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Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
May
31
comment Meaning of $\mathbb{R}^0$, $\mathbb{R}^{1/3}$, and $\mathbb{R}^{-2}$.
Sorry to digress from the main topic of discussion in this thread, but I have also seen notation like $\mathbb{R}^{a,b}$. What does this mean? For example, $\mathbb{R}^{2,0}$ represents the Euclidean two dim flat space.
May
28
comment Quick question about contravariant and covariant tensors
It makes sense - the notation is such to label a component of the tensor. There is no ambiguity in the Kronecker delta (since it is diagonal) so either notation is fine when dealing with that. Thanks Tom and Jonathan.
May
28
comment Quick question about contravariant and covariant tensors
Hi Jonathan. Thanks for your reply, but I understand that. My question is about the placement of the indices. In all cases, one of the indices is shifted to the left e.g $\omega_{v}^{\,\,\,\mu}$ vs $\omega_v^{\mu}$. I haven't seen the latter notation used anywhere, so I wanted to know if there are any differences.
May
28
asked Quick question about contravariant and covariant tensors
May
16
comment Understanding the irreducible representations of $D_3$
Yes, I think there must be some errors in that web page. For example, in D, the character of the 2D irrep is 0, not -1 as it should be as denoted in the table.
May
16
comment Understanding the irreducible representations of $D_3$
I meant $\rho$ to simply denote a representation. Jyrki, do you mean to say the other one dimensional representation is the alternating representation?
May
16
asked Understanding the irreducible representations of $D_3$
May
10
awarded  Popular Question
Mar
28
comment Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$
Or alternatively, going back to the original example, when I said that if $k <0$ we should take $R \rightarrow -\infty$, then that means $\sin t < 0$ so we too end up with a net positive in the exponent and the result then diverges?
Mar
28
comment Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$
I have seen some fourier transforms where $\exp(-ikz)$ is taken instead. In that case, if $z=R(\cos t, i \sin t)$, $|\exp(-ikz)| = |\exp(kR\sin t)|$ in which case if $k<0$, then $R$ should be taken to $+\infty$ since $sint > 0$ above the plane. If $k>0$ then how to make this term vanish? If $R$ is taken to $-\infty$ since $\sin t $ is then negative, we end up with a net positive. If $R$ is taken to positive infinity, then too, we end up with a positive. Thanks.
Mar
27
comment Invariants for the $SU(2)$ representation
With regard to your comment about dimension, the matrices for the $j=1/2$ rep are $2 \times 2$ so for the matrix product to make sense, the vectors would have to be $2$ dimensional, yes? So, what is the generalization of $\epsilon_{ijk}a_ib_jc_k$?
Mar
27
comment Invariants for the $SU(2)$ representation
Ok, so is it correct to say that the invariants are the same?
Mar
27
comment Invariants for the $SU(2)$ representation
I am not sure about this last statement because an explicit representation of the generators in $SU(2)$ are scaled Pauli matrices which are $2x2$. So the matrix product $U a_i$ say, would only make sense if $a_i$ was 2D. And, in that case, I am not sure how to generalise the $SO(3)$ invariant $\epsilon_{ijk}a_ib_jc_k.$ Thanks for your help!
Mar
27
comment Invariants for the $SU(2)$ representation
I did look at the notes, but it is not clear how the condition $U^{\dagger}U = 1$ is used. Seems to me it would work without this condition (i.e it would work with simply having $U^TU=1$) Also, the dimension of SU(2) is three, and given that the Kronecker and Levi-Civita tensors transform analogously under $U \in SU(2)$, are the invariants the same? Thanks for your help!
Mar
27
comment Invariants for the $SU(2)$ representation
In the line $...U^i_{\,k} U_j^{\,k} = \delta^i_{\, j}$, where are you using the fact that it is the condition $U^{\dagger}U = 1$ that holds rather than simply $U^TU=1$?
Mar
27
comment Invariants for the $SU(2)$ representation
Do you mean to say that the invariant quantities in the $j=1/2$ rep are the same? (i.e the Kronecker delta and the Levi-Civita tensor are both invariant under any transformation), so the corresponding invariants for the $SU(2)$ rep is exactly $\delta_{ij}a_ib_j$ and $\epsilon_{ijk}a_ib_jc_k$?
Mar
26
revised Invariants for the $SU(2)$ representation
deleted 19 characters in body
Mar
26
asked Invariants for the $SU(2)$ representation