All Questions
16
questions
1
vote
0
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84
views
Noether charge for dilatations in terms of creation and anihilation operators
I am trying to compute the conserved charge for a continuous diatation symmetry for the massless real scalar field in four dimensions terms of creation and annihilation operators. Then I have,
$$\...
- 368
1
vote
0
answers
102
views
Beta function from renormalized coupling
While trying to derive a specific beta function for a CFT a stumbled upon the following. I have some bare coupling $g_b$ and introduce a renormalized coupling $g$ as
$$g_b=\mu^\epsilon(g+\frac{zg^2}{\...
- 61
2
votes
0
answers
81
views
Mode expansion using generalised Fourier transform
I'm looking at a generalised Lagrangian $L = \frac{1}{2} \left[\dot{\phi}² + \phi \mathcal{D} \phi\right]$, where $\mathcal{D}u_n = -\omega_n²u_n$, where $\left\{ u_n | n \in \mathbb{Z} \right\}$ span ...
3
votes
0
answers
101
views
Overall constant for the scalar propagator in AdS background
I am trying to solve Exercise 3.3 in TASI Lectures on AdS/CFT by João Penedones. It is solving for the scalar propagator $\Pi(X,Y)$ in AdS, and states as follows:
$$
\begin{align}
\frac{1}{2} J_{AB}J^...
1
vote
0
answers
64
views
Traceless energy momentum tensor and energy spectrum
We have a $D$ dimensional flat minkowskian spacetime, and a field theory with $T_{\mu \nu}$ symmetric, traceless ($T^{\mu}_{\mu} = 0 $) and conserved ($\partial^{\mu} T_{\mu \nu} = 0$). We also assume ...
- 141
1
vote
0
answers
221
views
Commutation relations for a CFT
Exercise
To calculate the commutator $[L_m ; \phi_n]$ where:
$$T(z) = \sum \frac{L_m}{z^{m+2}} \hspace{5mm} and \hspace{5mm} \phi(w) = \sum \phi_n \frac{1}{w^{n+h}} \tag{1}$$
using contour ...
user256673
0
votes
1
answer
64
views
Question solving tensor problems for the Special Conformal Killing Equation
Background
I know that following index notation, these are true:
$$\partial_\mu x^\nu = \delta^\mu _\nu \hspace{5mm} and \hspace{5mm} \partial_\mu x_\nu = \eta_{\mu\nu} \tag{1}$$
Exercise
Knowing ...
- 753
-1
votes
2
answers
102
views
Is it valid to change the order of tensors by changing their sign?
Can I change the order of tensors in an equation by changing their sign?
So for example if I have something like:
$$-x^2\partial_\nu \partial_\mu $$
Can I do the following?
$$-x^2\partial_\nu \...
user256673
1
vote
1
answer
163
views
How do I get from the conformal transformation equation to the conformal killing equation?
I am unable to obtain the conformal killing equation:
$$2\kappa(x) \eta_{\mu\nu}= \partial_\mu \xi _\nu + \partial_\nu \xi_\mu\tag{1}$$
Theory:
I understand that the conformal transformation is:
$$...
- 753
5
votes
0
answers
188
views
Importance of an extra total derivative term in Liouville theory
In this paper on boundary Liouville theory, the authors have introduced an extra term, $-\partial_{\sigma}^2\phi$, (the last term in the equation below) in defining the stress tensor of the Liouville ...
- 544
-1
votes
1
answer
262
views
Computing the OPE of $T : \mathrm{e}^{ikX} : $ [closed]
I've hit a stumbling block where I'm just not seeing how to get from line to line in the following calculation from David Tong's strings notes. Can someone spell out how line 1 becomes line 2 in the $\...
- 1,050
3
votes
0
answers
409
views
OPE of normal ordered operators
In what follows I use $\mathcal{N}\{\ldots\}$ for normal ordering, $\langle\ldots\rangle$ for contraction and $\operatorname{Reg}\{\ldots\}$ for the complete sequence of regular terms which is ...
- 2,921
3
votes
0
answers
545
views
Calculating OPE of Graviton Vertex Operator
Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
- 113
4
votes
1
answer
695
views
Operator Dimension and Field Transformation under Rescaling
In conformal field theory the operator dimension $\Delta$ determines how fields and thus correlation functions behave under rescaling. I am having trouble seeing how this number arises from a scale ...
- 83
5
votes
0
answers
2k
views
How to derive the scale factor for special conformal transformation? [closed]
By definition a conformal transformation of the coordinates is an invertible mapping $x\rightarrow x'$ which leaves the metric invariant upto a scale factor:
\begin{equation}
g_{\mu\nu}'(x') = \...
- 59