Nice question, this issue also baffles me. I've taken QFT before, it seems to me that there is no particular book/resource that deals specifically with the "math" of QFT. It's apparent to me that the math used in QFT is quite diverse and there is no obvious structure that tells you, e.g. it's...
I see, your response clarified even further some concepts. I'll summarize again what I understood up until now. Please correct me if I'm wrong.
Elements of the Lie group are not the same as elements of the Lie algebra. So, the mechanism of representations for Lie groups and Lie algebras is not...
I'm sorry, I understand the calculation from the first step to the last step, but what I do not get is the main point of this and the big picture.
Let me summarize what I do understand.
Scalar, Vector (Fundamental), and Tensor Representation - given a Lie group ##SO(N)##, the scalar furnishes...
Some errata:
p. 200: (1) each of which is a ##N \times N## matrix. (2) ##T' \approx (I + \theta_a \mathcal{J}_a) T (I - \theta_a \mathcal{J}_a)## (3) ##\delta T = \theta_a [\mathcal{J}_a, T] = \theta_a [\mathcal{J}_a, A_b \mathcal{J}_b] = \theta_a A_b [\mathcal{J}_a, \mathcal{J}_b]## (4)...
I see, great response. So basically, we can construct the fundamental representation of the ##\mathfrak{so(4)}## algebra which is 4-dimensional, on the other hand, we can also construct an alternate representation called the adjoint representation which is 6-dimensional.
However, I still do not...
The way you wrote the fundamental rep and the adjoint rep, there will be no difference in their dimensions, i.e., both are 6-dimensional, but fundamental rep should be n-dimensional, i.e., n=4.
I believe it must be the latter, but physics books make it look/feel like it's the first one.
Can you elaborate ##\mathfrak{so}(n)\cong \mathfrak{ad}(\mathfrak{so}(n))##? I thought as in post #8, ##\mathfrak{so}(6)\cong \mathbb{R}^6##, i.e., the adjoint rep acts on the vector space...
I'm currently trying to learn Clifford algebra or more specifically spinors, in higher dimensions. My goal is to study AdS/CFT, but an essential part of learning it is to understand SUSY which then needs some element of Clifford algebra in higher dimensions.
I have consulted,
Introduction to...
I have some clarifications on the discussion of adjoint representation in Group Theory by A. Zee, specifically section IV.1 (beware of some minor typos like negative signs).
An antisymmetric tensor ##T^{ij}## with indices ##i,j = 1, \ldots,N## in the fundamental representation is...
So you are saying that the paper is wrong for saying that it is 2-dimensional, right?
What you are saying can be seen in the Kruskal diagram (courtesy of Wikipedia: Kruskal-Szekeres coordinates), right?
The dashed line represents the horizon ##r=r_H##, and that is a null line which of course...
I understand, however, that is why I wanted to clarify the 2-dimensional surface that the paper is pertaining to which states that the surface of the black hole is the same for all observers. What is this surface that is being talked about?
Exactly where part of my confusion comes in. From the statement in the abstract, it means that the 2-dimensional surface orthogonal to the light ray IS the black hole surface. But shouldn't the black hole surface be the same as the event horizon surface (if you could call it that). I'm confused...
I think I might be confusing the 2-dimensional surface which should be the black hole area that the paper is talking about (?) and the event horizon represented by ##r_H## for which the area is proportional to ##r_H^2##.
Ah, I'm sorry, for the null normal vector of the event horizon, indeed one of the orthogonal vectors to it is the null tangent vector but there are two additional orthogonal vectors for which one of them is spacelike.
So what does the orthogonality of the light ray to the spatial surface have...