San José State University |
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applet-magic.comThayer WatkinsSilicon Valley & Tornado Alley USA |
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The Physical Impossibilitiesof Point Particles |
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Charged particles like the electron and the proton consist of two parts: a core and a field. The core consists of the distribution of charge over a limited range. Here is the empirically determined charge distributions for protons and neutrons.

As can be seen above, a neutron does have a charges distribution even though it has zero net charge and no field.

of a Charged Particle

Consider a stationary particle with charge Q. The electrical field intensity E at a distance r from the center of that particle is given by

where ε_{0} is a constant called the *permitivity* of space.

For a particle in empty space the energy density U is

The energy in a spherical shell of radius r and thickness dr is equal to 4πr²(Q²/(32π²r^{4})
which reduces to Q²/(8πε_{0}r²). The integration of these terms from R to ∞ gives
a total energy T_{R}

If R→0, then T_{R} → ∞ no matter how small is the magnitude of the charge so long as it is nonzero.
Thus a charged point particle, if such existed, would have infinite energy.

The absurdityof the notion of a single electron having more energy than all of kinetic energy of all particles in all of the galaxies of the Universe need no further elaboration.

It is worth noting that there can be no conservation of energy if there is an infinite source of energy that can be drawn from.

Since a particle and its antiparticle can annihilate each other and their fields, it follows that such annihilation of point particles should produce photons of infinite energy. The mutual annihilation of an electron and a positron produces gamma ray(s) of finite energy. Therefore electrons and positrons cannot be point particles.

Pointedness of Particles Can Arise

There is a wonderful theorem in mathematical physics that says that the effect of a spherical charge uniformly distributed
is the same as if the charge were concentrated at the center of the sphere. That is for points outside of sphere of the charge;
inside that sphere the effect is zero. Therefore the deflection of probe particles impinging upon a spherical shell of charge
is the same as if the charge of the shell were concentrated at its center provided that the probe particles do not have enough
energy to penetrate within the shell. The effect on probe particles within the shell would be called *asymptotic freedom*.
Thus evidence of a particle having the same effect over a limited range of energies as a point particle is not evidence of the
existence of point particles.

in Quantum Field Theories of Fields which

which are those of Point Particles

Quantum Field Theories are plagued with infinities. Various devices such as renormalization group theory were developed to successfully eliminate those infinities. The interesting intellectual question is whether those infinitinities arise because the fields assumed in QFT are the same as those of point particles. But that is a topic for a different webpage.

(To be continued.)

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