I've defined A and B to be two affine transformations on \mathbb{R}^2. Then, I defined C and D to be some kinds of compositions of A and B, for example:
C = Composition[A,B,B,A,A][{x,y}]
D = Composition[B,A,B,A,B][{x,y}]
Now, I want to evaluate expressions like:
X =...
Homework Statement
Evaluate
\lim_{z \to 0}:e^{ik \cdot X(z)}:|0\rangle
where X(z) is a free chiral scalar field in the complex plane.
Homework Equations
In Conformal Field Theory, the free chiral scalar field in the complex plane is given by:
\begin{array}{rcl} X(z) &=& \frac{1}{2}q -...
Thanks... so for the example I gave, if \theta_{(x,y)}(a,b) = (xa-yb,xb+ya), then
(\theta_{(x,y)})_* = \left[\begin{matrix} x & -y \\ y & x \end{matrix}\right]
so
X_{(x,y)} = \left[\begin{matrix} x & -y \\ y & x \end{matrix}\right]\left[\begin{matrix} 1 \\ 0\end{matrix}\right] =...
I need help calculating the exponential map of a general vector.
Definition of the exponential map
For a Lie group G with Lie algebra \mathfrak{g}, and a vector X \in \mathfrak{g} \equiv T_eG, let \hat{X} be the corresponding left-invariant vector field. Then let \gamma_X(t) be the maximal...
What are the general features you should look for when classifying an arbitrary particle interaction according to strong, weak, or electromagnetic forces? Cheers
A black hole is just a collapsed star, so the gravitational field of a black hole is just like any star really. If we were a safe enough distance away from a black hole, Earth could orbit a black hole as if it were just another star. So to say that a black hole would consume all the matter and...
Thanks, yeah I also thought that having a maximal atlas wouldn't be that important as long as you had compatible charts. But anyway every text I read has it as a condition, which I thought was interesting:
"A (smooth) differential manifold M^m of dimension m is a topological manifold of...
I've been looking in various books in differential geometry, and usually when they show that a smooth manifold has a differentiable structure, they just show that the atlas is C^\infty compatible, and forget about showing it is maximal.
Which got me thinking. Given an atlas, how DOES one show...