diff options
| -rw-r--r-- | marginal.tex | 18 |
1 files 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 |