# Understanding the Coulomb Force Through Quantum Electrodynamics

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## Chapter 1: Introduction to the Coulomb Force

In elementary school, you likely encountered the fundamental principle that electrically charged objects can either attract or repel each other. This interaction between charged particles is known as the Coulomb force. While we often overlook this force, various theoretical models help elucidate its workings. One such model is quantum electrodynamics (QED), a highly validated framework for understanding the electromagnetic force and the behavior of charged particles.

In contemporary physics, the Coulomb force is mediated by a particle known as a photon. The interaction between two electrons is depicted as a scattering process, where they repel each other through the exchange of an intermediate photon. This interaction can be visualized using Feynman diagrams, where the photon’s movement is represented by a wavy line illustrating the probability amplitude of its propagation through spacetime. To derive the propagator of the photon, physicists calculate the correlation function between two points in spacetime, providing a probability amplitude from the contraction of two photon fields.

The photon field is denoted by the letter A, which comprises four components: one for time and three for spatial dimensions. The mathematical representation of the propagator in position space can be understood as the amplitude derived from the contraction of the photon field at two distinct points.

To determine the form of this propagator, we can employ a mathematical technique from classical physics known as a Green's function. Photons, like all particles, follow equations of motion derived from a Lagrangian. In classical physics, the propagator acts as the inverse of this equation of motion—a concept that will be elaborated on in a future discussion. In the quantum realm, the photon propagator is calculated by expanding the photon field using creation and annihilation operators, followed by collapsing the terms together as shown in the equation below. Further simplifications can be applied to this expression.

The photon field A is expressed as the Fourier expansion of creation and annihilation operators. The first term represents the annihilation operator, noted with a lowercase 'a', while the second term corresponds to the creation operator. The coefficients for these terms are polarization vectors, reflecting the degrees of freedom a photon possesses in its orientation. The existence of only two vectors is significant and ties back to the symmetry properties in electromagnetism. Regardless of whether we evaluate the propagator classically or quantum mechanically, the results remain consistent. In momentum space, the propagator is proportional to 1/p², with an additional term that depends on the chosen gauge.

As previously discussed, this scattering amplitude provides us with a value in momentum space, constructed from the Feynman rules corresponding to our physical theory. Working in momentum space simplifies calculations, but we must perform a Fourier transform on the scattering amplitude to translate it into position space. To find the force related to this amplitude, we need to follow several steps, which I will elaborate on next.

## Chapter 2: The Born Approximation

In non-relativistic quantum mechanics, governed by the Heisenberg equation, we determine a particle's wave function based on the potential influencing it. The cross-section of a scattering event indicates how a particle deviates when colliding with another object. The Born approximation allows us to express the cross-section of a specific scattering process as a function of this potential. It asserts that the scattering amplitude correlates with the Fourier transform of the system's potential, converting the representation from position space to momentum space.

The validity of the Born approximation in non-relativistic quantum mechanics provides a useful benchmark for assessing whether our quantum field theory calculations hold true at lower energies.

In quantum field theory, the cross-section is directly related to the square of the scattering amplitude. This scattering amplitude is precisely what we compute using Feynman rules in a Feynman diagram. Typically, the letter M represents this scattering amplitude derived from a Feynman diagram. Notably, the scattering amplitude from the Feynman diagram and the Born approximation share proportionality to the same cross-section, indicating that the Fourier transform of the scattering amplitude should yield the potential in the non-relativistic context.

In the preceding section, we derived the photon propagator in momentum space by employing Feynman rules, finding that it is proportional to 1/p². Utilizing this knowledge, we can ascertain the potential by performing a Fourier transform on the photon propagator. Executing a Fourier transform in momentum space entails integrating the function with a phase factor. This integration can be complex, but we can simplify it by switching to spherical coordinates and aligning the z-axis with the x-direction. Ultimately, we recognize that the integral converges—a notable result stemming from the integration of sin(p)/p, which spans from zero to infinity. The specifics of this integral will be explored in greater detail in future discussions, but for now, it suffices to illustrate the overall potential shape we are analyzing.

## Wrap-up

In conclusion, this discussion has provided insights into the quantum electrodynamics model related to electrostatic repulsion. We have demonstrated its consistency with the expectations set forth in the non-relativistic limit.

### References

[1] Quantum Field Theory and the Standard Model by Schwartz, Matthew D. (ISBN: 8601406905047)

The first video, "Coulomb's Law - Net Electric Force & Point Charges," offers a comprehensive explanation of Coulomb's law, detailing how electric forces operate between point charges.

The second video, "Coulomb Force Between a Point and a Line," delves into the specifics of how the Coulomb force is applied in scenarios involving different geometries.