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...
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 :=...
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
$$
\begin{align*}
0&=(\gamma^\mu...
Consider Moller scattering, that is $$e^-(\vec p_1, \alpha)+e^-(\vec p_2, \beta) \quad\longrightarrow\quad e^-(\vec q_1, \gamma)+e^-(\vec q_2, \delta),$$
where the ##\vec{p}_i,\vec q_i## label the momenta of the in and outgoing electrons and the greek letter the spin state.
The two relevant...
I have trouble understanding the solution to a homework problem.
Consider the interaction Lagragian ##\mathcal{L}_{\rm int} = -iqA_\mu \bar{\psi}\gamma^\mu \psi##, i.e. photon-electron/positron interaction. We want to focus on the Compton scattering
$$e^-(\vec p_1, \alpha) + \gamma(\vec p_2...
I'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...
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##:
$$
\begin{align*}
\frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\
\frac{\partial...
In the following I will try to deduce the scattering amplitude for a specific interaction. My question is at the bottom, the entire rest is my reasoning to explain how I came to the results I present.
My working
Let's assume I would like to calculate the second order scattering amplitude in ##...
My attempt was to first rewrite ##S_M## slightly to make it more clear where ##g_{\mu\nu}## appears
$$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...
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
$$\delta\phi(x)= \phi'(x')-\phi(x)=...
My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...
I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have
$$\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\\...
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 earths movement, can I also neglect the movement of the satellite with respect to the far away observer? If so, I...
I'm struggling in the details of this exercise. Let ##S'## be the reference frame where the acceleration of the spaceship is constant, in which case we have ##u'(t')= a' t'## (since we assume no acceleration at the beginning). The rest frame of the rocket ##S## is connected to ##S'## via a...
I think the solution to this problem is a straightforward calculation and I think I was able to make reasonable progress, but I'm not sure how to finish this...
$$\begin{align*} \vec{P}&=-\int dx^3 \pi \nabla \phi\\
&= -\int\int\int dx^3\frac{dp^3}{(2\pi)^3 2e(p)} \frac{du^3}{(2\pi)^3}...
As far as I know we can express the position and momentum operators in terms of ladder operators in the following way
$${\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}.$$...
First of, I have no idea what I'm supposed to do with the neutrinos and the photons. Can somebody explain how to handle these? The rest of what I tried is quite straight forward
$$\begin{align*}\Delta E &= 4M_p - M_{He} - 2 M_e + E_{\text{Neutrino and Photons}}\\&= 4M_p - (2[M_p+M_n]-E_B) - 2...
First of all, disclaimer: This isn't an official assignment or anything, so I'm not even sure if there is a resonably simple solution.
Consider the following sketch.
(Forgive me if it isn't completely clear, I didn't want to fiddle around for too long with tikz...)
Let us assume that we can...
some notes:
There was actually no proof given why ##u^s(p)## or ##v^s(p)## should solve the Dirac equation, only a statement that one could prove it using the identity
$$(\sigma\cdot p)(\bar\sigma\cdot p)=p^2=m^2.$$
We were using the Wely-representation of the ##\gamma##-matrices, if this should...