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...