us make an educated guess and conjecture that the portion of negative frequencies produced in a reflection goes like the black body spectrum.

If the proton would consist just of three quarks, then each reflection of a quark would produce a significant portion of negative frequencies, because there would be a significant amount of random acceleration a in each reflection. But since the mass of an individual quark is just 1 / 200 of the proton mass, the random acceleration is just 1 / 200 of a. The temperature is thus 1 / 200 of the temperature.

Planck's law says that black body radiation is proportional to

            f^3 / (exp(h f / (k T)) - 1),

where f is the frequency, h is the Planck constant and k is the Boltzmann constant. The maximum is when h f / k T is roughly 5. In our quark case, the frequency of the quark was 5X the frequency of the maximum Unruh effect. Thus, h f / (k T) might be 25. When we drop T by a factor 200, the ratio is 5,000. We conclude that the flux of negative frequencies is not significant by a factor 10^2,500. The proton lifetime is in the ballpark of 10^2,500 years.

How can the Compton wavelength of a quark be much bigger than the proton radius?

There is a problem in the proton model. If the Compton wavelength of a quark, or 1 / 3 of the proton mass, is much larger than the proton radius, how can it be localized inside the 9 *10^-14 m radius of the proton? Maybe the particle is in a potential well, and its wavelength inside that well is shorter than it would be as a free particle. Localization inside 10^-15 m suggests that the energy of each particle with respect to the floor of the potential well is actually in the GeV range.

If we try to pull a quark out of a proton, then according to the rubber band model of QCD, the rubber band which pulls the quark to the proton will eventually break, and two hadrons will be formed. This means that each quark inside the proton really is sitting in a potential well whose depth is in the GeV range, if we set the zero potential at the rubber band break position. But why should we set the zero level there?