### Virtual Photons

G. C. Johnson

#### Abstract

This analysis explores the properties of the virtual photons that give rise to the electric force between quantum charges.  In contrast to real photons, which are emitted and absorbed by atoms, virtual photons appear to have a characteristic angular momentum that is smaller than Planck's constant by a factor of 1/861.  This constant, (symbolized here by h'), multiplied by the frequency, c/R, defines the potential energy between any two quantum charges.  (The number 861 represents the ratio of the Compton wavelength to the classical electron radius.  Currently, this ratio has an accepted value of 861.023.)

The quantum states of single electron atoms are interpreted geometrically as quantum displacements occurring at a frequency of Zc/R.  (Each of the Z protons in the nucleus interact with the electron at the frequency c/R.)

Finally, quantum integers are identified for each of the planets, demonstrating the quantum structure of the solar system.

#### Real Photons

Planck's constant can be expressed as the product of electron mass, the electron's Compton wavelength, and the speed of light.

$\large h=m_{e}\lambda _{c}c$

The photon's energy is given by:

$\large E=h\frac{c}{\lambda }$

#### Virtual Photons

Let "Planck-prime", or h', be defined as the product of electron mass, the classical electron radius, and the speed of light.

$\large h'=m_{e}r_{e}c$

Here, the classical electron radius, re, replaces the Compton wavelength.  In both cases, however, each is the limiting wavelength with an energy of mec2.

The potential energy between two quantum charges is:

$\large PE=\pm h'\frac{c}{R}$

R, the distance between the charges, is the virtual photon's wavelength.  Virtual photons can not be observed directly, but much can be inferred from what we know about single electron atoms.

#### Single Electron Atoms

As observed by Bohr, the angular momentum of these electrons always occur in integer multiples of h-bar.

$\large m_{e}vR=n\hbar=n\frac{h}{2\pi }$

As a function of h', Bohr's relationship is:

$\large m_{e}vR=n\frac{861h'}{2\pi }=\frac{n}{\alpha }m_{e}r_{e}c$

Alpha is Sommerfeld's constant, also known as the fine structure constant.

Moving all quantities to the left hand side:

$\large \frac{\alpha }{n}\times \frac{v}{c}\times \frac{R}{r_{e}}=1$

The product of the above three dimensionless ratios must be unity.

#### Hydrogen ground state, (Z=1, n=1)

The above equation is satisfied for the hydrogen ground state by:

$\large \frac{v}{c}=\alpha$

and,

$\large \frac{R}{r_{e}} =\frac{1}{\alpha ^{2}}$

#### All ground states, (Z=>1, n=1)

For all single electron ground states, Bohr's condition is satisfied by increasing velocity by a factor of Z and reducing the radius by 1/Z.

$\large \frac{v}{c} =Z\alpha$

Increasing velocity by a factor of Z increases kinetic energy by Z2.

$\large \frac{R}{r_{e}} =\frac{1}{Z\alpha ^{2}}$

Decreasing radius by 1/Z increases (negatively) potential energy per proton by a factor of Z.  For Z protons, absolute value of potential energy is also increased by Z2.

#### Excited states, ( n>1)

For excited states, unity is preserved in the Bohr equation by dividing velocity by n, and multiplying R by n2.

$\large \frac{v}{c} =\frac{Z\alpha}{n}$

Decreasing velocity by a factor of 1/n decreases kinetic energy by 1/n2.

$\large \frac{R}{r_{e}} =\frac{n^{2}}{Z\alpha ^{2}}$

Increasing radius by n2 decreases potential energy by a factor of 1/n2

Assuming that the atomic electron does not move smoothly, but instead makes quantum displacement jumps at a frequency of Zc/R, we can construct a vector diagram of these displacements:

#### Conclusions

Plank's constant is not the smallest quantity of action between charged particles.  It, (h), is 861 times greater than h', the corresponding quantity that describes virtual photons.

Integer multiples of angular momentum is an intrinsic property of the electric field, independent of the number of protons contributing to the field of the nucleus.  (In the unity equation, the atomic number, Z, multiplies the velocity and divides the radius, leaving the angular momentum the same for all values of n.) This suggests that the gravitational field might also have preferred orbits with integer multiples of a specific angular momentum.  The solar system appears to confirm this hypothesis with a quantum specific angular momentum of 1.3038 AU2/yr.