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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 what's going on in the provided solution). It's possible that...
  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})...
  31. M

    Transformation from de Sitter to flat spacetime coordinates

    You were right, there is a minus sing missing and with it everything works out... Thanks for the help!
  32. M

    Transformation from de Sitter to flat spacetime coordinates

    Let me begin by stating that I'm aware of the fact that this is a metric of de Sitter spacetime, aka I know the solution, my problem is getting there. My idea/approach so far: in the coordinates ##(u,v)## the metric is given by $$g_{\mu\nu}= \begin{pmatrix}1 & 0\\ 0 & -u^2\end{pmatrix}.$$ The...
  33. M

    Wick contraction in scalar QED

    While writing out the Dyson series due to the time ordering above I encountered the two expressions $$T(\mathcal{L}_{int}(x))\quad \text{and}\quad T(\mathcal{L}_{int}(x)\mathcal{L}_{int}(y))$$ I was able to write out the first term in terms of contractions using Wick's theorem and then finally...
  34. M

    A Confused about notation regarding the covariant derivative

    I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***): $$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...
  35. M

    Dyson's series and the time derivative

    I'm having a hard time understanding how exactly to evaluate the expression} $$\partial_t \mathcal{T}\exp\left(-i S(t)\right)\quad \text{where}\quad S(t)\equiv\int_{t_0}^tdu \,H(u) .$$ The confusing part for me is that if we can consider the following: $$\partial_t \mathcal{T}\exp\left(-i...
  36. M

    Nother current of given symmetry

    Thank you very much for the detailed answer! I tried using your approach on a problem which I already solved, to see if I really understood it.. Unfortunately I'm still struggling. Lets say I have a vector field ##A_\mu## and I consider only a translation, aka something of the form...
  37. M

    Nother current of given symmetry

    First of all, thank you for the reply. I'd like to split the following question into two parts: Assuming that ##\mathcal{L}## is scale-invariant (aka ##m## and ##\mu## are zero and the values of ##\lambda## and ##\Delta## chosen accordingly). In this case we have ##\delta\mathcal{L}=0##, which...
  38. M

    Show that the given Green Function is the propagator of a certain Lagrangian

    After some trying out I was able to obtain a partial result I think. First derive the EoM for the given Lagrangian, which results in $$\partial^2 A^\rho -\partial^\rho\partial^\lambda A_\lambda + \xi^{-1}\partial^\rho\partial^\lambda A_\lambda=0.$$ Now we can apply the following trick to get...
  39. M

    Nother current of given symmetry

    This is my first time dealing with scaling symmetry, so I'm sorry if the following is fundamental wrong. My approach was the same as if I was trying to show the same for translation or Lorentz symmetry. We have $$\delta\phi(x)= \phi'(x')-\phi(x)=...
  40. M

    Show that the given Green Function is the propagator of a certain Lagrangian

    My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...
  41. M

    Covariant derivative of a (co)vector field

    Alright, so if I don't misunderstand we have $$\begin{align*} (\nabla_{\partial_l}dx^k)(\partial_i) &= \nabla_{\partial_l}(dx^k(\partial_i)) - dx^k(\nabla_{\partial_l}\partial_i)\\ &= \nabla_{\partial_l}\delta^k_i - dx^k(\Gamma^u_{li}\partial_u)\\ &= \nabla_{\partial_l}\delta^k_i - \Gamma^k_{li}...
  42. M

    Covariant derivative of a (co)vector field

    I don't know. Usually I'd say the "derivative" of a constant is zero, but I'm really not sure if this is true when talking about the covariant derivative... The only things I know about this derivative are written above, and we defined ##\nabla_{\partial_i}\partial_j :=...
  43. M

    Covariant derivative of a (co)vector field

    I'm sorry, I can't follow you... I mean we then would have ##\nabla _X (dx^a (\partial_b))= \nabla_X \delta^a_b = X^i\nabla_{\partial_i}\delta^a_b##. But what now? Also, why exactly are we considering this? I mean, there is a difference between ##\nabla _X (dx^a (\partial_b)) ## and ##(\nabla _X...
  44. M

    Covariant derivative of a (co)vector field

    I think it's ##\nabla_X (dx^a(\partial_a)) = \nabla_X \delta^a_a = \nabla_X##, or isn't it?
  45. M

    Covariant derivative of a (co)vector field

    My attempt so far: $$\begin{align*} (\nabla_X Y)^i &= (\nabla_{X^l \partial_l}(Y^k\partial_k))^i=(X^l \nabla_{\partial_l}(Y^k\partial_k))^i\\ &\overset{2)}{=} (X^l (Y^k\nabla_{\partial_l}(\partial_k) + (\partial_l Y^k)\partial_k))^i = (X^lY^k\Gamma^n_{lk}\partial_n + X^lY^k{}_{,l}\partial_k)^i\\...
  46. M

    The sphere in general relativity

    Thank you very much! I really appreciate it. As you said, the notation so far just seems like a mess.. I completely missed that we have ##x(\lambda) \equiv (\psi^+_1)^{-1}(\lambda)##, which in hindsight should have been pretty clear (I mean, if I'm not misunderstanding this is the whole point of...
  47. M

    The sphere in general relativity

    I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have $$\begin{align*}(\psi^+_1)^{-1}:\begin{array}{rcl} \{\lambda^1,\lambda^2\in [a,b]\mid (\lambda^1)^2+(\lambda^2)^2<1\}&\longrightarrow& \{\pm x_1>0\}\subset \mathbb{S}^2\\...
  48. M

    Global Positioning System / Clocks in Space

    Thank you for the answer and the hints. Up until now we have only been doing special relativity, slowly starting to move towards general relativity by introducing acceleration in inertial frames (e.g. the exercise you helped me solve last weekend about orbits of particles with constant...
  49. M

    Global Positioning System / Clocks in Space

    I'm a bit lost at how to exactly start this exercise... As far as I understand we need to first determine ##d\tau_E## and ##d\tau_S##. First question: Since we can neglect the Earth's movement, can I also neglect the movement of the satellite with respect to the far away observer? If so, I...
  50. M

    Acceleration in special relativity

    Thanks for the hint, but this time I didn't work because I again didn't pay enough attention to the details... The differential equation, with the same boundary condition, is actually $$\frac{dv}{dt} = {\left(1-\frac{v^2}{c^2}\right)^{3/2}}a'\quad\Longrightarrow \quad...
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