by Ciro Santilli (@cirosantilli, 32)
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.
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.
There appear to be two main equivalent formulations of quantum field theory:
Video 1. 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.
Video 2. Quantum Fields: The Real Building Blocks of the Universe by David Tong (2017) Source. Boring, does not give anything except the usual blabla everyone knows from Googling:
Video 3. Quantum Field Theory: What is a particle? by Physics Explained (2021) Source. Gives some high level analogies between high level principles of non-relativistic quantum mechanics and special relativity in to suggest that there is a minimum quanta of a relativistic quantum field.
TODO holy crap, even this is hard to understand/find a clear definition of.
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).
The formulation of QFT also appears to be a form of infinite-dimentional calculus.
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.
The term and idea was first introduced initialized by Hermann Weyl when he was working on combining electromagnetism and general relativity to formulate Maxwell's equations in curved spacetime in 1918 and published as gravity and electricity by Hermann Weyl (1918). Based on perception that symmetry implies charge conservation. The same idea was later adapted for quantum electrodynamics, a context in which is has even more impact.
A random field you add to make something transform locally the way you want. See e.g.: Video "Deriving the qED Lagrangian by Dietterich Labs (2018)".
Video 1. Lawrence Krauss explains Gauge symmetry by Joe Rogan (2017) Source.
While most of this is useless as you would expect from the channel, it does give one key idea: you can change charge locally, but things somehow still work out.
And this has something to do with the general intuition of special relativity that only local measures make much sense, as evidenced by Einstein synchronization.
Yup, this one Focks you up.
Video 1. What's a Fock space? by Physics Duck (2023) Source.
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:
Bibliography:
Basically a synonym for second quantization.
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.
As mentioned at: physics.stackexchange.com/questions/212726/a-quantum-particle-moving-from-a-to-b-will-take-every-possible-path-from-a-to-b/212790#212790, classical gravity waves for example also "take all possible paths". This is just what waves look like they are doing.
Thought experiment that illustrates the path integral formulation of quantum field theory.
Video 1. The Biggest Ideas in the Universe | 11. Renormalization by Sean Carroll (2020) Source. Gives a very quick and high level overview of renormalization. It is not enough to satisfy Ciro Santilli as usual for other Sean Carroll videos, but it goes some way.
www.youtube.com/watch?v=WB8r7CU7clk&list=PLUl4u3cNGP60TvpbO5toEWC8y8w51dtvm by Iain Stewart. Basically starts by explaining how quantum field theory is so generic that it is hard to get any numerical results out of it :-)
But in particular, we want to describe those subtheories in a way that we can reach arbitrary precision of the full theory if desired.
  • www.youtube.com/watch?v=-_qNKbwM_eE Unsolved: Yang-Mills existence and mass gap by J Knudsen (2019). Gives 10 key points, but the truly hard ones are too quick. He knows the thing though.
Video 1. Yang-Mills 1 by David Metzler (2011) Source.
A bit disappointing, too high level, with very few nuggests that are not Googleable withing 5 minutes.
Breakdown:
Video 2. Millennium Prize Problem: Yang Mills Theory by David Gross (2018) Source. 2 hour talk at the Kavli Institute for Theoretical Physics. Too mathematical, 2021 Ciro can't make much out of it.
Video 3. Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem. Source. Unknown year. He almost gets there, he's good. Just needed to be a little bit deeper.
Theory that describes electrons and photons really well, and as Feynman puts it "accounts very precisely for all physical phenomena we have ever observed, except for gravity and nuclear physics" ("including the laughter of the crowd" ;-)).
Learning it is one of Ciro Santilli's main intellectual fetishes.
While Ciro acknowledges that QED is intrinsically challenging due to the wide range or requirements (quantum mechanics, special relativity and electromagnetism), Ciro feels that there is a glaring gap in this moneyless market for a learning material that follows the Middle Way as mentioned at: the missing link between basic and advanced. Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) is one of the best attempts so far, but it falls a bit too close to the superficial side of things, if only Feynman hadn't assumed that the audience doesn't know any mathematics...
The funny thing is that when Ciro Santilli's mother retired, learning it (or as she put it: "how photons and electrons interact") was also one of her retirement plans. She is a pharmacist by training, and doesn't know much mathematics, and her English was somewhat limited. Oh, she also wanted to learn how photosynthesis works (possibly not fully understood by science as that time, 2020). Ambitious old lady!!!
Combines special relativity with more classical quantum mechanics, but further generalizing the Dirac equation, which also does that: Dirac equation vs quantum electrodynamics. The name "relativistic" likely doesn't need to appear on the title of QED because Maxwell's equations require special relativity, so just having "electro-" in the title is enough.
Before QED, the most advanced theory was that of the Dirac equation, which was already relativistic but TODO what was missing there exactly?
As summarized at: youtube.com/watch?v=_AZdvtf6hPU?t=305 Quantum Field Theory lecture at the African Summer Theory Institute 1 of 4 by Anthony Zee (2004):
  • classical mechanics describes large and slow objects
  • special relativity describes large and fast objects (they are getting close to the speed of light, so we have to consider relativity)
  • classical quantum mechanics describes small and slow objects.
  • QED describes objects that are both small and fast
That video also mentions the interesting idea that:
Therefore, for small timescales, energy can vary a lot. But mass is equivalent to energy. Therefore, for small time scale, particles can appear and disappear wildly.
QED is the first quantum field theory fully developed. That framework was later extended to also include the weak interaction and strong interaction. As a result, it is perhaps easier to just Google for "Quantum Field Theory" if you want to learn QED, since QFT is more general and has more resources available generally.
Like in more general quantum field theory, there is on field for each particle type. In quantum field theory, there are only two fields to worry about:
Video 1. Lecture 01 | Overview of Quantum Field Theory by Markus Luty (2013) Source. This takes quite a direct approach, one cool thing he says is how we have to be careful with adding special relativity to the Schrödinger equation to avoid faster-than-light information.
Experiments explained by QED but not by the Dirac equation:
2s/2p energy split in the hydrogen emission spectrum, not predicted by the Dirac equation, but explained by quantum electrodynamics, which is one of the first great triumphs of that theory.
Note that for atoms with multiple electrons, 2s/2p shifts are expected: Why does 2s have less energy than 1s if they have the same principal quantum number?. The surprise was observing that on hydrogen which only has one electron.
Initial experiment: Lamb-Retherford experiment.
On the return from the train from the Shelter Island Conference in New York, Hans Bethe managed to do a non-relativistic calculation of the Lamb shift. He then published as The Electromagnetic Shift of Energy Levels by Hans Bethe (1947) which is still paywalled as of 2021, fuck me: journals.aps.org/pr/abstract/10.1103/PhysRev.72.339 by Physical Review.
The Electromagnetic Shift of Energy Levels Freeman Dyson (1948) published on Physical Review is apparently a relativistic analysis of the same: journals.aps.org/pr/abstract/10.1103/PhysRev.73.617 also paywalled as of 2021.
TODO how do the infinities show up, and how did people solve them?
Video 1. Lamb shift by Dr. Nissar Ahmad (2020) Source. Whiteboard Lecture about the phenomena, includes description of the experiment. Seems quite good.
Video 2. Murray Gell-Mann - The race to calculate the relativistic Lamb shift by Web of Stories (1997) Source. Quick historical overview. Mentions that Richard Feynman and Julian Schwinger were using mass renormalization and cancellation if infinities. He says that French and Weisskopf actually managed to do the correct calculations first with a less elegant method.
www.mdpi.com/2624-8174/2/2/8/pdf History and Some Aspects of the Lamb Shift by G. Jordan Maclay (2019)
Video 3. Freeman Dyson - The Lamb shift by Web of Stories (1998) Source.
Mentions that he moved to the USA from the United Kingdom specifically because great experiments were being carried at Columbia University, which is where the Lamb-Retherford experiment was done, and that Isidor Isaac Rabi was the head at the time.
He then explains mass renormalization briefly: instead of calculating from scratch, you just compare the raw electron to the bound electron and take the difference. Both of those have infinities in them, but the difference between them cancels out those infinities.
Video 4. Hans Bethe - The Lamb shift (1996) Source.
Ahh, Hans is so old in that video, it is sad to see. He did live a lot tough. Mentions that the shift is of about 1000 MHz.
The following video: www.youtube.com/watch?v=vZvQg3bkV7s Hans Bethe - Calculating the Lamb shift.
Video 5. Lamb shift by Vidya-mitra (2018) Source.
Published as "Fine Structure of the Hydrogen Atom by a Microwave Method" by Willis Lamb and Robert Retherford (1947) on Physical Review. This one actually has open accesses as of 2021, miracle! journals.aps.org/pr/pdf/10.1103/PhysRev.72.241
Microwave technology was developed in World War II for radar, notably at the MIT Radiation Laboratory. Before that, people were using much higher frequencies such as the visible spectrum. But to detect small energy differences, you need to look into longer wavelengths.
This experiment was fundamental to the development of quantum electrodynamics. As mentioned at Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "Shrinking the infinities", before the experiment, people already knew that trying to add electromagnetism to the Dirac equation led to infinities using previous methods, and something needed to change urgently. However for the first time now the theorists had one precise number to try and hack their formulas to reach, not just a philosophical debate about infinities, and this led to major breakthroughs. The same book also describes the experiment briefly as:
Willis Lamb had just shined a beam of microwaves onto a hot wisp of hydrogen blowing from an oven.
It is two pages and a half long.
They were at Columbia University in the Columbia Radiation Laboratory. Robert was Willis' graduate student.
Previous less experiments had already hinted at this effect, but they were too imprecise to be sure.
This was one of the first two great successes of quantum electrodynamics, the other one being the Lamb shift.
In youtu.be/UKbp85zpdcY?t=52 from freeman Dyson Web of Stories interview (1998) Dyson mentions that the original key experiment was from Kusch and Foley from Columbia University, and that in 1948, Julian Schwinger reached the correct value from his calculations.
Bibliography:
Published on Physical Review by Polykarp Kusch and Foley.
TODO: in high level terms, why is QED more general than just solving the Dirac equation, and therefore explaining quantum electrodynamics experiments?
Also, is it just a bunch of differential equation (like the Dirac equation itself), or does it have some other more complicated mathematical formulation, as seems to be the case? Why do we need something more complicated than
Advanced quantum mechanics by Freeman Dyson (1951) mentions:
A Relativistic Quantum Theory of a Finite Number of Particles is Impossible.
(1)
where:
Note that this is the sum of the:
  • Dirac Lagrangian, which only describes the "inertia of bodies" part of the equation
  • the electromagnetic interaction term , which describes term describes forces
Note that the relationship between and is not explicit. However, if we knew what type of particle we were talking about, e.g. electron, then the knowledge of psi would also give the charge distribution and therefore
As mentioned at the beginning of Quantum Field Theory lecture notes by David Tong (2007):
Video 1. Particle Physics is Founded on This Principle! by Physics with Elliot (2022) Source.
Like the rest of the Standard Model Lagrangian, this can be split into two parts:
Video 1. Deriving the qED Lagrangian by Dietterich Labs (2018) Source.
As mentioned at the start of the video, he starts with the Dirac equation Lagrangian derived in a previous video. It has nothing to do with electromagnetism specifically.
He notes that that Dirac Lagrangian, besides being globally Lorentz invariant, it also also has a global invariance.
However, it does not have a local invariance if the transformation depends on the point in spacetime.
He doesn't mention it, but I think this is highly desirable, because in general local symmetries of the Lagrangian imply conserved currents, and in this case we want conservation of charges.
To fix that, he adds an extra gauge field (a field of matrices) to the regular derivative, and the resulting derivative has a fancy name: the covariant derivative.
Then finally he notes that this gauge field he had to add has to transform exactly like the electromagnetic four-potential!
So he uses that as the gauge, and also adds in the Maxwell Lagrangian in the same go. It is kind of a guess, but it is a natural guess, and it turns out to be correct.
TODO find/create decent answer.
I think the best answer is something along:
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.
Bibliography:
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.
No, but why?
What they presented on richard Feynman's first seminar in 1941. Does not include quantum mechanics it seems.
fafnir.phyast.pitt.edu/py3765/ Phys3765 Advanced Quantum Mechanics -- QFT-I Fall 2012 by E.S. Swanson mentions several milestone texts including:
Lecture notes that were apparently very popular at Cornell University. In this period he was actively synthesizing the revolutionary bullshit Richard Feynman and Julian Schwinger were writing and making it understandable to the more general physicist audience, so it might be a good reading.
We shall not develop straightaway a correct theory including many particles. Instead we follow the historical development. We try to make a relativistic quantum theory of one particle, find out how far we can go and where we get into trouble.
Oh yes, see also: Dirac equation vs quantum electrodynamics.
Julian Schwinger's selection of academic papers by himself and others.
Talk title shown on intro: "Today's Answers to Newton's Queries about Light".
6 hour lecture, where he tries to explain it to an audience that does not know any modern physics. This is a noble effort.
Part of The Douglas Robb Memorial Lectures lecture series.
Feynman apparently also made a book adaptation: QED: The Strange Theory of Light and Matter. That book is basically word by word the same as the presentation, including the diagrams.
According to www.feynman.com/science/qed-lectures-in-new-zealand/ the official upload is at www.vega.org.uk/video/subseries/8 and Vega does show up as a watermark on the video (though it is too pixilated to guess without knowing it), a project that has been discontinued and has has a non-permissive license. Newbs.
4 parts:
  • Part 1: is saying "photons exist"
  • Part 2: is amazing, and describes how photons move as a sum of all possible paths, not sure if it is relativistic at all though, and suggests that something is minimized in that calculation (the action)
  • Part 3: is where he hopelessly tries to explain the crucial part of how electrons join the picture in a similar manner to how photons do.
    He does make the link to light, saying that there is a function which gives the amplitude for a photon going from A to B, where A and B are spacetime events.
    And then he mentions that there is a similar function for an electron to go from A to B, but says that that function is too complicated, and gives no intuition unlike the photon one.
    He does not mention it, but P and E are the so called propagators.
    This is likely the path integral formulation of QED.
    On Quantum Mechanical View of Reality by Richard Feynman (1983) he mentions that is a bessel function, without giving further detail.
    And also mentions that:
    (1)
    where m is basically a scale factor. such that both are very similar. And that something similar holds for many other particles.
    And then, when you draw a Feynman diagram, e.g. electron emits photon and both are detected at given positions, you sum over all the possibilities, each amplitude is given by:
    (2)
    summed over all possible Spacetime points.
    TODO: how do electron velocities affect where they are likely to end up? suggests the probability only depends on the spacetime points.
    Also, this clarifies why computations in QED are so insane: you have to sum over every possible point in space!!! TODO but then how do we calculate anything at all in practice?
  • Part 4: known problems with QED and thoughts on QCD. Boring.
This talk has the merit of being very experiment oriented on part 2, big kudos: how to teach and learn physics
Video 1. Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) uploaded by Trev M (2015) Source. Single upload version. Let's use this one for the timestamps I guess.
Video 2. Richard Feynman Lecture on Quantum Electrodynamics 1/8. Source.
Basically the same content as: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), but maybe there is some merit to this talk, as it is a bit more direct in some points. This is consistent with what is mentioned at www.feynman.com/science/qed-lectures-in-new-zealand/ that the Auckland lecture was the first attempt.
Some more information at: iucat.iu.edu/iub/5327621
By Mill Valley, CA based producer "Sound Photosynthesis", some info on their website: sound.photosynthesis.com/Richard_Feynman.html
They are mostly a New Age production company it seems, which highlights Feynman's absolute cult status. E.g. on the last video, he's not wearing shoes, like a proper guru.
Feynman liked to meet all kinds of weird people, and at some point he got interested in the New Age Esalen Institute. Surely You're Joking, Mr. Feynman this kind of experience a bit, there was nude bathing on a pool that oversaw the sea, and a guy offered to give a massage to the he nude girl and the accepted.
youtu.be/rZvgGekvHest=5105 actually talks about spin, notably that the endpoint events also have a spin, and that the transition rules take spin into account by rotating thing, and that the transition rules take spin into account by rotating things.
This book has formulas on it, which is quite cool!! And the formulas are basically not understandable unless you know the subject pretty well already in advance. It is however possible to skip over them and get back to the little personal stories.
Explains beta decay. TODO why/how.
Maybe a good view of why this force was needed given beta decay experiments is: in beta decay, a neutron is getting split up into an electron and a proton. Therefore, those charges must be contained inside the neutron somehow to start with. But then what could possibly make a positive and a negative particle separate?
www.thestargarden.co.uk/Weak-nuclear-force.html gives a quick and dirty:
Beta decay could not be explained by the strong nuclear force, the force that's responsible for holding the atomic nucleus together, because this force doesn't affect electrons. It couldn't be explained by the electromagnetic force, because this does not affect neutrons, and the force of gravity is far too weak to be responsible. Since this new atomic force was not as strong as the strong nuclear force, it was dubbed the weak nuclear force.
Also interesting:
While the photon 'carries' charge, and therefore mediates the electromagnetic force, the Z and W bosons are said to carry a property known as 'weak isospin'. W bosons mediate the weak force when particles with charge are involved, and Z bosons mediate the weak force when neutral particles are involved.
Video 1. Weak Nuclear Force and Standard Model of particle physics by Physics Videos by Eugene Khutoryansky (2018) Source. Some decent visualizations of the field lines.
Video 1. Electroweak Theory and the Origin of the Fundamental Forces by PBS Space Time (2020) Source. Unsatisfactory, as usual.
This is quite mind blowing. The laws of physics actually differentiate between particles and antiparticles moving in opposite directions!!!
Only the weak interaction however does it of the fundamental interactions.
Some historical remarks on Surely You're Joking, Mr. Feynman section "The 7 Percent Solution".
It gets worse of course with cP Violation.
Formulated as a quantum field theory.
Video 1. Quarks, Gluon flux tubes, Strong Nuclear Force, & Quantum Chromodynamics by Physics Videos by Eugene Khutoryansky (2018) Source. Some decent visualizations of how the field lines don't expand out like they do in electromagnetism, suggesting color confinement.
Video 2. PHYS 485 Lecture 6: Feynman Diagrams by Roger Moore (2016) Source. Despite the title, this is mostly about QCD.
TODO experimental discovery.
Force carrier of quantum chromodynamics, like the photon is the force carrier of quantum electrodynamics.
One big difference is that it carrier itself color charge.
Can be thought as being produced from gluon-gluon lines of the Feynman diagrams of quantum chromodynamics. This is in contrast to quantum electrodynamics, in which there are no photon-photon vertices, because the photon does not have charge unlike gluons.
This phenomena makes the strong force be very very different from electromagnetism.
TODO why is it so hard to find anything non perturbative :-(
On a quantum computer...:
Video 1. Are we living in the matrix? by David Tong (2020) Source. Talks about how the Nielsen-Ninomiya theorem means it is impossible to simulate QFT on a computer in the case of a lattice gauge theory.
As mentioned at Video "Are we living in the matrix? by David Tong (2020)" somehow implies that it is difficult or impossible to simulate physics on a computer. Big news!!!
The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":
From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.
Recommendations by friend P. C.:
Lecture notes found by Googling "quantum field theory pdf":
These seem very direct and not ultra advanced, good read.
Author: David Tong.
Number of pages circa 2021: 155.
It should also be noted that those notes are still being updated circa 2020 much after original publication. But without Git to track the LaTeX, it is hard to be sure how much. We'll get there one day, one day.
Some quotes self describing the work:
A follow up course in the University of Cambridge seems to be the "Advanced QFT course" (AQFT, Quantum field theory II) by David Skinner: www.damtp.cam.ac.uk/user/dbs26/AQFT.html
Author affiliation: University of California, Santa Barbara.
Number of pages: 616!
Don't redistribute clause, and final version by Cambridge University Press, alas, so corrections will never be merged back: web.physics.ucsb.edu/~mark/qft.html. But at least he's collecing erratas for the published (and therefore draft) versions there.
The book is top-level organized in spin 0, spin half, and spin 1. Quite ominous, really.
The preface states that one of its pedagogical philosophies is to "Illustration of the basic concepts with the simplest examples.", so maybe there is hope after all.
45 1 hour lessons. The Indian traditional music opening is the best.
10 2-hour lessons.
Lecturer: Leonard Susskind.
14 1 hours 20 minute lectures.
The video resolution is extremely low, with images glued as he moves away from what he wrote :-) The beauty of the early Internet.
This is a bit "formal hocus pocus first, action later". But withing that category, it is just barely basic enough that 2021 Ciro can understand something.
Lecture notes transcribed by a student: github.com/avstjohn/qft
18 1h30 lectures.
Bibliography review:
Course outline given:
Non-relativistic QFT is a limit of relativistic QFT, and can be used to describe for example condensed matter physics systems at very low temperature. But it is still very hard to make accurate measurements even in those experiments.
Defines "relativistic" as: "the Lagrangian is symmetric under the Poincaré group".
Mentions that "QFT is hard" because (a finite list follows???):
There are no nontrivial finite-dimensional unitary representations of the Poincaré group.
But I guess that if you fully understand what that means precisely, QTF won't be too hard for you!
Notably, this is stark contrast with rotation symmetry groups (SO(3)) which appears in space rotations present in non-relativistic quantum mechanics.
  • something about finding a unitary representation of the poincare group
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