Update ph229-notes.tex eq (2.12) missing -e^{2K) in sqrt.

ph229-notes.tex

@@ -171,7 +171,7 @@ \subsection{QFT and emergent symmetry}
 
 Statistical systems are described by QFTs in Euclidean signature, e.g.\ on $\R^d$. Such QFTs capture properties of the equilibrium state. By contrast, condensed matter and particle systems are described by QFTs in Lorentzian signature, e.g.\ on $\R^{d-1,1}$. Such QFTs encode time-dependent quantum dynamics.
 
-We will be interested in QFTs with rotational symmetry, by which we mean $\SO(d)$ symmetry in Euclidean signature and $\SO(d-1,1)$ symmetry in Lorentzian signature. In particle physics and string theory, this symmetry is built into the microscopic theory. However in lattice systems, rotational symmetry is emergent. This means that correlation functions become rotationally invariant in the limit of large distances, even though microscopic correlation functions are not rotationally-invariant.\footnote{Emergent rotational symmetry is very familiar: we� often cannot determine the orientation of a microscopic lattice using macroscopic observations. Some examples of materials {\it without\/} emergent $\SO(d)$ symmetry are crystals like salt. A very exotic example is the Haah code \cite{}.} In particular, for condensed matter systems, the effective ``speed of light" associated with $\SO(d-1,1)$-invariance is an emergent property and has nothing to do with the speed of actual light. (We will see some explicit examples later.)
+We will be interested in QFTs with rotational symmetry, by which we mean $\SO(d)$ symmetry in Euclidean signature and $\SO(d-1,1)$ symmetry in Lorentzian signature. In particle physics and string theory, this symmetry is built into the microscopic theory. However in lattice systems, rotational symmetry is emergent. This means that correlation functions become rotationally invariant in the limit of large distances, even though microscopic correlation functions are not rotationally-invariant.\footnote{Emergent rotational symmetry is very familiar: weÊ often cannot determine the orientation of a microscopic lattice using macroscopic observations. Some examples of materials {\it without\/} emergent $\SO(d)$ symmetry are crystals like salt. A very exotic example is the Haah code \cite{}.} In particular, for condensed matter systems, the effective ``speed of light" associated with $\SO(d-1,1)$-invariance is an emergent property and has nothing to do with the speed of actual light. (We will see some explicit examples later.)
 
 Under general conditions, $\SO(d)$-invariant Euclidean QFTs are in one-to-one correspondence with $\SO(d-1,1)$-invariant Lorentzian QFTs. The map between them is called Wick rotation, and we will discuss it in detail. Because of this correspondence, we can focus mostly on Euclidean QFTs, and later understand Lorentzian QFTs by Wick rotating what we learned in Euclidean signature.
 
@@ -534,7 +534,7 @@ \subsection{The 1d Ising model and the transfer matrix}
 \ee
 where
 \be
-\l_\pm &= e^K \cosh h \pm \sqrt{e^{2K} \sinh^2 h + e^{-2K}} \nn\\
+\l_\pm &= e^K \cosh h \pm \sqrt{e^{2K} \sinh^2 h - e^{2K} + e^{-2K}} \nn\\
 &\to \begin{cases}
 2\cosh K \\
 2\sinh K