Exotic and hard to find experimentally.
Topological quantum computation by Jason Alicea (2021)
Source. In informal contexts, it usually refers to the phase of ice observed in atmospheric pressure, Ice Ih.
Theoretical framework on which quantum field theories are based, theories based on framework include:so basically the entire Standard Model
The basic idea is that there is a field for each particle particle type.
E.g. in QED, one for the electron and one for the photon: physics.stackexchange.com/questions/166709/are-electron-fields-and-photon-fields-part-of-the-same-field-in-qed.
And then those fields interact with some Lagrangian.
One way to look at QFT is to split it into two parts:Then interwined with those two is the part "OK, how to solve the equations, if they are solvable at all", which is an open problem: Yang-Mills existence and mass gap.
- deriving the Lagrangians of the Standard Model: S. This is the easier part, since the lagrangians themselves can be understood with not very advanced mathematics, and derived beautifully from symmetry constraints
- the qantization of fields. This is the hard part Ciro Santilli is unable to understand, TODO mathematical formulation of quantum field theory.
There appear to be two main equivalent formulations of quantum field theory:
Quantum Field Theory visualized by ScienceClic English (2020)
Source. Gives one piece of possibly OK intuition: quantum theories kind of model all possible evolutions of the system at the same time, but with different probabilities. QFT is no different in that aspect.- youtu.be/MmG2ah5Df4g?t=209 describes how the spin number of a field is directly related to how much you have to rotate an element to reach the original position
- youtu.be/MmG2ah5Df4g?t=480 explains which particles are modelled by which spin number
The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.
But what the heck is the mathematical model for a quantum field theory? TODO someone was saying it is equivalent to an infinite set of PDEs somehow. Investigate. Related:
The path integral formulation might actually be the most understandable formulation, as shown at Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:
The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.
Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:
- convenient for quantum field theory because of particle creation and annihilation changes the number of particles all the time
- convenient for condensed matter physics because there you have a gazillion particles occupying entire energy bands
Bibliography:
- www.youtube.com/watch?v=MVqOfEYzwFY "How to Visualize Quantum Field Theory" by ZAP Physics (2020). Has 1D simulations on a circle. Starts towards the right direction, but is a bit lacking unfortunately, could go deeper.
This one might actually be understandable! It is what Richard Feynman starts to explain at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
The difficulty is then proving that the total probability remains at 1, and maybe causality is hard too.
The path integral formulation can be seen as a generalization of the double-slit experiment to infinitely many slits.
Feynman first stared working it out for non-relativistic quantum mechanics, with the relativistic goal in mind, and only later on he attained the relativistic goal.
TODO why intuitively did he take that approach? Likely is makes it easier to add special relativity.
This approach more directly suggests the idea that quantum particles take all possible paths.
Mentioned for example in quantum field theory in a nutshell by Anthony Zee (2010) page 8.
What does it mean that photons are force carriers for electromagnetism? by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16TODO find/create decent answer.
I think the best answer is something along:
- local symmetries of the Lagrangian imply conserved currents. gives conserved charges.
- OK now. We want a local symmetry. And we also want:Given all of that, the most obvious and direct thing we reach a guess at the quantum electrodynamics Lagrangian is Video "Deriving the qED Lagrangian by Dietterich Labs (2018)"
- Dirac equation: quantum relativistic Newton's laws that specify what forces do to the fields
- electromagnetism: specifies what causes forces based on currents. But not what it does to masses.
A basic non-precise intuition is that a good model of reality is that electrons do not "interact with one another directly via the electromagnetic field".
A better model happens to be the quantum field theory view that the electromagnetic field interacts with the photon field but not directly with itself, and then the photon field interacts with parts of the electromagnetic field further away.
The more precise statement is that the photon field is a gauge field of the electromagnetic force under local U(1) symmetry, which is described by a Lie group. TODO understand.
This idea was first applied in general relativity, where Einstein understood that the "force of gravity" can be understood just in terms of symmetry and curvature of space. This was later applied o quantum electrodynamics and the entire Standard Model.
From Video "Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem":
- www.youtube.com/watch?v=pCQ9GIqpGBI&t=1663s mentions this idea first came about from Hermann Weyl.
- youtu.be/pCQ9GIqpGBI?t=2827 mentions that in that case the curvature is given by the electromagnetic tensor.
Bibliography:
- www.youtube.com/watch?v=qtf6U3FfDNQ Symmetry and Quantum Electrodynamics (The Standard Model Part 1) by ZAP Physics (2021)
- www.youtube.com/watch?v=OQF7kkWjVWM The Symmetry and Simplicity of the Laws of Nature and the Higgs Boson by Juan Maldacena (2012). Meh, also too basic.
Hans-Beat Bürgi is a Swiss mathematician known for his contributions to various areas of mathematics, particularly in the field of number theory, combinatorics, and mathematical education. He has been recognized for his work in explaining complex mathematical concepts in an accessible manner and has published numerous papers and educational materials.
4 K. Enough for to make "low temperature superconductors" like regular metals superconducting, e.g. the superconducting temperature of aluminum if 1.2 K.
Contrast with liquid nitrogen, which is much cheaper but only goes to 77K.
I think they are a tool to calculate the probability of different types of particle decays and particle collision outcomes. TODO Minimal example of that.
And they can be derived from a more complete quantum electrodynamics formulation via perturbation theory.
At Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), an intuitive explanation of them in termes of sum of products of propagators is given.
- www.youtube.com/watch?v=fG52mXN-uWI The Secrets of Feynman Diagrams | Space Time by PBS Space Time (2017)
Surprisingly, it can also become a superfluid even though each atom is a fermion! This is because of Cooper pair formation, just like in superconductors, but the transition happens at lower temperatures than superfluid helium-4, which is a boson.
aps.org/publications/apsnews/202110/history.cfm: October 1972: Publication of Discovery of Superfluid Helium-3 contains comments on the seminal paper and a graph which we must steal.
Also sometimes called helium II, in contrast to helium I, which is the non-superfluid liquid helium phase.
Pinned article: Introduction to the OurBigBook Project
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