I'm going through the "Advanced Lectures on General Relativity" by G. Compère and got stuck with solving one set of conditions on the subject of asymptotic flatness. Let ##(M,g)## be ##4##-dimensional spacetime and ##(u,r,x^A)## be a chart such that the coordinate expression of ##g## is in Bondi...
There is an important point about Eq. (4) in the OP that I forgot to mention when starting the thread, which is the definition of the ##f(\bar{\theta},\theta)## map appearing in Eq. (5) when defining the matrix ##h^a_{\phantom{a}b}(\theta)##. It is the coordinate representation of the group...
This seems to be what Robert Ticciati does on the book "QFT for Mathematicians". If I understand, given the Lie algebra representation ##D : \mathfrak{g}\to \mathfrak{gl}({\cal H})## he tentatively defines ##U:G\to GL({\cal H})## to be $$U(\exp tX) = \exp tD(X),\tag{1}$$ on which the ##\exp## on...
In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...
This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...
Let us consider Ashtekar's definition of asymptotic flatness at null infinity:
I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition.
In fact, a distinct approach to asymptotic flatness already starts with...
I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented.
Now, I'm having a hard time to grasp the idea/motivation behind the...
I did that. The one I have already spoken to seems to be the only available advisor in the moment which has interest in QFT/gravity in the department. He asked me to come up with something concrete, however, so he asked for suggestion of some objective.
One remark is that on that department...
I have a major in Mathematics and Mathematical Physics and I'm finishing a masters in Physics (just finishing to write down the dissertation really). I have also already enrolled the PhD course so that I need now to pick an advisor and a theme before june.
My main interest since the early days...
That's exactly my problem. I don't know how to define the S matrix. It may really be the problem that this doesn't fit scattering theory at all, although I have one impression that it can be seen as scattering. In Hawking's papers for instance, one considers ##\mathcal{I}^-## to be one "initial...
Yes, the "Particle Creation by Black Holes" as well as some other references like Parker's QFT in Curved Spacetimes book. But all of them seem to discuss a different matter: how one observer at ##\mathcal{I}^+## perceives the natural vacuum for an observer at ##\mathcal{I}^-##. This is answered...
Here we consider a black hole formed by gravitational collapse classically. We also consider a scalar massless Klein-Gordon field propagating on this background.
To quantize the field we expand it in appropriate modes. The three sets of modes required are:
The incoming modes, appropriate for...
So in the end, ##\psi(t,\mathbf{x})## can be seen as the free evolution - with the Klein-Gordon equation - of a single particle in the initial state ##\psi(0,\mathbf{x})## as one would do with relativistic quantum mechanics without fields?
As you say this has a few interpretational problems...
Let us consider QFT in Minkowski spacetime. Let ##\phi## be a Klein-Gordon field with mass ##m##. One way to construct the Hilbert space of this theory is to consider ##L^2(\Omega_m^+,d^3\mathbf{p}/p^0)## where ##\Omega_m^+## is the positive mass shell. This comes from the requirement that there...
I have taken one first QFT course last year which used Matthew Schwartz "Quantum Field Theory and the Standard Model" book. The course went all the way to renormalization of QED, although path integrals weren't discussed.
Now I want to continue learning QFT and also I want to make a second...
@martinbn I thought the same when I've read section 2 of Sachs' paper the first time. But notice that Strominger points out that any geometry can be locally written in these coordinates with that metric tensor. I actually have the impression that it is true. My problem is that if any geometry...
I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates.
In the paper Lectures on the Infrared Structure of Gravity and Gauge Theories Andrew Strominger points out in section 5.1 the following:
In the previous sections, flat...
Although the question came to my mind while studying Weinberg's QFT books, the doubt is much more general than that, and is not a doubt about physics, but rather about how to actually study and learn the topic alone from the book.
From one point I agree that coming up with this doubt nearly...
Thanks @rubi. I've seem this theorem explained on nLab as follows:
Fell’s theorem is about a property of vector states of a C-star algebra, it says that if the kernels of two representations of the algebra coincide, then the vector states are mutually weak-* dense. This has a profound...
In the algebraic approach, a quantum system has associated to it one ##\ast##-algebra ##\mathscr{A}## generated by its observables and a state is a positive and normalized linear functional ##\omega : \mathscr{A}\to \mathbb{C}##.
Given the state ##\omega## we can consider the GNS construction...
So let's see if I got this straight: the ##\ast##-algebra is generated by the observables of the theory, hence we can define the necessary Hamiltonian in terms of them. In one usual one-particle non-relativistic QM problem, we would necessarily have ##X,P\in \mathscr{A}## such that...
Thanks for the response A. Neumaier ! What still somehow bothers me is that on the algebraic setting we pick one distinguished state to generate the whole representation, so that this state becomes distinguished somehow.
I can see one parallel with usual perturbative QFT, because there the...
Studying QFT on curved spacetimes I've found the algebraic approach, based on ##\ast##-algebras. In that setting, a quantum system has one associated ##\ast##-algebra ##\mathscr{A}## generated by its observables.
Here we have the algebraic states. These are defined as linear functionals...
In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##)
To discuss general properties of tensor symmetries, we shall use the representation theory of the...
Yes I'll do a Phd after getting the master's degree. I don't know if what I'm doing now constrains my Phd, but certainly it doesn't help. What I think is that in the master's I could learn things that would allow me a better work on the Phd, in the sense that I would start it with some of the...
You mean talk to another advisor I know and trust to see what they think about it?
This is really a point. The only issue is that I feel that on my master's and my Phd I should be learning what I require for the research area I want to go. I believe the learning part is what bothers me the...
I have a major in mathematical physics and mathematics and I've started on graduate school january of the last year to get a master's degree in theoretical physics.
My real interest is fundamental physics, specialy related to general relativity and quantum field theory. I've talked to a...
I have a major in mathematical physics and mathematics and currently I'm on a graduate course in Physics working on a master's thesis. When I started the graduate course I was going to work on General Relativity and Quantum Field Theory on Curved Spacetimes (QFTCS). It turns out that by several...
I believe I got your point. The Lorentz transformation's actually take place on the tangent spaces of spacetime relating tangent vectors, they are not transformations on the spacetime manifold itself. So what I did is fine for the four-potential because it is actually a tangent vector, but for...
But then I can't see why the ##\mathbf{R}## appears on the result. Notice that by my derivation we have (upon writing explicitly ##\mathbf{x}'=\mathbf{x}-\mathbf{v}t## that comes from the Lorentz transformation)...
I believe I got it, expanding \tanh^{-1}\beta \approx \beta I have \Lambda(-\vec{\beta})=e^{\vec{\beta}\cdot \mathbf{K}}. Then expanding the exponential I get \Lambda(-\vec{\beta})\approx 1 + \vec{\beta}\cdot \mathbf{K} which is first order in \beta.
With this I can derive that A^{\mu}(x) =...
Homework Statement
An electric dipole instantaneously at rest at the origin in the frame K' has potentials \Phi'=\mathbf{p}\cdot\mathbf{r}'/r'^3 and \mathbf{A}'=0 (and thus only an electric field). The frame K' moves with uniform velocity \mathbf{v}=\vec{\beta }c in the frame K.
Show that in...
I'm in a graduate course in Physics to obtain a master's degree. I have a major in mathematical physics and my main interests are General Relativity (GR), Quantum Field Theory on Curved Spacetimes (QFTCS), and usual Quantum Field Theory (QFT) itself.
My interest is in the fundamental physics...
Homework Statement
A uniformly magnetized and conducting sphere of radius R and total magnetic moment m = 4\pi MR^3/3 rotates about its magnetization axis with angular speed \omega. In the steady state no current flows in the conductor. The motion is nonrelativistic; the sphere has no excess...
I'm reading the book Quantum Field Theory and the Standard Model by Matthew Schwartz and currently I'm studying the chapter 17 titled "The anomalous magnetic moment" which is devoted to computing the corrections due to QFT to the g factor.
My main issue is in the beginning of the chapter, where...
Homework Statement
Show that the vacuum polarization \Pi^{\mu\nu}_2(p) in 1-loop is transverse. Decide whether you want to use Ward's identity and prove this to be true in all orders or only prove for 1-loop.
Homework Equations
Ward's identity q_\mu \mathcal{M}^{\mu}=0 which must hold where...
Homework Statement
Let \Gamma^\mu be the three-point vertex in scalar QED and \Gamma^{\mu\nu} be the four-point vertex. Use Feynman's rule at tree level and verify that the Ward-Takahashi identities are satisfied:
q^\mu \Gamma_\mu(p_1,p_2)=e[D_F^{-1}(p_1)-D_F^{-1}(p_2)],\\...
Not only that. As I said, my main interest is both GR and QFT. It is not that someone has told that things must be quantum (nor do I think like that by the way), is just what I'm interested in.
I still think that if one has decided to produce such a thesis, it must be something they actually...
Last year I've finished the undergraduate course in Mathematical-Physics and Mathematics and this year I've started on graduate school on Physics in order to obtain a master's degree. What I'm really interested are two main topics: general relativity and quantum field theory. I also like...
I'm studying Quantum Field Theory and the main books I'm reading (Peskin and Schwartz) present Feynman diagrams something like this: one first derive how to express with perturbation theory the n-point correlation functions, and then represent each term by a diagram. It is then derived the...
Homework Statement
Consider two real scalar fields \phi,\psi with masses m and \mu respectively interacting via the Hamiltonian \mathcal{H}_{\mathrm{int}}(x)=\dfrac{\lambda}{4}\phi^2(x)\psi^2(x).
Using the definition of the S-matrix and Wick's contraction find the O(\lambda) contribution to...
Actually after thinking a little bit more about it, I believe I started to get the idea. The initial and final states are both two-particle states of definite momentum. My guess is that the author is saying that spin can be introduced here as another degree of freedom of these two particle...
I'm reading the book "Quantum Field Theory and the Standard Model" by Matthew Schwartz and I'm finding it quite hard to understand one derivation he does. It is actually short - two pages - so I find it instructive to post the pages here:
The point is that the author is doing this derivation...
Not much examples really. After the definition it immediately gets to classify reference frames according to synchronizability. Actually in one exercise he defines a reference frame and asks to (i) show its a reference frame and (ii) show it is not synchronizable according to his definition...
Homework Statement
Consider the process of decay of a muon into one electron, one electron antineutrino and one muon neutrino using the Fermi theory. Assume the matrix element is, ignoring the electron's and the two neturino's masses,
|\mathcal{M}|^2 = 32G_F^2(m^2-2mE)mE
being E the electron...
Actually some time after I've posted the thread I found out a simple way to do it. It was just a matter of rewriting the fields in a more convenient way:
\phi(x)=\int \dfrac{d^3k}{(2\pi)^3} \dfrac{1}{2\omega_k}(a_k+a_{-k}^\dagger)e^{i\mathbf{k}\cdot \mathbf{x}}
where now the...
In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field Z. In that sense a reference frame is a collection of observers, since its integral lines...
Homework Statement
Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators.
Homework Equations
Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
Homework Statement
Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...