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  1. M

    I How to determine matching coefficient in Effective Field Theory?

    Assume that I have the Lagrangian $$\mathcal{L}_{UV} =\frac{1}{2}\left[\left(\partial_{\mu} \phi\right)^{2}-m_{L}^{2} \phi^{2}+\left(\partial_{\mu} H\right)^{2}-M^{2} H^{2}\right] -\frac{\lambda_{0}}{4 !} \phi^{4}-\frac{\lambda_{2}}{4} \phi^{2} H^{2},$$ where ##\phi## is a light scalar field...
  2. M

    Showing that this identity involving the Gamma function is true

    My attempt at this: From the general result $$\int \frac{d^Dl}{(2\pi)^D} \frac{1}{(l^2+m^2)^n} = \frac{im^{D-2n}}{(4\pi)^{D/2}} \frac{\Gamma(n-D/2)}{\Gamma(n)},$$ we get by setting ##D=4##, ##n=1##, ##m^2=-\sigma^2## $$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...
  3. M

    How to translate expression into momentum-space correctly

    This seems rather straight forward, but I can't figure out the details... Generally speaking and ignoring prefactors, the Fourier transformation of a (nicely behaved) function ##f## is given by $$f(x)= \int_{\mathbb{R}^{d+1}} d^{d+1}p\, \hat{f}(p) e^{ip\cdot x} \quad\Longleftrightarrow \quad...
  4. M

    Mass correction in ##\phi^4##-theory

    @vanhees71 Thanks a lot for the explanations and I will be sure to check out your lecture notes! Just as a quick check, the issue is that I basically conflated the following, right? i.e. I assumed that ##\phi^4## has this one extra loop diagram that appears due to a ##\phi^3## interaction...
  5. M

    Mass correction in ##\phi^4##-theory

    Alright, this makes sense. Then we have $$m_{\text{ren}}^2=m^2[1+I(m_{\text{ren}}^2)] \approx m^2[1+I(m^2)].$$ When exactly did that happen? Where in post #1 did I make a mistake so that I ended up in ##\phi^3## theory?
  6. M

    Mass correction in ##\phi^4##-theory

    I'm sorry, but I don't understand how to do that... What I have tried (thought about) so far: $$ \frac{1}{p^{2}-m^{2}-m^2I(p^2)} \approx \frac{1}{p^2-m^2} + \frac{1}{p^2-m^2}m^2I(p^2)\frac{1}{p^2-m^2}.$$ Can we use this maybe like this: $$\frac{i Z...
  7. M

    Mass correction in ##\phi^4##-theory

    Thank you very much for the response! I hope you mean the ##\log## that will eventually show up in ##I(p^2)##, if not, I'm not really sure what you mean. I just went back to my QFT1 lecture notes (Chp. 11.2) one more time to check, and my Prof. got for this integral two different expression...
  8. M

    Mass correction in ##\phi^4##-theory

    Before I start, let me say that I have looked into textbooks and I know this is a standard problem, but I just can't get the result right... My attempt goes as follows: We notice that the amplitude of this diagram is given by $$\begin{align*}K_2(p) &= \frac{i(-i...
  9. M

    Conservation law for FRW metric

    My attempt: Realize we can work in whatever coordinate system we want, therefore we might as well work in the rest frame of the fluid. In this case ##u^a=(c,\vec{0})##. The conservation law reads ##\nabla^a T_{ab}=0##. Let us pick the Levi-Civita connection so that we don't have to worry about...
  10. M

    Covariant derivative and the Stress-enegery tensor

    Perfect, thanks a lot for checking and looking up the references!
  11. M

    Equation of motion in curved spacetime

    1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities $$Q^t :=...
  12. M

    Covariant derivative and the Stress-enegery tensor

    Thanks for spotting the typo. I'm rather new to this entire GR-formalism, i.e. the covariant derivatives, etc., so I was just a bit unsure if I'm really doing operations that are permitted. Also, ##C=1## seemed a bit odd in the first moment, but if you think this works, then I'm happy!
  13. M

    Covariant derivative and the Stress-enegery tensor

    My try: $$ \begin{align*} \nabla^a T_{ab} &= \nabla^a \left(\nabla_{a} \phi \nabla_{b} \phi-\frac{C}{2} g_{a b} \nabla_{c} \phi \nabla^{c} \phi\right)\\ &\overset{(1)}{=} \underbrace{(\nabla^a\nabla_{a} \phi)}_{=0} \nabla_{b} \phi + \nabla_{a} \phi (\nabla^a\nabla_{b} \phi)-\frac{C}{2}...
  14. M

    I Condtion on transformation to solve the Dirac equation

    @PeroK Could you elaborate? I'm asking because I don't see how what you write adds up with the very next line in my professors reasoning, i.e. $$\begin{aligned} \left(\gamma^{\mu} \partial_{\mu}^\prime-m\right) \psi^{\prime}(x^\prime)\neq\left(\gamma^{\mu} \partial_{\mu}-m\right) S \psi(\Lambda...
  15. M

    I Condtion on transformation to solve the Dirac equation

    The problem is given in the summary. My attempt: Assume that ##\psi^\prime (x^\prime)## is a solution of the Dirac equation in the primed frame, given the transformation ##x\mapsto x^\prime :=\Lambda^{-1}x## and ##\psi^\prime (x^\prime)=S\psi(x)##, we have $$ \begin{align*} 0&=(\gamma^\mu...
  16. M

    I How to determine Spinor in Feynman diagram

    Isn't this just a matter of definition? My lecturer demands that we use ##v_\alpha## for electrons (created and annihilated by ##a^\dagger## and ##a##) and ##u_\alpha## for positrons (created and annihilated by ##b^\dagger## and ##b##), which unfortunately makes looking up stuff sometimes really...
  17. M

    I How to determine Spinor in Feynman diagram

    Consider Moller scattering, that is $$e^-(\vec p_1, \alpha)+e^-(\vec p_2, \beta) \quad\longrightarrow\quad e^-(\vec q_1, \gamma)+e^-(\vec q_2, \delta),$$ where the ##\vec{p}_i,\vec q_i## label the momenta of the in and outgoing electrons and the greek letter the spin state. The two relevant...
  18. M

    I Normal order and overlap of states

    Thank you very much @HomogenousCow I think this clears up my confusion about the topic.
  19. M

    I Normal order and overlap of states

    The second sentence is exactly what confuses me! When you say "we need [...] to contract with the creation annihilation operators outside of the time ordering sign", what exactly do you mean with the "contract"? Up to now I thought that contractions can only arise in the context of Wick's...
  20. M

    I Normal order and overlap of states

    @HomogenousCow Thank you for the answer. Maybe I'm misunderstanding you, but the exercise was supposed to be solved in the way I presented above, so I cannot just change that (I technically could, but I would like to understand whats going on in the provided solution). It's possible that I...
  21. M

    I Normal order and overlap of states

    I have trouble understanding the solution to a homework problem. Consider the interaction Lagragian ##\mathcal{L}_{\rm int} = -iqA_\mu \bar{\psi}\gamma^\mu \psi##, i.e. photon-electron/positron interaction. We want to focus on the Compton scattering $$e^-(\vec p_1, \alpha) + \gamma(\vec p_2...
  22. M

    Weird condition describing symmetry transformation

    I think I've got the answer, feel free to correct me if I'm wrong. The point that I missed is that we require ##\phi^\prime (x^\prime) = \phi(x)## only for Lorentz transformations, i.e. we want the scalar field to transform like a scalar under a Lorentz transformation, but we don't make any...
  23. M

    Weird condition describing symmetry transformation

    I'm a bit confused about the condition given in the description of the symmetry transformation of the filed. Usually, given any symmetry transformation ##x^\mu \mapsto \bar{x}^\mu##, we require $$\bar\phi (\bar x) = \phi(x),$$ i.e. we want the transformed field at the transformed coordinates to...
  24. M

    I Consfused about the workflow for calculating scattering amplitudes with Feynman diagrams

    Perfect, this is what I was looking for. So once the Feynman rules are know I can just draw "all" permissible diagrams and use the rules to compute them instead of going through the detailed computations, i.e. computing the ##F^{(n)}## individually. Thanks for the help!
  25. M

    I Consfused about the workflow for calculating scattering amplitudes with Feynman diagrams

    It's not about deriving them. You would do that by going through the calculations that I did for the "elemental building blocks" of the Feynman diagrams. Maybe I can rephrase my question to make it clearer. Where do you get the Feynman diagrams from? The information you have is ##e^-e^- \to...
  26. M

    I Consfused about the workflow for calculating scattering amplitudes with Feynman diagrams

    Thanks for the answer! You're right. Sorry about that, things got a little bit messy towards the end... I know the Feynman rules for QED, I also know how the diagrams look like and I also know how to "convert" Feynman diagrams using the Feynman rules into ##M_{\alpha\beta\gamma\delta}##. The...
  27. M

    Solving the same question two ways: Parallel transport vs. the Lie derivative

    a) I found this part to be quite straight forward. From the Parallel transport equation we obtain the differential equations for the different components of ##X^\mu##: $$ \begin{align*} \frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\ \frac{\partial...
  28. M

    I Consfused about the workflow for calculating scattering amplitudes with Feynman diagrams

    In the following I will try to deduce the scattering amplitude for a specific interaction. My question is at the bottom, the entire rest is my reasoning to explain how I came to the results I present. My working Let's assume I would like to calculate the second order scattering amplitude in ##...
  29. M

    Calculating Energy-Momentum Tensor in GR

    You are absolutely right, there should be a ##1/2## in front of the first term... I completely overlooked this. With this in mind we have $$ \begin{align*} \delta S_M &= \int d^4x (\delta\sqrt{-g}) (\frac{1}{2}g^{\alpha\beta} \nabla_\alpha\phi\nabla_\beta\phi-\frac{1}{2}m^2\phi^2) + \int d^4x...
  30. M

    Calculating Energy-Momentum Tensor in GR

    My attempt was to first rewrite ##S_M## slightly to make it more clear where ##g_{\mu\nu}## appears $$S_M = \int d^4x \sqrt{-g} (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2).$$ Now we can apply the variation: $$\begin{align*} \delta S_M &= \int d^4x (\delta\sqrt{-g})...
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