"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...
I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states.
How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...
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...
3) Taylor expansion question in the context of Lie algebra elements:
Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g(
\alpha)...
1) Likely an Einstein summation confusion.
Consider Lorentz transformation's defined in the following matter:
Please see image [2] below.
I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...
Hello,
I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course.
I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...
Hello there,
Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way?
In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the...
Let us for a moment look a field transformations of the type
$$\phi(x)\longmapsto \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right)\phi(x),$$
where ##\omega## is anti-symmetric and ##S^{\mu\nu}## satisfy the commutation relations of the Lorentz group, namely
$$\left[S_{\mu \nu}, S_{\rho...
Given a representation of a Lie Group, is there a equivalence between possible electric charges and projections of the roots? For instance, in the standard model Q is a sum of hypercharge Y plus SU(2) charge T, but both Y and T are projectors in root space, and so a linear combination is. But I...
Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example:
$$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...
Homework Statement
For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else?
Homework Equations
The Attempt at a Solution
I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups.
Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e,
Lgh = gh where g,h ∈ G
Which corresponds to a translation by g.
Is this an example of a module over a ring...
I am trying to decompose some exponential operators in quantum optics. The interesting thing is that the operators includes operators from Su(1,1) algebra $$ [K_+,K_-]=-2K_z \quad,\quad [K_z,K_\pm]=\pm K_\pm.$$
For example this one: $$ (K_++K_-)^2.$$ But as you can see they are squares of it.
I...
Good Day
I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc.
But I am making a lot of progress.
There is, however, one issue that continues to elude me.
I often read how Lie developed Lie Groups to study symmetries of PDE's
May I ask if someone...
Homework Statement
How can you find all inequivalent (non-isomorphic) 2D Lie algebras just by an analysis of the commutator?
Homework Equations
$$[X,Y] = \alpha X + \beta Y$$
The Attempt at a Solution
I considered three cases: ##\alpha = \beta \neq 0, \alpha = 0## or ##\beta = 0, \alpha =...
Homework Statement
Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras...
Homework Statement
Given for one-dimensional Galilean symmetry the generators ##K, P,## and ##H##, with the following commutation relations: $$[K, H] = iP$$ $$[H,P] = 0$$ $$[P,K] = 0$$
Homework Equations
Show that the Lie algebra for the generators ##K, P,## and ##H## is isomorphic to the...
I would like to know where one may operate with tensor quantities in quantum field theory: Minkowski tensors, spinors, effective lagrangians (for example sigma models or models with four quark interaction), gamma matrices, Grassmann algebra, Lie algebra, fermion determinants and et cetera.
I...
Hello.
I will be attending a course on Group theory and the book that the professor suggests is Georgi's Lie Algebras in Particle Physics.
As I liked Zee's book on General Relativity, I thought that it would be a blast to also use his Group theory textbook for the course.
Problem is that I don't...
I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy)
I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix.
(I understand how I obtain this equation... that is not the issue.)
Now I am making the leap to learning about...
Hi!
I'm studying Lie Algebras and Lie Groups. I'm using Brian Hall's book, which focuses on matrix lie groups for a start, and I'm loving it. However, I'm really having a hard time connecting what he does with what physicists do (which I never really understood)... Here goes one of my questions...
Hello! I am sorry that this questions is not actually directly related to physics, but, can anyone recommend me a good book about abstract algebra (basically lie algebra, representation theory etc.) used in physics? I have tried for a long time to find something online but I haven't find a...
No question this time. Just a simple THANK YOU
For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups.
My math background was very deficient: I am a 55 year old retired (a good life) professor of...
I recently got confused about Lie group products.
Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...
Hello,
As you might discern from previous posts, I have been teaching myself:
Calculus on manifolds
Differential forms
Lie Algebra, Group
Push forward, pull back.
I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...
The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices.
I know that there exists Lie Groups, one of which is SO(3).
And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix.
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First, can someone edit...