- #1
rubbergnome
- 14
- 0
Hello everyone, I'm an high school student, but I try to study maths and physics at a decent level anyways. I have some questions to pose; to give you an idea of how could you guys answer me, i recently finished viewing David Tong's QED video lectures and notes, and I'm stuck at the CFT sections of his string theory notes. These are the questions:
1) Superpoincarè algebra:
I read in various references that the SUSY extension of the poincarè algebra is the simplest SUSY algebra one can get. I get why the generators have to be spinors, but nobody ever gives a derivation of the defining anticommutator
\left\{ Q_\alpha , \bar{Q}_{\dot \beta}\right\}=2\sigma_{\alpha \dot \beta}^{\mu} P_\mu
simply defining the SUSY transformation to be such and such. Also, why are dots on the \beta index? I read this is a type of notation, but I don't know its usefulness. How is SUSY a continuous transformation? Doesn't it map bosons (integer spin) to fermions (half integer) etc. in a ladder?
2) Infinitesimal generators of general lie algebras:
I think I can safely say to understand what generators are, but it seems there isn't an unique an precise way to compute them. I found various ways but, especially in case of SO(p,q) lorentz transformations, I'm confused because I found two different kinds of generators. The first are
M_{\mu \nu} = x_\nu \partial_\mu - x_\mu \partial_\nu
which are obviously related to angular momentum, and then the others are simply a basis for antisymmetric matrices, because imposing the lorentz condition on the metric with infinitesimal transformations one gets antisymmetric matrices. The generators above are in fact antisymmetric, but I'm talking about combinations of the Minkowski metric (you can see them in Tong's QED notes). What's the difference? Is the first one local and the second global or what? Is it because they belong to different representations?
3) Classification of representations:
I often find names of representations such as direct sums of (1/2, 0), (0,1/2), (0, 1), and also of "integers written in bold" as names. Why is that? What do they represent and what's the usefulness?
4) Derivation of conformal killing equation in CFT:
I think i understood the basics of CFT quite well, but when I try to derive the equation for flat space killing vectors
\partial_\mu \epsilon_\nu + \partial_\nu \epsilon_\mu = \frac{2}{d}(\partial \cdot \epsilon) \eta_{\mu \nu}
I get an annoying extra term. I start with equating the transformation of the metric (the generic one) with the infinitesimal conformal transformation. I get a quadratic term in derivatives of \epsilon which shouldn't be there; perhaps it is zero, because of orthogonality of different partial derivatives? I'd like to see a complete derivation.
Thanks in advance for responses, sorry if I disturbed you. :)
1) Superpoincarè algebra:
I read in various references that the SUSY extension of the poincarè algebra is the simplest SUSY algebra one can get. I get why the generators have to be spinors, but nobody ever gives a derivation of the defining anticommutator
\left\{ Q_\alpha , \bar{Q}_{\dot \beta}\right\}=2\sigma_{\alpha \dot \beta}^{\mu} P_\mu
simply defining the SUSY transformation to be such and such. Also, why are dots on the \beta index? I read this is a type of notation, but I don't know its usefulness. How is SUSY a continuous transformation? Doesn't it map bosons (integer spin) to fermions (half integer) etc. in a ladder?
2) Infinitesimal generators of general lie algebras:
I think I can safely say to understand what generators are, but it seems there isn't an unique an precise way to compute them. I found various ways but, especially in case of SO(p,q) lorentz transformations, I'm confused because I found two different kinds of generators. The first are
M_{\mu \nu} = x_\nu \partial_\mu - x_\mu \partial_\nu
which are obviously related to angular momentum, and then the others are simply a basis for antisymmetric matrices, because imposing the lorentz condition on the metric with infinitesimal transformations one gets antisymmetric matrices. The generators above are in fact antisymmetric, but I'm talking about combinations of the Minkowski metric (you can see them in Tong's QED notes). What's the difference? Is the first one local and the second global or what? Is it because they belong to different representations?
3) Classification of representations:
I often find names of representations such as direct sums of (1/2, 0), (0,1/2), (0, 1), and also of "integers written in bold" as names. Why is that? What do they represent and what's the usefulness?
4) Derivation of conformal killing equation in CFT:
I think i understood the basics of CFT quite well, but when I try to derive the equation for flat space killing vectors
\partial_\mu \epsilon_\nu + \partial_\nu \epsilon_\mu = \frac{2}{d}(\partial \cdot \epsilon) \eta_{\mu \nu}
I get an annoying extra term. I start with equating the transformation of the metric (the generic one) with the infinitesimal conformal transformation. I get a quadratic term in derivatives of \epsilon which shouldn't be there; perhaps it is zero, because of orthogonality of different partial derivatives? I'd like to see a complete derivation.
Thanks in advance for responses, sorry if I disturbed you. :)
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