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...
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##
$$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...
This seems rather straight forward, but I can't figure out the details... Generally speaking and ignoring prefactors, the Fourier transformation of a (nicely behaved) function ##f## is given by
$$f(x)= \int_{\mathbb{R}^{d+1}} d^{d+1}p\, \hat{f}(p) e^{ip\cdot x} \quad\Longleftrightarrow \quad...
@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...
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?
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:
$$\frac{i Z...
Thank you very much for the response!
I hope you mean the ##\log## that will eventually show up in ##I(p^2)##, if not, I'm not really sure what you mean. I just went back to my QFT1 lecture notes (Chp. 11.2) one more time to check, and my Prof. got for this integral two different expression...
Before I start, let me say that I have looked into textbooks and I know this is a standard problem, but I just can't get the result right...
My attempt goes as follows:
We notice that the amplitude of this diagram is given by $$\begin{align*}K_2(p) &= \frac{i(-i...
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...
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
$$Q^t :=...
Thanks for spotting the typo. I'm rather new to this entire GR-formalism, i.e. the covariant derivatives, etc., so I was just a bit unsure if I'm really doing operations that are permitted. Also, ##C=1## seemed a bit odd in the first moment, but if you think this works, then I'm happy!
@PeroK Could you elaborate? I'm asking because I don't see how what you write adds up with the very next line in my professors reasoning, i.e.
$$\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...
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
$$
\begin{align*}
0&=(\gamma^\mu...
Isn't this just a matter of definition? My lecturer demands that we use ##v_\alpha## for electrons (created and annihilated by ##a^\dagger## and ##a##) and ##u_\alpha## for positrons (created and annihilated by ##b^\dagger## and ##b##), which unfortunately makes looking up stuff sometimes really...
Consider Moller scattering, that is $$e^-(\vec p_1, \alpha)+e^-(\vec p_2, \beta) \quad\longrightarrow\quad e^-(\vec q_1, \gamma)+e^-(\vec q_2, \delta),$$
where the ##\vec{p}_i,\vec q_i## label the momenta of the in and outgoing electrons and the greek letter the spin state.
The two relevant...
The second sentence is exactly what confuses me! When you say "we need [...] to contract with the creation annihilation operators outside of the time ordering sign", what exactly do you mean with the "contract"? Up to now I thought that contractions can only arise in the context of Wick's...
@HomogenousCow
Thank you for the answer.
Maybe I'm misunderstanding you, but the exercise was supposed to be solved in the way I presented above, so I cannot just change that (I technically could, but I would like to understand what's going on in the provided solution).
It's possible that...
I have trouble understanding the solution to a homework problem.
Consider the interaction Lagragian ##\mathcal{L}_{\rm int} = -iqA_\mu \bar{\psi}\gamma^\mu \psi##, i.e. photon-electron/positron interaction. We want to focus on the Compton scattering
$$e^-(\vec p_1, \alpha) + \gamma(\vec p_2...
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...
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...
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!
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...
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...
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##:
$$
\begin{align*}
\frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\
\frac{\partial...
In the following I will try to deduce the scattering amplitude for a specific interaction. My question is at the bottom, the entire rest is my reasoning to explain how I came to the results I present.
My working
Let's assume I would like to calculate the second order scattering amplitude in ##...
You are absolutely right, there should be a ##1/2## in front of the first term... I completely overlooked this. With this in mind we have
$$
\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...
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:
$$\begin{align*}
\delta S_M
&= \int d^4x (\delta\sqrt{-g})...
Let me begin by stating that I'm aware of the fact that this is a metric of de Sitter spacetime, aka I know the solution, my problem is getting there. My idea/approach so far: in the coordinates ##(u,v)## the metric is given by
$$g_{\mu\nu}= \begin{pmatrix}1 & 0\\ 0 & -u^2\end{pmatrix}.$$
The...
While writing out the Dyson series due to the time ordering above I encountered the two expressions
$$T(\mathcal{L}_{int}(x))\quad \text{and}\quad T(\mathcal{L}_{int}(x)\mathcal{L}_{int}(y))$$
I was able to write out the first term in terms of contractions using Wick's theorem and then finally...
I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***):
$$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...
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:
$$\partial_t \mathcal{T}\exp\left(-i...
Thank you very much for the detailed answer! I tried using your approach on a problem which I already solved, to see if I really understood it.. Unfortunately I'm still struggling.
Lets say I have a vector field ##A_\mu## and I consider only a translation, aka something of the form...
First of all, thank you for the reply. I'd like to split the following question into two parts:
Assuming that ##\mathcal{L}## is scale-invariant (aka ##m## and ##\mu## are zero and the values of ##\lambda## and ##\Delta## chosen accordingly). In this case we have ##\delta\mathcal{L}=0##, which...
After some trying out I was able to obtain a partial result I think.
First derive the EoM for the given Lagrangian, which results in
$$\partial^2 A^\rho -\partial^\rho\partial^\lambda A_\lambda + \xi^{-1}\partial^\rho\partial^\lambda A_\lambda=0.$$
Now we can apply the following trick to get...
This is my first time dealing with scaling symmetry, so I'm sorry if the following is fundamental wrong. My approach was the same as if I was trying to show the same for translation or Lorentz symmetry.
We have
$$\delta\phi(x)= \phi'(x')-\phi(x)=...
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...
Alright, so if I don't misunderstand we have
$$\begin{align*}
(\nabla_{\partial_l}dx^k)(\partial_i) &= \nabla_{\partial_l}(dx^k(\partial_i)) - dx^k(\nabla_{\partial_l}\partial_i)\\
&= \nabla_{\partial_l}\delta^k_i - dx^k(\Gamma^u_{li}\partial_u)\\
&= \nabla_{\partial_l}\delta^k_i - \Gamma^k_{li}...
I don't know. Usually I'd say the "derivative" of a constant is zero, but I'm really not sure if this is true when talking about the covariant derivative... The only things I know about this derivative are written above, and we defined ##\nabla_{\partial_i}\partial_j :=...
I'm sorry, I can't follow you... I mean we then would have ##\nabla _X (dx^a (\partial_b))= \nabla_X \delta^a_b = X^i\nabla_{\partial_i}\delta^a_b##. But what now? Also, why exactly are we considering this? I mean, there is a difference between ##\nabla _X (dx^a (\partial_b)) ## and ##(\nabla _X...
Thank you very much! I really appreciate it. As you said, the notation so far just seems like a mess.. I completely missed that we have ##x(\lambda) \equiv (\psi^+_1)^{-1}(\lambda)##, which in hindsight should have been pretty clear (I mean, if I'm not misunderstanding this is the whole point of...
I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have
$$\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\\...
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...
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...
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...