This topic reminds me of decades ago, when I read
“The Tao of Physics” by Fritjof Capra.
He explains and diagrams many of the priciples discussed here, but it was still difficult to wrap my mind around these processes. He even discussed how theorectically, one could move forward and backward in time, or at least a proton and electron seemed to be demostrating this feat.
QFT was built on QM which created much of the math difficulties. QFT define creation and anihilation operators expressed as matrices that modeled the raising and lowering of energy states from the ground vacuum state. These operators form the nodes of the quantum field and their time evolution is known as the Schroedinger wave equation of the Universe.
QM was born when Einstein demonstrated that energy was not continuous like every classical scientist believed but came in packets called quanta. Because these packets were the product of a constant (Plank’s) and a wavelenth (change in the angle of a unit circle/time) the mathmeticians used trig substitutions to define the basic units in QM. Rather than do everything in sines and cosines they leveraged the work of the mathmetician David Hilbert and simplified the equations using complex numbers.
What this did was it changed the way classical physicts treated objects with single sets of parameters into objects with two parameters (real and imaginary). This took QM from classical set theory logic to its own algebra of squaring the two parts to get a part that we can measure in the world of real numbers where we live.
The problems with infinities comes in because there is theoretically no limit to how short a wavelength can be and the shorter the wavelength (higher frequency) the higher the energy it models. Since we can’t measure these energies yet scientists simply define a “cutoff” point beyond which we ignore frequencies which is called renormalization. This is why we call QFT an “effective field theory” because its good enough without being the whole explanation.
I think that some concepts from QFT were used to explain rogue waves out in the oceans. Everything is just waves interacting and sometimes they interact in ways that are not “ordinary”.
Cool stuff but completely beyond my comprehension.