# Quantum Field Theory

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## General

This course is tought at VUB by professor Sevrin. [OUTDATED] Five points are awarded for excercises during the year. It is mainly important that you make them, whether they are right or not. In any case, it's a good idea to solve the exercises, since this is the only way to get familiar with all of the new calculation techniques.

In a previous year: The exam had two questions. The goal was that the exam wouldn't take more than 3 hours, but it eventually ran out of time for a total of 4 hours. Professor Sevrin is against heavy calculations on the exam, and if you head in the right direction immediately, you won't have to calculate much.

Professor Sevrin is quite generous with marks. During the exam he will most of the time talk about extra features outside of the scope of the course (but which are interesting), en ask some small additional questions. Don't hesitate to ask for a hint if needed, you won't lose a lot of points with it.

## Old exams

Good to know: this year (2020-2021), the assignments counted for half of the score, the exam was online and lasted only 1 hour. Additionally, students were asked to write a little paper on a subject they liked.

### 29 january 2021 (Morning)

The extra question were all the same as that from 28 January 2021 (18h30).

### 28 January 2021 (18h30)

• We shortly went through my paper (do not worry too much about this, he is not going to ask you hard questions here and in my case we were discussing around 10min only).

Questions:

• In QED, does the running coupling constant become greater as we increase momentum? Give a conceptual explanation (think of the dipoles-in-vacuum explanation he gave).
• What happens when you try to naively combine SR with QM? It is important that you talk about the three examples he argued (i.e. the stability issue with the KG equation, the failure of Born's statistical interpretation when applied to the KG equation and the issue with microcausality).
• Is QED's perturbative expansion summable?
• He went like this: 'in lecture 2 I stablished the equal-time commutation relations for KG fields (EQ. 3.6) and I missed to add something... you had as homework to read about this... what was it?' That was arguing the covariance of the commutation relations and what is the connection between [\phi(\vec x, t), \phi(\vec y, t)]=0 and microcausality (study section 3.3. in M&S).
• Precisely the problem posted in Wina for the 26th and 28th January. He says 'give the asymptotic states'. That means that you need to find the two linear combinations of the E.O.M. that are solutions of Dirac equation (AND NOT THE ADJOINT). That linear combination is \psi_1 + i \psi_2 and \psi_1 - i \psi_2.

### 28 january 2021 (12h30)

${\displaystyle {\mathcal {L}}=i{\bar {\psi }}_{1}\gamma ^{\mu }\partial _{\mu }\psi _{1}-m{\bar {\psi }}_{1}\psi _{1}+i{\bar {\psi }}_{2}\gamma ^{\mu }\partial _{\mu }\psi _{2}-m{\bar {\psi }}_{2}\psi _{2}-i{\tilde {m}}{\bar {\psi }}_{1}\psi _{2}+i{\tilde {m}}{\bar {\psi }}_{2}\psi _{1}}$

• If we write ${\displaystyle -i{\tilde {m}}{\bar {\psi }}_{1}\psi _{2}}$, why do we have to include ${\displaystyle i{\tilde {m}}{\bar {\psi }}_{2}\psi _{1}}$ as well?
• Find the asymptotic states
• Something with asymptotic series...
• Where does the non-interactive part of the Lagrangian stand for. For what is it used.

### 26 january 2021 (afternoon)

${\displaystyle {\mathcal {L}}=i{\bar {\psi }}_{1}\gamma ^{\mu }\partial _{\mu }\psi _{1}-m{\bar {\psi }}_{1}\psi _{1}+i{\bar {\psi }}_{2}\gamma ^{\mu }\partial _{\mu }\psi _{2}-m{\bar {\psi }}_{2}\psi _{2}-i{\tilde {m}}{\bar {\psi }}_{1}\psi _{2}+i{\tilde {m}}{\bar {\psi }}_{2}\psi _{1}}$

• If we write ${\displaystyle -i{\tilde {m}}{\bar {\psi }}_{1}\psi _{2}}$, why do we have to include ${\displaystyle i{\tilde {m}}{\bar {\psi }}_{2}\psi _{1}}$ as well?
• Find the asymptotic states

### 15 january 2021 (morning)

The calculation with the Maxwell field with extra term ${\displaystyle {\frac {M^{2}}{2}}A^{\mu }A_{\mu }}$ in the Lagrangian. The additional questions were all of the questions that are listed for this year, and also the following questions.

• What is the physical interpretation of the cutoff parameter?
• Does the running coupling constant e(k) increase or decrease at small distances? Why? What about the strong interaction?
• Should we worry about the Landau divergence in the running coupling constant?

### 8 january 2021 (afternoon)

#### Calculation

Given the following free Lagrangian density ${\displaystyle {\mathcal {L}}_{0}={\frac {-1}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {M^{2}}{2}}A_{\mu }A^{\mu }}$, give the equations of motion using the variation of the action (Euler-Lagrange equations). Discuss your result for ${\displaystyle M=0}$ and ${\displaystyle M\neq 0}$ concerning the gauge invariances and number of degrees of freedom. Generalise the discussion of the degrees of freedom to the document on Poincaré.

#### Question

Question was about whether the sum of terms in the Dyson expansion diverges or not and the consequences.

### 7 January 2021 (morning)

#### Calculation

You are given a free Lagrangian density ${\displaystyle {\mathcal {L}}_{0}=i{\bar {\psi _{1}}}\not {\partial }\psi _{1}+i{\bar {\psi _{2}}}\not {\partial }\psi _{2}-m{\bar {\psi _{1}}}\psi _{1}-m{\bar {\psi _{2}}}\psi _{2}-i{\tilde {m}}{\bar {\psi _{1}}}\psi _{2}+i{\tilde {m}}{\bar {\psi _{2}}}\psi _{1}}$ and are asked to find the asymptotic states. Here ${\displaystyle \psi _{1}}$ and ${\displaystyle \psi _{2}}$ are fermion fields and ${\displaystyle {\tilde {m}}.

#### Question

Is photon scattering possible in QFT?

### 5 January 2021

#### Morning

##### Calculation

Given the Lagrangian density ${\displaystyle {\mathcal {L}}_{0}={\frac {1}{2}}\partial _{\mu }\phi _{1}\partial ^{\mu }\phi _{1}+{\frac {1}{2}}\partial _{\mu }\phi _{2}\partial ^{\mu }\phi _{2}-{\frac {m^{2}}{2}}\phi _{1}^{2}-{\frac {m^{2}}{2}}\phi _{2}^{2}-{\tilde {m}}^{2}\phi _{1}\phi _{2}}$, ${\displaystyle {\tilde {m}}^{2} What are the asymptotic states?

##### Question

What conceptual problems arise when we naively try to combine quantum mechanics and special relativity? (e.g.: some wavefunctions yield non-zero probabilities to observe particles at all points in space, whereas all particles must stay inside their light cone). Is this/Are these problem(s) still present in quantum field theory? If not, how are they solved?

#### Afternoon

##### Calculation

Question 1.2 and 1.3 from January 7 2019 where the discussion of the result was mostly about the number of degrees of freedom (polarization vectors). Connect your comparison between the massive and massless case to the document on Poincaré.

##### Question

In classical mechanics photon-photon scattering isn't possible since photons don't have any electrical charge. Is it possible for two photons to scatter in QFT?

### 16 januari 2012

#### Vraag 1

Vraag 1 van 2008.

#### Vraag 2

Vraag 2 van 2011.

### 19 januari 2011

#### Vraag 1

(Lorentz-invariantie) Gegeven een Dirac spinor veld ${\displaystyle \psi }$ en een bosonisch veld ${\displaystyle \phi }$ met Lagrange-dichtheid

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {m^{2}}{2}}\phi ^{2}+i{\bar {\psi }}\not {\partial }\psi -M{\bar {\psi }}\psi +i\lambda \phi {\bar {\psi }}\gamma _{5}\psi }$

1. Hoe moet ${\displaystyle \phi }$ transformeren onder Lorentztransformaties opdat de Lagrange dichtheid invariant zou zijn onder de volledige Lorentz groep?
2. Stel nu dat de term ${\displaystyle \lambda '\phi \hbar {\bar {\psi }}\psi }$ aan bovenstaande dichtheid wordt toegevoegd. Kunnen we dan nog invariantie onder de hele Lorentzgroep hebben?
3. Wat is de dimensie in n.u. van ${\displaystyle \lambda }$ en ${\displaystyle \lambda '}$ ?
4. Als we naar de laatste twee interactietermen van de Lagrangedichtheid met de in 2) toegevoegde term kijken, dan zien we dat de laatste een reÃ«le coÃ«fficiÃ«nt heeft en de voorlaatste een imaginaire. Hoe komt dit?

#### Vraag 2

(IJkinvariantie van Feynman amplitudes)

1. Waarom is ${\displaystyle e^{+}e^{-}\to \gamma }$ geen fysisch proces terwijl ${\displaystyle e^{+}e^{-}\to \gamma \gamma }$ dat wel is?
2. Beschouw het laatstgenoemde fysische proces. Het positron heeft (moment, chiraliteit) ${\displaystyle (p_{1},r_{1})}$, het elektron ${\displaystyle (p_{2},r_{2})}$ en de fotonen hebben ${\displaystyle (k_{1},s_{1})}$ en ${\displaystyle (k_{2},s_{2})}$. Geef de twee Feynmandiagrammen die dit proces in leidende orde beschrijven. Geef expliciet de bijbehorende Feynmanamplitudes (polarisatie-indices, momenta, etc expliciet schrijven).
3. Vervang nu in bovenstaande uitdrukking de polarisatievector ${\displaystyle \varepsilon _{s_{1}}({\vec {k}}_{1})}$ door k_1 en toon aan dat de beide bijdrages tegen over elkaar wegvallen.
4. Men zegt dat dit een gevolg van ijkinvariantie is. Leg uit.

### 15 januari 2009

Exact, maar dan ook exact dezelfde vragen als in 2008, alleen moest je in de oefening 'electron' door 'positron' vervangen...

### 17 januari 2008

#### Vraag 1

Bekijk volgende Lagrange dichtheid ${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\frac {m^{2}}{2}}A^{\mu }A_{\mu }}$ waarbij ${\displaystyle F^{\mu \nu }}$ op de gebruikelijke manier gedefinieerd is.

• Overtuig uzelf ervan dat deze Lagrange dichtheid niet ijkinvariant is.
• Bepaal de bewegingsvergelijkingen. Toon aan dat ondanks de afwezigheid van een ijksymmetrie deze toch de Lorentconditie impliceren.
• Geef een volledig stel oplossingen en interpreteer het resultaat (vergelijk met het massaloze geval).

#### Vraag 2

Oefening 8.7 blz 160. Het volstaat om deze oefening op te lossen voor enkel de transformatie voor ${\displaystyle \epsilon }$, deze voor ${\displaystyle \epsilon '}$ is analoog.