From 405f6727a6915c61e09160fba52dd8832c2207e3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:19:16 +0100
Subject: Slightly modified convention for superbases.

---
 marginal.tex | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

diff --git a/marginal.tex b/marginal.tex
index 442e7e2..c8d0b8b 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1804,26 +1804,26 @@ where the usual role of the determinant is replaced by the superdeterminant.
 The superdeterminant can be defined using the ordinary determinant by writing a
 block version of the matrix $M$. If $\mathbf e(1)=\{1,i\bar\theta_1\theta_1\}$ is
 the basis vector of the even subspace of the superspace and $\mathbf
-f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{\theta_1,-\bar\theta_1\}$ can be defined by the requirement that
+f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{-\theta_1,\bar\theta_1\}$ can be defined by the requirement that
 \begin{align}
-  \int d1\,\mathbf e(1)\mathbf e^\dagger(1)=iI
+  &\int d1\,e_i^\dagger(1)e_j(1)=i\delta_{ij}
   &&
-  \int d1\,\mathbf f(1)\mathbf f^\dagger(1)=iI \\
-  \int d1\,\mathbf e(1)\mathbf f^\dagger(1)=0
+  \int d1\,f_i^\dagger(1)f_j(1)=i\delta_{ij} \\
+  &\int d1\,e_i^\dagger(1)f_j(1)=0
   &&
-  \int d1\,\mathbf f(1)\mathbf e^\dagger(1)=0
+  \int d1\,f_i^\dagger(1)e_j(1)=0
 \end{align}
 With such bases and dual bases defined, we can form a
 block representation of $M$ in analogy to the matrix form of an operator in quantum mechanics by
 \begin{equation}
   \int d1\,d2\,\begin{bmatrix}
-    \mathbf e(1)M(1,2)\mathbf e^\dagger(2)
+    \mathbf e^\dagger(1)M(1,2)\mathbf e(2)
     &
-    \mathbf e(1)M(1,2)\mathbf f^\dagger(2)
+    \mathbf e^\dagger(1)M(1,2)\mathbf f(2)
     \\
-    \mathbf f(1)M(1,2)\mathbf e^\dagger(2)
+    \mathbf f^\dagger(1)M(1,2)\mathbf e(2)
     &
-    \mathbf f(1)M(1,2)\mathbf f^\dagger(2)
+    \mathbf f^\dagger(1)M(1,2)\mathbf f(2)
   \end{bmatrix}
   =i\begin{bmatrix}
     A & B \\ C & D
-- 
cgit v1.2.3-70-g09d2