Ciro Santilli $$Sponsor Ciro$$ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱

# Quantum field theory (QFT)

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:

## Mathematical formulation of quantum field theory

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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.

### Gauge theory

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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.

#### Gauge field

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A random field you add to make something transform locally the way you want. See e.g.: Video 15. "Deriving the qED Lagrangian by Dietterich Labs (2018)".

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### Fock space

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Yup, this one Focks you up.

### Second quantization

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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:

#### Canonical quantization

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Basically a synonym for second quantization.

### Path integral formulation

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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.

#### Quantum particles take all possible paths

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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.

#### Infinitely many slits thought experiment

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Thought experiment that illustrates the path integral formulation of quantum field theory.

### Renormalization

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#### Effective field theory

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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.

#### Yang-Mills theory

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##### Yang-Mills existence and mass gap
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• 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.

## Quantum electrodynamics (QED)

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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:

### Quantum electrodynamics experiment

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Experiments explained by QED but not by the Dirac equation:

#### Lamb shift (1947)

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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?
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?
www.mdpi.com/2624-8174/2/2/8/pdf History and Some Aspects of the Lamb Shift by G. Jordan Maclay (2019)
##### Lamb-Retherford experiment (1947, Lamb shift experiment)
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Published as Fine Structure of the Hydrogen Atom by a Microwave Method by Willis Lamb and Robert Retherford (1947) on Physical review.
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.
This one has open accesses as of 2021: journals.aps.org/pr/pdf/10.1103/PhysRev.72.241
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.

#### Electron magnetic moment

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##### Anomalous magnetic dipole moment
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###### Anomalous magnetic dipole moment of the electron
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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:
###### The Magnetic Moment of the Electron by Kusch and Foley (1948)
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Published on Physical review by Polykarp Kusch and Foley.

#### Dirac equation vs quantum electrodynamics

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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.

### Quantum electrodynamics Lagrangian (QED Lagrangian)

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$$LQED​=ψˉ​(iℏcD/ −mc2)ψ−4μ0​1​Fμν​Fμν (1)$$ 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):

#### Derivation of the quantum electrodynamics Lagrangian (Derivation of the QED Lagrangian)

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Like the rest of the Standard Model Lagrangian, this can be split into two parts:

### What does it mean that photons are force carriers for electromagnetism?

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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:

### Feynman diagram

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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.

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No, but why?

### Wheeler-Feynman absorber theory

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What they presented on richard Feynman's first seminar in 1941. Does not include quantum mechanics it seems.

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### Quantum electrodynamics bibliography

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fafnir.phyast.pitt.edu/py3765/ Phys3765 Advanced Quantum Mechanics -- QFT-I Fall 2012 by E.S. Swanson mentions several milestone texts including:

#### Advanced quantum mechanics by Freeman Dyson (1951)

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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.

#### Selected Papers on Quantum Electrodynamics by Julian Schwinger (1958)

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Julian Schwinger's selection of academic papers by himself and others.

#### Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979)

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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: $$E=f(1,2,m)P=f(1,2,0) (2)$$ 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: $$c×E(A,D)×E(D,B)×P(B,C) (3)$$ 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
##### Quantum Mechanical View of Reality by Richard Feynman (1983)
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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.
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.

#### QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994)

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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.

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## Weak interaction

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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.

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### Parity violation

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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.

## Quantum chromodynamics (QCD)

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Formulated as a quantum field theory.

### Quark

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TODO experimental discovery.

### Gluon

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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.

### Color confinement

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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.

## Quantum field theory simulations

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TODO why is it so hard to find anything non perturbative :-(
On a quantum computer...:

### Nielsen-Ninomiya theorem

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As mentioned at Video 22. "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!!!

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## Internal and spacetime symmetries

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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.

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## Quantum field theory bibliography

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Ciro Santilli's favorites so far:
Recommendations by friend P. C.:

### Quantum field theory lecture notes

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Lecture notes found by Googling "quantum field theory pdf":

#### An Introduction to QED and QCD by Jeff Forshaw (1997)

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These seem very direct and not ultra advanced, good read.

#### Quantum Field Theory lecture notes by David Tong (2007)

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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

#### Quantum Field Theory book by Mark Srednicki (2006)

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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.

### Quantum field theory lectures

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#### Relativistic Quantum Mechanics by Apoorva D Patel (2014)

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45 1 hour lessons. The Indian traditional music opening is the best.

#### New Revolutions in Particle Physics by Leonard Susskind (2009)

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10 2-hour lessons.
Lecturer: Leonard Susskind.

#### David Tong's 2009 Quantum Field Theory lectures at the Perimeter Institute

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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.

#### Quantum field theory courses by Tobias Osborne

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##### Quantum field theory lecture by Tobias Osborne (2017)
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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.
###### Lecture 1
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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.
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###### Lecture 5
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• something about finding a unitary representation of the poincare group
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Interactions.
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Dirac field.
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Dirac equation.
##### Advanced quantum field theory lecture by Tobias Osborne (2017)
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When the word "advanced" precedes QFT, you know that the brainrape is imminent!!!
Big goal: explain the Standard Model.
###### Lecture 2
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Gaussian path integrals.

### Quantum field theory book

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#### No-Nonsense Quantum Field Theory by Jakob Schwichtenberg (2020)

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This book really tries to recall basic things to ensure that the reader will be able to understand the more advanced ones.
Sometimes it goes a little bit overboard, like defining what a function does several times.
But Ciro Santilli really prefers it when authors error on the side of obvious.

#### Quantum Field Theory for The Gifted Amateur by Tom Lancaster (2015)

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People are mostly saying you have to be a more of a genius amateur to read it.

#### Quantum field theory in a nutshell by Anthony Zee (2010)

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ISBN-13: 978-0691140346
lecture 1 mentions that this book is quick and dirty, as one might guess from the title. Ciro Santilli thinks he's gonna like this one.
First edition: from 2003, www.amazon.com/dp/0691010196, ISBN-13: 978-0691010199.
Summary:

#### Quantum Field Theory Demystified by David McMahon (2008)

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This didn't really deliver. It does start from the basics, but it is often hard to link those basics to more interesting or deeper points. Also like many other Quantum field theory book, it does not seem to contain a single comparison between a theoretical result and an experiment.

#### An Introduction To Quantum Field Theory by Peskin and Schroeder (1995)

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This is very widely used in courses as of 2020, it became kind of the default book.
Unfortunately, this approach bores Ciro Santilli to death. Or perhaps is too just advanced for him to appreciate. Either of those.
800+ pages.