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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]
@@ Line 100: / Line 100: @@
 ===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=====
+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=====

CFT Zoo