Question on Witten's paper: Perturbative Gauge Theory As A String Theory

In summary, the conversation revolved around a question regarding formula 2.12 in Witten's paper "Perturbative Gauge Theory As A String Theory In Twistor Space". The formula was not fully understood and there was difficulty in deriving it. However, after discussing the amplitude and its polarization vectors, it was discovered that the formula could be easily understood by considering the action of the operator H_i. This solved the initial problem and provided a better understanding of the formula.
  • #1
earth2
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Question on Witten's paper: "Perturbative Gauge Theory As A String Theory..."

Hi guys,

I have a question regarding formula 2.12 of Witten's paper hep-th/0312171
"Perturbative Gauge Theory As A String Theory In Twistor Space". He just states this formula but i don't really understand his 'justification' nor do I see a why to derive it myself. I tried to play around with the Pauli-Lubanski-Vector since it is in the massless limit related to helicity but with no success...

Do any of you guys have an idea?
Thanks,
earth2
 
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  • #2


It's easy to see that formula if you think about the amplitude in precisely the manner in which Witten doesn't want to, namely in terms of the momenta and polarization vectors. Expand the amplitude in a series

[tex]\hat{A}(\lambda_i,\bar{\lambda}_i, h_i) = \sum C_{\mu_1\cdots \mu_n \nu_1\cdots \nu_m} p_1^{\mu_1} \cdots p_n^{\mu_n} \epsilon_1^{\nu_1} \cdots \epsilon_m^{\nu_m} ,[/tex]

where [tex]C_{\mu_1\cdots \mu_n \nu_1\cdots \nu_m}[/tex] are some coefficients that take into account however the momenta and polarizations are contracted. It is an important fact that each particle in the amplitude contributes one and only one polarization vector, so no factor of [tex]\epsilon_i[/tex] is repeated.

Now under the maps

[tex] p_i^\mu \rightarrow \lambda^i_a \tilde{\lambda}^i_{\dot{a}} [/tex]
[tex] \epsilon_i^\nu \rightarrow \frac{\lambda^i_a \tilde{\mu}^i_{\dot{a}}}{\langle \tilde{\lambda}^i,\tilde{\mu}^i\rangle} ~\tex{or}~\frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle},[/tex]

we consider the action of

[tex]H_i = \lambda^a_i \frac{\partial}{\partial\lambda^a_i} - \tilde{\lambda}^{\dot{a}}_i \frac{\partial}{\partial\tilde{\lambda}^{\dot{a}}_i}.[/tex]

When [tex]H_i[/tex] acts on a momentum factor, we obtain zero, since acting on [tex]\tilde{\lambda}[/tex] cancels the action on [tex]\lambda[/tex]. When [tex]H_i[/tex] acts on a positive helicity polarization vector we find

[tex] - \frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle^2}\langle \lambda,\mu\rangle - \frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle} = -2(+1) \frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle} [/tex]

Similarly on a negative helicity polarization vector, we obtain a factor of [tex]2 = -2(-1)[/tex]. So we can conclude that

[tex]H_i \hat{A}(\lambda_i,\bar{\lambda}_i, h_i) = - 2 h_i \hat{A}(\lambda_i,\bar{\lambda}_i, h_i) .[/tex]
 
  • #3


Thanks for your help! That solved my problem :)
 

What is Witten's paper about?

Witten's paper, titled "Perturbative Gauge Theory As A String Theory", explores the connection between perturbative gauge theory and string theory. It proposes a new approach to understanding the dynamics of gauge theories by using concepts from string theory.

Why is Witten's paper significant?

Witten's paper is significant because it suggests a new way of thinking about gauge theories, which are fundamental in understanding the interactions between particles in the Standard Model of particle physics. It also helps bridge the gap between two seemingly unrelated theories, gauge theory and string theory.

What is perturbative gauge theory?

Perturbative gauge theory is a mathematical framework used to study the dynamics of subatomic particles. It involves breaking down the interactions between particles into smaller, more manageable parts, and then using perturbation theory to calculate the effects of these interactions.

What is string theory?

String theory is a theoretical framework that attempts to reconcile quantum mechanics and general relativity by describing particles as one-dimensional vibrating strings instead of point-like objects. It is still a work in progress and has not been experimentally proven.

How does Witten's paper connect perturbative gauge theory and string theory?

Witten's paper suggests that perturbative gauge theory can be understood as a string theory in certain cases. This means that the calculations and predictions made using perturbative gauge theory can also be interpreted and solved using the tools and techniques of string theory. This provides a new perspective and potentially new avenues for further research in both fields.

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