# Quantum field theory

## Table of Contents

- STRT Tong qft book qft
- qft
shit. ok, I guess I sort of figured that out
- TODO this still doesn't explain how to swap variation and partial derivatives properly…. I have to figure that out.. qft
- TODO what's the lesson I learnt here? not sure. I still haven't found decent justifications to that crap. qft
- qft ok, I skimmed it. the book is actually really good, I hightlighted some stuff and should read it again…

- qft
shit. ok, I guess I sort of figured that out
- Klen-Gordon equation qftkleingordon
- qftkleingordon ok, so Klein-Gordon equation is for spin 0, such as Higgs, or pions
- qftkleingordon Is it true that the Schrödinger equation only applies to spin-1/2 particles? - Physics Stack Exchange
- [C] (11) Deriving The Klein Gordon Equation (Relativistic Quantum) - YouTube qftkleingordonquantum
- [C] (11) Deriving The Klein Gordon Equation (Relativistic Quantum) - YouTube qftkleingordon
- [B] The Klein-Gordon equation doesn't seem to be complete. How can I use it for a general-purpose simulation? /r/Physics qftkleingordon
- [B] Why does the Klein-Gordon equation describe bosons with spin equal to zero? /r/AskPhysics qftkleingordon
- qftkleingordon PBS space time guy also recomments it!

- qftkleingordonspinor Where can I go to learn more about spinors? /r/math
- [C] How To Solve The Klein-Gordon Equation For The Hydrogen Atom - YouTube qftkleingordon
- [C] Klein-Gordon equation in nLab qftkleingordon
- qftkleingordon Klein Gordon simulation 1 - YouTube

- STRT [C] Peskin – recommended by Baez qftbook
- qftbook p. 17 We can think of the three terms, respectively, as the energy cost of "moving"in time, the energy cost of "shearing" in space, and the energy cost of havingthe field around at all. We will investigate this Hamiltonian much further inSections 2.3 and 2.4.

- [C] Quantum Field Queries /r/AskPhysics qft
- [C] Why can precomputed sets of lattice QFT field configurations be used to measure arbitrary observables? /r/askscience qft
- TODO [C] [Daniel and Jorge Explain the Universe] What is the real charge of the electron? danielAndJorgeExplainTheUniverse https://podcastaddict.com/episode/106315566 via @PodcastAddict qft
- TODO cccccccccccchrome-extension://bjfhmglciegochdpefhhlphglcehbmek/content/web/viewer.html?file=https%3A%2F%2Farxiv.org%2Fpdf%2F1410.6753.pdf qft
- [B] David Skinner – Advanced Quantum Field Theory – University of Cambridge qft
- [B] Объясняем шутку из твиттера про дедушку и лагранжиан квантовой теории поля qft
- qft (12) Quantum Fields: The Real Building Blocks of the Universe - with David Tong - YouTube
- qft A Brief Intro to Topological Quantum Field Theories. - YouTube
- [C] Self-energy - Wikipedia qft
- [C] Zee's book is a bit hard to read.. lots of jumping around and notation abuse… qftphysics
- [B] https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange qft
- TODO SU 2 = Spin 3 qftdrill
- TODO QFT in a nutshell qftstudy
- TODO Path Integral Salesman on Twitter: "As the pope of climate science, I declare as our holy book: https://t.co/nrvStKMcS4 https://t.co/nsHhCxX2XA" / Twitter qft
- DONE [B] The Inaugural Atiyah Lecture: Jean-Pierre Bourguignon - What is Spinor? - Media Hopper Create qfttowatchspinor

## ¶STRT Tong qft book qft

index notation:

lover index: covariant, upper index: contravariant

∂f,ᵤ ≡ ∂ᵤf ≡ ∂f/∂xᵘ

gradient: ∇fᵤ ≡ ∂f/∂xᵤ – ok, makes sense…

gradient is covariant

field EL equation:

∂ᵤ ( ∂L/∂(∂ᵤϕ) ) - ∂L/∂ϕ = 0

free lagrangian: L = 1/2 ∂ᵤϕ ∂ᵘϕ - 1/2 m² ϕ²

or, L = 1/2 ηᵤᵥ ∂ᵤϕ ∂ᵥϕ - 1/2 m² ϕ²

∂L/∂ϕ = - m²ϕ

the other term: equal to ∂ᵘϕ hmm. if you treat ∂ᵤϕ as formal expression, that makes sense. there is nothing more to it, right? since it's a partial derivative with respect to expression.

wait, no, that doesn't explain the vanishing 1/2 term

### ¶ shit. ok, I guess I sort of figured that out qft

express L = 1/2 (∂ϕ/∂t)² - 1/2 ( (∂ϕ/∂x)² + (∂ϕ/∂y)² + (∂ϕ/∂z)²) - 1/2 m ϕ²

∂L/∂(∂ᵤ ϕ) is actually a [co??]vector of (∂L/∂(∂ₜϕ), ∂L/∂(∂ₓϕ), ∂L/∂(∂_{y} ϕ), ∂L/∂(∂_{z} ϕ)), where ALL PARTIAL DERIVATIVES ARE KINDA LIKE INDEPEDENT SYMBOLS

soo, if we calculate each of them, we get (∂ₜ, -∂ₓ, -∂_{y}, -∂_{z}), which is ∂ᵘ ϕ

to calculate it the 'proper', tensor way:

TODO ??? now use ∂xᵢ/∂xⱼ = xᵢ,ⱼ = δᵢⱼ

∂L/∂(∂ᵤϕ) = (ignoring term with mass, which is although, actually an inner product, so nevertheless sort of interesting)

(note that L is a scalar!)

1/2 ∂(ηᵢⱼ ∂ᵢϕ ∂ⱼϕ)/∂ (∂ᵤϕ) = (chain rule, and use the fact that metric is constant for each index)

= 1/2 ηᵢⱼ ( ∂∂ᵢϕ/∂∂ᵤ ∂ⱼϕ + ∂ᵢϕ ∂∂ⱼϕ/∂∂ᵤ ϕ)

https://physics.stackexchange.com/a/3006/40624

use that ∂∂ᵢφ/∂∂ⱼφ = δʲᵢ

= 1/2 ηᵢⱼ (δᵘᵢ ∂ⱼϕ + ∂ᵢϕ δᵘⱼ )

this thing with delta works exactly as raising and picking the component?

= 1/2 δᵘᵢ ∂ⁱϕ + 1/2 δᵘⱼ ∂ʲϕ = 1/2 ∂ᵘϕ + 1/2 ∂ᵘϕ = ∂ᵘϕ

dunno, I did it kinda formally. how to figure that shit out quickly in general? probably no easier way especially considering that metric tensor is not constant in general.

http://www.continuummechanics.org/tensornotationadvanced.html

this helped a bit

https://www.physicsforums.com/threads/applying-euler-lagrange-to-real-klein-gordon-lagrangian.901389/

https://www.physicsforums.com/threads/klein-gordon-lagrangian.684071/

https://physics.stackexchange.com/questions/418382/derivation-of-klein-gordon-equation-in-general-relativity

https://physics.stackexchange.com/questions/340009/derivatives-in-euler-lagrange-for-fields

#### ¶TODO this still doesn't explain how to swap variation and partial derivatives properly…. I have to figure that out.. qft

#### ¶TODO what's the lesson I learnt here? not sure. I still haven't found decent justifications to that crap. qft

#### ¶ ok, I skimmed it. the book is actually really good, I hightlighted some stuff and should read it again… qft

## ¶Klen-Gordon equation qftkleingordon

### ¶ ok, so Klein-Gordon equation is for spin 0, such as Higgs, or pions qftkleingordon

### ¶ Is it true that the Schrödinger equation only applies to spin-1/2 particles? - Physics Stack Exchange qftkleingordon

Well, this should not surprise us, since the Schrodinger equation is the nonrelativistic limit to the Klein-Gordon equation. And we should recall the Klein-Gordon equation describes spin-0 bosons! The nonrelativistic limit c→∞c→∞ should not affect the spin of the particles involved. (That's why the Pauli model is the nonrelativistic limit of the Dirac equation!)

### ¶[C] (11) Deriving The Klein Gordon Equation (Relativistic Quantum) - YouTube qftkleingordonquantum

https://www.youtube.com/watch?v=bPlP9EkGixk

so apparently we can get klein-gordon if we replace energy and momentum with quantum operator in einsteins identity

### ¶[C] (11) Deriving The Klein Gordon Equation (Relativistic Quantum) - YouTube qftkleingordon

https://www.youtube.com/watch?v=bPlP9EkGixk

covariant has got overloaded meaning. covariant vector – lower index; covariant thing – transforms according to symmetry group

### ¶[B] The Klein-Gordon equation doesn't seem to be complete. How can I use it for a general-purpose simulation? /r/Physics qftkleingordon

The Klein Gordon equation does not describe a quantum mechanical wavefunction. It describes a classical relativistic free field. To get quantum mechanical behavior out of this, you need to quantize this theory, moving to the world of quantum field theory. That's a physics answer... In terms of how this fits into your game it would depend on what you're trying to do. But the bottom line is that QFT was essentially created to describe quantum mechanical effects in a relativistic setting.

### ¶[B] Why does the Klein-Gordon equation describe bosons with spin equal to zero? /r/AskPhysics qftkleingordon

>Why can it still describe particles? It can still describe particles if you give up the notion that the wavefunction represents a probability density. Instead, what you called the wavefunction should be promoted to a field operator in a second-quantized field theory. >Why does it specifically describe bosons with spin 0, where does it say in the equation that particles with spin 0 use the KG-equation? Because it's a scalar field. This implies that it's got no angular momentum. For instance in the Dirac equation, the field operators are bi-spinors, meaning that you've got electrons and positrons both with spin-1/2. For the photon field, the field operator is a vector (the vector potential), so the photon has spin-1. It's all about how the field transforms under rotations. That tells you how many units of angular momentum the particles have. >What is the physically reason why spin 1/2 have Dirac equation and Klein gordon equation spin 0? This sort of ties in with the above. If you haven't already, I'd suggest reading the last chapter of Sakurai. He goes into details about relativistic QM and shows the pitfalls that push you towards actual field theories.

#### ¶ PBS space time guy also recomments it! qftkleingordon

### ¶ Where can I go to learn more about spinors? /r/math qftkleingordonspinor

The mathematics behind spinors is pretty daunting. I think it's a realization of Clifford algebras (I could be mistaken; it's been some time since I've really been a physicist). You can learn about where they come from without the mathematical overhead. In quantum, spin is usually presented as some ad hoc property of particles. The way it came about was noting from experiment that basic particles like electrons had quantities which behaved a lot like *orbital* angular momentum without actually being bound to another particle! The orbital angular momentum operators were well understood and they can be abstractly viewed (via certain commutation relations). If wavefunctions were just scalars, they couldn't possibly relate to the commutation relations they were seeing from orbital angular momentum so the next best thing is vectors. Well with this idea, the simplest (non scalar) vector is a two dimensional vector. Representing the algebra of spin operators on C^2 ended up giving the Pauli matrices and the theory gave results similar to that in experiment. This however is pretty unsatisfactory since it's very ad hoc. The theory didn't tell you which particles had spin, which didn't and it didn't really explain where it came from. Parallel to the development of regular quantum mechanics was the development of relativistic quantum mechanics. (Yes they were developed nearly at the same time and in fact Schrodinger envisioned relativistic quantum mechanics before his eponymous equation.) From Einstein's energy momentum relation E^2 = p^2 c^2 + m^2 c^4 came the Klein Gordon equation by replacing E with - i\*d/dt and p with i\*grad, neglecting factors of hbar. This looks pretty close to the usual wave equation for an electromagnetic wave, but with a mass term (which changes everything!). The Klein Gordon equation had some serious issues with it. Even though it seemed to be a reasonable relativistic equation which actually reduces to the Schrodinger equation in a certain limit, it didn't preserve probabilities. Particularly the integral of psi\* psi over all space changed from one time to another, which is really bad! This breaks one of the foundational aspects of the Copenhagen interpretation: the integral of psi\* psi over all space is always 1 since we expect to find the particle *somewhere*. Dirac pinpointed where the issue was. If you consider the Schrodinger equation, the time derivative is a first order derivative. In the Klein Gordon equation, it's a second derivative with respect to time. I think he was able to prove/argue that this is actually where the problems come in. (These issues with the Klein Gordon equation were fixed later, ironically by Dirac's sea of particles idea: the negative probabilities were actually because the particle turned into an anti particle.) So.. What he set out to do was to eliminate one of the time derivatives but there isn't a super obvious way of doing that. What he realized is that if you *factor* the equation into a product of two operators, you could in effect eliminate one of the time derivatives. The only way to do this is to have the operators actually be matrix valued (particularly they're 4 by 4 matrices or higher) and if the operators are matrices, then they must act on vectors.. So the solutions to his equation were vector valued wavefunctions! This is actually where spinors come from. There is one problem though. In the original quantum theory, they were two dimensional vectors. Dirac gave four dimensional vectors. What you find (after some relatively tedious analysis) is that in the non relativistic limit, two of the components are really small compared to the other two and their effects aren't really measured in a normal laboratory setting. For a self contained treatment of this, I *highly* recommend Griffiths' particle physics book. The Dirac equation is in chapter 7 if I recall correctly. I feel that the basics of relativistic quantum mechanics are very beautiful and are as close to an axiomatic (and painless) theory in physics as you're going to get.

### ¶[C] How To Solve The Klein-Gordon Equation For The Hydrogen Atom - YouTube qftkleingordon

https://www.youtube.com/watch?v=9_3uQdF-tEs

klein gordon solution for hydrogen

### ¶[C] Klein-Gordon equation in nLab qftkleingordon

https://ncatlab.org/nlab/show/Klein-Gordon+equation#FundamentalSolutions

where □g \Box_g denotes the wave operator on (X,g) (X,g)

### ¶ Klein Gordon simulation 1 - YouTube qftkleingordon

## ¶STRT [C] Peskin – recommended by Baez qftbook

- State "STRT" from "TODO"

### ¶ p. 17 We can think of the three terms, respectively, as the energy cost of "moving"in time, the energy cost of "shearing" in space, and the energy cost of havingthe field around at all. We will investigate this Hamiltonian much further inSections 2.3 and 2.4. qftbook

## ¶[C] Quantum Field Queries /r/AskPhysics qft

>I have a few conceptual questions about QFT. >1.) Is renormalization a problem that comes from the use of perturbation theory, or is it something more fundamental? Renormalization is not specific to perturbation theory. For a fully nonperturbative *description* of renormalization, consider the Wilsonian approach to the RG group flow. >Does lattice QFT require the techniques of renormalization? Lattice QFT *displays* renormalization as a feature. On a lattice a theory is simply defined as it is and scattering amplitudes are simply reduced (in principle) to an explicit finite QM problem - there are no field theory techniques to apply at all. Your system is not a QFT; the QFT appears (hopefully) in the limit of lattice spacing -> 0. If you put QED on a lattice you will see in the computed scattering amplitudes the running of α on the energy scale. >3.) It is said that one can derive Einstein-like equations from a massless spin-2 field. Is this done by just using the most important Feynman diagrams, and neglecting the problematic higher order diagrams? This is done keeping only tree-level diagrams, since GR is the classical limit of the quantum theory of gravitons. If you want, you can also add one-loop diags to obtain the hbar^(1) (or semiclassical) corrections to GR. However, you cannot go further since from two-loops onwards the theory is nonrenormalizable and so non-predictive.

## ¶[C] Why can precomputed sets of lattice QFT field configurations be used to measure arbitrary observables? /r/askscience qft

My knowledge of quantum mechanics is rusty and my understanding of (lattice) quantum field theory on a very novice level at best, so it is likely my whole question is based on completely wrong assumptions and a lack of understanding. Most introductory texts about QFT give some sort of a translation table between quantities in QM and QFT. (see e.g. [top of page 16 here](www.itp.uni-hannover.de/saalburg/Lectures/wiese.pdf) ). A given particle path (over all of which you integrate in the path integral formulation of QM to get the amplitude of a given process) is translated into a given field configuration in QFT (over all of which, again, you integrate in the path integral formulation of QFT). As far as I understand, lattice quantum field theory calculations on big high-power computing clusters are effectively generating lots of field configurations (for a given lattice size, spacing, boundary conditions etc, but independent of any "starting conditions"). These field configurations are generated using metropolis MC methods (or similar more advanced importance sampling schemes), based on the action calculated for a given field configuration. In the "list" of output field configurations, the occurence of field configurations is then already weighted by their effective action, so that summing up over them yields the most relevant results without near infinite amounts of practically irrelevant field configurations. To extract an observable from such a set of field configurations, one simply sums the value of the observable for all field configurations. I wonder why it is possible and reasonable to extract any observable from a pre-computed number of field configurations that did not include any information about what observable one would like to obtain. In other words: how can the action of a field configuration be independent of the process I want to extract afterwards? To maybe clarify a bit further, consider a simple double slit experiment: I want to calculate the QM amplitude of an electron at position A (on one side of the double slit) at time t0 to appear at position B (on the other side of the double slit) at time t1. For this I randomly generate a bunch of paths the satisfy the conditions of my observable (position A at t0, position B at t1) and evaluate their actions. If I want to be smart about it, I do some importance sampling of paths (Metropolis or whatever). However in this scenario, I only generated paths that were connected to the observable I knew I was looking for from the beginning (propagation A->B). I could not change the observable to a different transition amplitude afterwards and use the same paths. So how do I unify these two pictures in my head? The only thing I can think of would be to not restrict the generation of QM paths to starting point A and ending point B, instead generating QM paths for all possible starting and end points. Afterwards, to calculate the desired transition amplitude, I could only sum up over paths going from A to B, which would have made the very most of my generated paths absolutely unnecessary. If that should be the case, why to LQFT calculations not restrict the generation of field configurations to such that give a meaningful contribution to a predefined observable?

## ¶TODO [C] [Daniel and Jorge Explain the Universe] What is the real charge of the electron? danielAndJorgeExplainTheUniverse https://podcastaddict.com/episode/106315566 via @PodcastAddict qft

OK this is pretty fascinating

Stuff in the freezer analogy for an electron charge

Electron is surrounded/shielded by a virtual cloud of positrons. So to measure charge +1, the real charge had to be infinite (renormalization)

Electron gets mass from higgs interaction, but also from virtual photons. But they act as a multiplier? If it wasn't for higgs, it'd have zero mass

## ¶TODO cccccccccccchrome-extension://bjfhmglciegochdpefhhlphglcehbmek/content/web/viewer.html?file=https%3A%2F%2Farxiv.org%2Fpdf%2F1410.6753.pdf qft

A situation could arise where we know what the symmetries of the objectare, but we might not know what the object really is. This is sometimes enough to makepredictions. For example, we can have a rigid object which is symmetric under rotations,2 with the symmetries of a circle. Then we know that it will roll smoothly on a table. It canbe a solid cylinder or a hollow cylinder, but both will roll smoothly on a table. Of course,in other respects the hollow and solid cylinder can behave dierently. For example, onecan oat in water and the other might not. then one finds that the condition for banks not to run out of either ofthe currencies, or that the net ow of money across each bridge is zero, is equivalent toMaxwell's equations.

### ¶TODO man, this pdf is fucking cool. realltry to understand it. qft

## ¶[B] David Skinner – Advanced Quantum Field Theory – University of Cambridge qft

QFT in Zero Dimensions:

## ¶ (12) Quantum Fields: The Real Building Blocks of the Universe - with David Tong - YouTube qft

https://www.youtube.com/watch?v=zNVQfWC_evg&list=WL&index=69

pretty good lecture, also turns out Tong is quite hilarious!

## ¶ A Brief Intro to Topological Quantum Field Theories. - YouTube qft

https://www.youtube.com/watch?v=59uLGIrkMxM&list=WL&index=61&t=0s

I guess same stuff that Baez was writing about, didn't find talk particularly interesting

## ¶[C] Self-energy - Wikipedia qft

The photon and gluon do not get a mass through renormalization because gauge symmetry protects them from getting a mass. This is a consequence of the Ward identity. The W-boson and the Z-boson get their masses through the Higgs mechanism; they do undergo mass renormalization through the renormalization of the electroweak theory.

## ¶[C] Zee's book is a bit hard to read.. lots of jumping around and notation abuse… qftphysics

## ¶[B] https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange qft

There are limits to the validity of the virtual particle picture. The virtual-particle formulation is derived from a method known as perturbation theory which is an approximation assuming interactions are not too strong, and was intended for scattering problems, not bound states such as atoms. For the strong force binding quarks into nucleons at low energies, perturbation theory has never been shown to yield results in accord with experiments,[3] thus, the validity of the "force-mediating particle" picture is questionable. Similarly, for bound states the method fails.[4] In these cases the physical interpretation must be re-examined. As an example, the calculations of atomic structure in atomic physics or of molecular structure in quantum chemistry could not easily be repeated, if at all, using the "force-mediating particle" picture

## ¶TODO SU 2 = Spin 3 qftdrill

## ¶TODO QFT in a nutshell qftstudy

- 9 path integrals
- 12 Lagrangian

## ¶TODO Path Integral Salesman on Twitter: "As the pope of climate science, I declare as our holy book: https://t.co/nrvStKMcS4 https://t.co/nsHhCxX2XA" / Twitter qft

https://twitter.com/litgenstein/status/1210890946201739269

@litgenstein Such a great book, really clear description of renormalization