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 [Broken] 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...
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...
The norm of the normal vector is ##|\mathbf{N}| = \sqrt{ N_x^2 + N_y^2 }##. The unit normal is ##\mathbf{n} = \mathbf{N}/ |\mathbf{N}|##, so
$$ n_x = \frac{ N_x}{\sqrt{N_x^2 + N_y^2}}.$$
In particular, there are common factors of ##r+\rho## in the numerator and denominator that cancel out...
##Z## is generally complex, so the correct BPS bound is ##M\geq \sqrt{2} |Z|##. For the ##N=2## pure vector multiplet, if you explicitly construct the supercurrents, you can work out an expression
$$ Z = a ( n_e + \tau n_m),$$
where ##a## is the VEV of the complex scalar in the vector...
If you look up the explicit form of the spherical harmonics, you'll find that ##Y_{\ell,\pm m}## differ by a relative phase that is ##\pm e^{\pm 2 i \phi}##. So their complex modulus is the same. See also the normalization conventions at...
It is not really the case that we only consider powers of 2, it is that spinors in various dimensions have properties such that, when all requirements are considered, it might be that there are additional SUSYs present. For instance, there is the possibility of rigid ##N=3## in 4d, but once...
Yes, I tried to limit the hard numbers to academia since some statistics are available. I don't have hard numbers for government or industry. I would guess that they are not that large for government positions, probably on the order of <10% of the number of university positions and the...
The chance that a new PhD will go on to a tenured faculty position in their lifetime is very small. Obviously the chances are affected by the ratio of job seekers to open faculty positions in a subfield and by the quality of the applicant, but on average, the chance is less than 10% in the US...
From these equations, you have
$$f_{ij,mn}^{ks}A_{ks}=-(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j}) .$$
You can expand the RHS using identities like ##A_{jm} = \delta_{jk}\delta_{ms} A_{ks}## to derive the form of the structure constants.
Yes, except that I would rephrase
It's not quite that it would not work, it's that the classical theory of a vector field does not require gauge symmetry, But, of course, we need to include gauge invariance if we want the theory to describe classical electrodynamics.
Physical examples of...
The paper Hans-Jürgen Schmidt, Why do all the curvature invariants of a gravitational wave vanish?, http://arxiv.org/abs/gr-qc/9404037 gives an argument based on form-invariance of the metric under a boost along a null direction and the fact that the curvature invariants are continuous functions...
Let's go through the Lorentz-invariant terms that we can have from scratch. At linear order we have ##j^\mu A_\mu##, which is how we include the current. The gauge variation of this term is ##j^\mu \partial_\mu f= \partial_\mu (j^\mu f) - f \partial_\mu j^\mu##. The first term is a total...
From the classical perspective, we impose gauge invariance by hand, typically because we already know that it is a symmetry of the Maxwell equations. If we were doing quantum field theory, we would see that we really need gauge invariance, otherwise the quantum theory doesn't make sense...
I don't know if you have seen the Lagrangian description of the electromagnetic field itself. The Lagrangian (really Lagrangian density, but I won't make this distinction below) is
$$ L_\text{Max.} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} = \frac{1}{2}( \mathbf{E}^2 - \mathbf{B}^2 ),$$
where in...
I don't think that the Lagrangian for a charged particle is completely determined by invariance under gauge and Lorentz transformations. I think there is some discussion in Landau and Lifschitz about an additional term that would be consistent with the symmetries but appears to be absent...
I don't have access to the paper right now, but I can discuss generalities. First of all, ##K^0+X \rightarrow K^\pm +Y## implies that we are exchanging a ##d## or ##\bar{d}## for a ##u## or ##\bar{u}## quark, respectively. For the ##s\bar{d}## component, this is kind of easy, since we have...
Sure, I didn't seriously consider suggesting the contour integral because it is a bit rare to find someone comfortable with the method. I probably should have asked first. It's good that you were able to work it out yourself that way.
Since presumably ##\partial\cdot \epsilon = \partial_0 \epsilon_0+\partial_1 \epsilon_1## in your conventions, you should have found ##2\partial_0 \epsilon_0 = \partial_0 \epsilon_0+\partial_1 \epsilon_1## for the 00 component.
This actually turns out to be very complicated to do and I am having trouble giving hints that you can follow without giving too much of the answer away, so please bear with me. At least using Mathematica seems like a legitimate solution to the problem and I don't believe that many people would...
I am fairly certain that mfb knows more particle physics than I do, so you shouldn't jump to such conclusions, which seem disrespectful in any case.
The Higgs boson looks like a tachyon if you expand the field around the false vacuum where ##\langle H\rangle = 0##, because of the wrong sign of...
I'm sorry, I misread your 4-vector in post #3 as a badly formatted matrix. Taking a pair of 2-vectors to build a 4-vector is precisely the tensor product rather than the dyadic product (which would have given the 2x2 matrix instead). So your method seems correct (I haven't checked the...
There are a few motivations for considering SUSY extensions of the Standard Model. One is the hierarchy problem that asks why the Higgs mass could be small compared to the scale at which new physics occurs, be that a grand unification, or gravity, or something else. Another motivation is that...
More precisely, the vacuum expectation values of the fields are the maps. A nonzero VEV takes the coordinate ##x^\mu## as input and returns the coordinate ##\phi^i## as output.
A linear function of a vector ##\mathbf{f(v)}## satisfies the conditions that ##\mathbf{f(v+w)}=...
Let's talk about this geometrically. The fields ##\phi^i(x^\mu)## are maps from ##B\rightarrow \mathcal{M}##. Then the objects ##\partial_\mu\phi^i## are the components of what is called the pushforward map. We don't need to dwell too much on the significance apart from the fact that this is...
You are using the summation convention, so
$$\delta_{jj} \equiv \sum_{j=1}^3 \delta_{jj}.$$
This gives a different numerical factor in front of the first ##\delta_{km}## term.
If you just had the operator ##\mathcal{O}_A = (\sigma^2)_A## and a state ##|0\rangle_A##, I'm sure that you know how to compute ##\mathcal{O}_A |0\rangle_A##. You could do the computation separately for each side of the tensor product in the formula for ##\mathcal{O}_{AB}## and then take the...
Well there is probably a way to make sense of what you've done, but you have a 4x4 matrix that won't directly act in a natural way on either the 2d vectors or that 2x2 matrix you wrote above. I would suggest working things out using the formalism in post #4. Afterwards, you can try to figure...
What you've written is sometimes called the dyadic product of vectors. While it is true that the dyadic product is directly related to the tensor product of vectors, I don't believe that you want to use that here.
The properties of the tensor product of Hilbert spaces that you will want to...
I think OP is referring to the delirium associated with certain patients presenting with urinary retention, that was referred to as cystocerebral syndrome. AFAICT, this phenomenon is not completely understood, but it has been suggested that the central nervous response to the stress involves...
No, unfortunately we are using ##\delta## for two different things. The Kronecker delta ##\delta^a_b## is not directly related to the variations ##\delta h## or ##\delta h_{ab}##. What I meant is that you had an expression that we can write
$$\frac{\delta h}{\delta h_{\alpha_1\beta_1}}...
You have to compute the variation with respect to something, so what you're computing here is
$$\delta h=\frac{1}{(n-1)!}\epsilon^{\alpha_1...\alpha_n}\epsilon^{\beta_1...\beta_n} \delta h_{\alpha_1\beta_1} h_{\alpha_2\beta_2}...h_{\alpha_n\beta_n}.$$
There's some additional argument needed to...