Spinor notation exercise with grassman numbers
I'm checking a term when squaring a vector superfield in Wess-Zumino gauge, but its really just an excercise in index/spinor notation:
I need to square the term...
From Peskin and Schroeder:
The finite-dimentional representations of the rotation group correspond precisely to the allowed values for the angular momentum: integers or half integers.
From the Lorentz commutation relations:
\left[J^{\mu \nu},J^{\rho \sigma}\right]=i \left(g^{\nu \rho}J^{\mu...
Paraphrasing Peskin and Schroeder:
By repeated use of
\left\{ \gamma^{\mu} , \gamma^{\nu} \right\}= 2 g^{\mu\nu} \times \textbf{1}_{n \times n} (Clifford/Dirac algebra),
verify that the n-dimensional representation of the Lorentz algebra,
S^{\mu...
I'm having some trouble with a concept in group theory. I'm reading Howard Georgi's book on Lie Algebra, this is from the 1st chapter. Really sorry to have to use a picture but I don't know how to TeX a table:
There's a couple things I don't quite understand but mainly, I don't see how he...
Having some trouble following my notes in QFT. Any help greatly appreciated.
We have the Klein Gordon equation for a real scalar field \phi\left(\overline{x},t\right); \partial_{\mu}\partial^{\mu}\phi + m^{2}\phi = 0.
To exhibit the coordinates in which the degrees of freedom decouple...
Not a particularly direct question, just something I don't mathematically understand and would very much appreciate help with.
For some scalar field \phi, what would \partial_{\mu} \phi^{*}\partial^{\mu} \phi mean in mathematical terms. ie how would I calculate it?
From what I understand...
Homework Statement
Part (e) of the attached question. Sorry for using a picture, and thanks to anyone who can help.
Homework Equations
the answer to part (d) is that the eigenvalue is
\hbar^{2}\left(l\left(l+1\right) + s\left(s+1\right)+2m_{l}m_{s}\right)
where, for this part of the...
A side note on the choice of substitution:
If we had a minus infront of the l^{-2}, so;
\int \frac{da}{\sqrt{\frac{H_{0}^{2}\Omega_{0}}{a}-l^{-2}}}=dx^{0}-dx^{0}_{*}
then I would be suggested the substitution
\frac{a}{l^{2}H^{2}_{0}\Omega_{0}}=\sinh^{2}(\frac{u}{2})
and this would give the...
AH! so sorry, I've misstyped the equation it should read
\int \frac{da}{\sqrt{\frac{H_{0}^{2}\Omega_{0}}{a}+l^{-2}}}=dx^{0}-dx^{0}_{*}
which is where my algabra comes from
So sorry, i will alter it in the OP now..
thank you very much for you're quick reply.
I've checked the...
Homework Statement
Hi, this situtation arises in my cosmology lectures, but its purely mathematical:
I need to evaluate the LHS of \int \frac{da}{\sqrt{\frac{H_{0}^{2}\Omega_{0}}{a}+l^{-2}}}=dx^{0}-dx^{0}_{*}
using the substitution \frac{a}{l^{2}H^{2}_{0}\Omega_{0}}=\sin^{2}(\frac{u}{2})...
Homework Statement
(firstly, Apologies for having to use a picture..)
If u^{i} is the 4-velocity of a point on a manifold, then we use affine parameterisation g_{ij}u^{i}u^{j}=1.
The attached picture shows our rest frame, ie x^{0}=const and a point ("us") on this surface. If our velocity is...
Homework Statement
If I have a two curves \gamma_{1}, \gamma_{2} with the same start and end points, lying on a smooth manifold M. For a vector v at the "start" point, if I parallelly transport down both curves to the "end" point, will the two vectors at the "end" be different or the same...
Hi, thanks so much for your help - very much appreciated.
I get the Debye wavevector from N=\int^{k_{D}}_{0}g(k)dk=\int^{k_{D}}_{0}\frac{L}{\pi}dk=\frac{k_{D}L}{\pi}
ie k_{D}=\frac{N\pi}{L}
putting this into the dispersion relation gives...