Quantum theory solved many problems with classical theory - the photoelectric effect solved by Einstein, atomic model solved by Niels Bohr.
But quantum theory had a high price - determinism. it introduced randomness. Events were not deterministic anymore but probabilistic. We don’t see this in our macro world because it gets smoothed out.
Small things move at very high speeds. And so to describe them at velocities near the speed of light, Einstein’s Special relativity must apply. The integration of quantum mechanics with electromagnetism is called Quantum electrodynamics, or QED. QED replaces the classical theory of electromagnetism, which involves continuous electromagnetic fields and puts it in terms of discrete quantities.
In 1928 the first big step to QED was taken by British physicist Paul Dirac, when he published the Dirac equation. It blended quantum mechanics and special relativity. His equation was similar to the Schrodinger equation. Psi which is the wave function is present in both equations. Dirac made a Hamiltonian, which is the sum of all the energy, in terms of space-time. It also has the terms MC^2, and p the momentum on the left side - from special relativity.
Schrodinger equation treats time and space as independent coordinates, but Dirac integrated space-time, into the equation. There are some formulations that result in negative solutions. His equation was predicting anti-matter. No one had ever thought of that before. Dirac believed antimatter had to exist based simply on the math. 4 years later, 1932 the positron was discovered by American physicist Carl Anderson.
Later, three scientists began formulating a quantum theory of electromagnetism based on Dirac’s equation, and formulated QED - Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga. They won the 1965 Nobel prize in physics. The result was a Lagrangian. This equation holds information about the electromagnetic field and about whatever charged fermions one wants to investigate, like electrons and quarks.
In the 19th century, it was thought that two electrons near each other would simply repel similar to the way that two like poles of a magnet repel. Paul Dirac and others showed that there is no continuous field, but that the forces are mediated by the exchange of discrete photons.
When two charged particles travel near each other toward a colliding path, as they get close they will repel each other. In classical theory, this would be thought of as the particles repelling due to a continuous field. But in QED, the path can be simplified to look like a Feynman diagram. These diagrams were developed by American physicist Richard Feynman around 1949 and they help to intuitively understand the mathematics of QED.
The direction of the arrow on the fermion line tells us if it's a matter, or the anti-matter particle. If the arrow goes forward in time (to the right), it is the matter, and if it goes to the left, it’s antimatter. From this simplest Feynman diagram, you can build all other diagrams for QED.
The lines with the arrows represent fermions, and they must be continuous. This is because of the law of conservation. The same number of fermions coming into a process must also come out. Note that a photon is a boson, not a fermion. But if a positron and electron come in, and only a photon comes out, then how are things conserved? The reason is because a matter particle is considered a +1 fermion and an antimatter particle is considered a -1 fermion. So an electron and positron coming in works out to be +1-1 = 0, so the rule still works.
The Key to understanding Feynman diagrams is to know that each diagram is really an equation. One of the simplest diagrams is where two electrons come in, two electrons coming out and a photon is being exchanged – this equation has 7 components. Each of these 7 components has an associated element in an equation.
Note that when two charged particles actually come near each other, they exchange many photons, not just one. Many different types of potential exchanges can occur. But by simplifying the equation to the exchange of a single photon, we can get a good approximation.
The lessons from the development of QED were later used for the development of Quantum Chromodynamics (QCD) & Quantum Field Theories starting in the 1960’s, like elecroweak theory.
Electromagnetism was the first force to be expressed in terms of a quantum field theory.
QED is electromagnetism expressed in terms of quantum fields.