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liouville_theory [2021/06/29 21:35]
Brian McPeak [Liouville theory] 		liouville_theory [2021/06/29 22:36] (current)
Brian McPeak [Bootstrap Picture]
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 {{tag>2d}} {{tag>2d}}
  
-Liouville theory is an interacting 2d CFT with a continuous spectrum. It is defined for any value of the central charge.+Liouville theory is an interacting 2d CFT with a continuous spectrum of scalar states.
  
 ====Bootstrap Picture==== ====Bootstrap Picture====
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 ===Some Remaining questions=== ===Some Remaining questions===
-  * This theory has no ground state. So how do I normalize the partition function?+  * How is the partition function normalized in a theory with no ground state?
  
 =====Lagrangian Picture===== =====Lagrangian Picture=====
 +
 +Alternatively, the theory may be defined through the path integral. This requires an action, which takes the form
 +
 +\begin{align}
 +    S =  \frac{1}{4 \pi} \int d^2 x (\partial_a \phi \partial^a \phi + 4 \pi \mu e^{2 b \phi} )
 +\end{align}
 +
 +$\phi$ is called the Liouville field. The equations of motion which result from this theory are 
 +
 +\begin{align}
 +  \Box \phi = - 4 \pi \mu b e^{2 b \phi}
 +\end{align}
 +
 +This is equivalent to Liouville's equation, which is the source of the theory's name. 
 +
 +It is instructive to consider the wavefunction of states which ``scatter" off the potential barrier $V(\phi)$. This takes the form
 +
 +\begin{align}
 +  \Psi_P(\phi) \sim e^{2 i P \phi} + R(P) e^{-2 i P \phi}
 +\end{align}
 +
 +Here $R(P)$ is the reflection amplitude for incoming plane-waves with momentum $P$, and may be computed from the equation of motion. The wave functions $\Psi$ describe the states which map to the Virarsoro primaries of Liouville theory under the state-operator correspondence. 
  
 =====External Links===== =====External Links=====
Last modified: 2021/06/29 21:35
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