Recent content by fzero

It's not exactly clear what you want. Do you mean a rank 2 tensor, which we could think of as a matrix? In that case, we can have two possible conditions: 1. ##\sum_j M_{ij} v_j =0##, or 2. ##\sum_j v_j M_{ji}=0##. You might also want both of these to be satisfied. These conditions are...

I looked in a few places that I thought it would be explained and it really wasn't. The idea of dimensional reduction is that we start with a field theory with spacetime coordinates ##X^M## and Lorentz group ##SO(D-1,1)##. The fields ##\Phi(X)## are in representations ##\mathbf{R}## of the...

The first few terms in the sequence for ##t## are {1., 1.55377, 1.73205, 1.78812}, so this certainly looks convergent. You might try to prove a bound on convergence for the expression where we replace 2 by ##n##.

You can apply Ramanujan's method to this. Set $$ t =\sqrt{1+\sqrt{2+\sqrt{2^2+\sqrt{2^3+\sqrt{2^4+ \cdots }}}}}.$$ Then $$ t^2 = 1 + \sqrt{2} t,$$ and we must take the positive root.

I get the same factor of 4. It turned out that I forgot some factors of 2 in the mapping I suggested. Let us define ##A_{ij} =\gamma \epsilon_{ijp} X_p## for some constant ##\gamma##. Then we can multiply this again by ##\epsilon## to show that ##X_p = (1/(2\gamma)) \epsilon_{pij} A_{ij}##...

As a short overview, I would suggest http://pdg.lbl.gov/2015/reviews/rpp2014-rev-guts.pdf from the PDG. For an exhaustive review of non-SUSY GUTs, look at Langacker, Grand Unified Theories and Proton Decay, Phys.Rept. 72 (1981) 185. For SUSY GUTs, Raby has a shorter review at...

For ##SO(3)## we can use the invariant ##\epsilon_{ijk}## to project a pair of indices onto a single index. So the expression that you should compute is ##\epsilon_{aks}{\epsilon_b}^{ij}{\epsilon_c}^{mn} f^{ks}_{ij,mn}##.

The Hilbert space is infinite dimensional when ##d>1##, so we can only make the direct geometrical connection between spinors and the exterior algebra in ##d=1##. However, it is still worthwhile to figure out the counting. In ##d=4##, as you say, the irreducible spinors are either Weyl (2...

The GUT scale would be determined by using the renormalization group to run the electromagnetic, weak, and strong coupling constants up to a high scale and looking for a scale ##M_\text{GUT}## at which all 3 become equal. Once this is done, you can determine an equivalent temperature as...

The choice where the phase of the Higgs field is set to zero is a particular gauge fixing, called unitarity gauge. It is possible to choose a different gauge, which can be useful in calculations, as the wiki article suggests. Setting unitarity gauge and then performing a gauge...

Yes, the generators are antisymmetric in both pairs of indices: $$(A_{ab})_{st} = - (A_{ba})_{st} = -(A_{ab})_{ts} .$$

The norm of a quantum state must be positive definite in order that the probability interpretation of quantum mechanics makes sense. For a nonsemisimple group, the Killing form is not definite, so we can't guarantee that there won't be any negative norm states. There are ways to use...

As I mentioned, I can't think of a way to connect these operators to the Hamiltonian that we might compute canonically by considering the sigma model Lagrangian or some other method. So I would be conservative and think of them as symmetries that we use to classify the states. Since they...

I had forgotten something that led to this confusion. I said that ##Z## was in general complex, but I believe that we can use an R-symmetry transformation to rotate the ##Q^A_\alpha## to make the central charge real. This is a unitary transformation, so it wouldn't make the...

We can put the metric ##ds^2 = |dz|^2 + |dw|^2## on ##\mathbb{C}^2##, so the sigma model for ##(z,w)## will be invariant under global phase changes of ##z## and ##w## independently. We can choose the generators of these symmetries to be ##J_1 = z \partial_z - \bar{z} \partial_{\bar{z}}## and...