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--- 수학:그라스만_대수 [2026/04/14 10:20] – created admin
+++ 수학:그라스만_대수 [2026/04/20 13:50] (current) – [적분] admin
@@ Line 4: / Line 4: @@
 $$\{ \theta_i, \theta_j \} = \theta_i \theta_j + \theta_j \theta_i = 0.$$
-=====지수함수=====
+======지수함수======
 지수함수를 전개했을 때 위의 성질로부터 이차항 이상의 고차항이 모두 사라지고 $\exp(\bar{\theta} \theta) = 1 + \bar{\theta} \theta$.
-=====미분 연산자=====
+======미분 연산자======
 먼저 아래와 같은 성질은 쉽게 생각할 수 있다:
 $$\frac{\partial}{\partial \theta_i} \theta_j = \delta_{ij}.$$
@@ Line 17: / Line 17: @@
 $$\left\{ \frac{\partial}{\partial \theta_i}, \frac{\partial}{\partial \theta_j} \right\} = 0.$$
-=====적분=====
+======적분======
 적분구간 끝에서의 기여분이 없다면
 $$\int d\theta \frac{\partial}{\partial \theta} f(\theta) = 0.$$
@@ Line 24: / Line 24: @@
 즉 사실상 적분은 미분과 동일하다:
 $$\int d\theta = \frac{\partial}{\partial \theta}.$$
+여기에 따라오는 성질로서 다음과 같은 것들이 있다:
+$$\int d\theta = 0$$
+$$\int \theta d\theta = 1.$$
+======가우스 적분======
+$n\times n$ 행렬 $M$에 대해
+$$\det M = \int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \ldots d\theta_n d\bar{\theta}_n \exp\left( \sum_{i,j=1}^n \bar{\theta}_i M_{ij} \theta_j \right).$$
+$n=2$인 경우를 예로 들면 이렇게 계산한다:
+\begin{eqnarray*}
+\int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \ldots d\theta_n d\bar{\theta}_n \exp\left( \sum_{i,j=1}^n \bar{\theta}_i M_{ij} \theta_j \right) &=&
+\int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \exp\left( \bar{\theta}_1 M_{11} \theta_1 + \bar{\theta}_1 M_{12} \theta_2 + \bar{\theta}_2 M_{21} \theta_1 + \bar{\theta}_2 M_{22} \theta_2 \right)\\
+&=& \int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \frac12 \left( \bar{\theta}_1 M_{11} \theta_1 + \bar{\theta}_1 M_{12} \theta_2 + \bar{\theta}_2 M_{21} \theta_1 + \bar{\theta}_2 M_{22} \theta_2 \right)^2\\
+&=& \int d\theta_1 d\bar{\theta}_1 d\theta_2 d\bar{\theta}_2 \frac12 \left( \bar{\theta}_1 M_{11} \theta_1 \bar{\theta}_2 M_{22} \theta_2 + \bar{\theta}_1 M_{12} \theta_2 \bar{\theta}_2 M_{21} \theta_1 + \bar{\theta}_2 M_{21} \theta_1 \bar{\theta}_1 M_{12} \theta_2 + \bar{\theta}_2 M_{22} \theta_2 \bar{\theta}_1 M_{11} \theta_1 \right)\\
+&=& \frac{\partial}{\partial\theta_1} \frac{\partial}{\partial\bar{\theta}_1} \frac{\partial}{\partial \theta_2} \frac{\partial}{\partial\bar{\theta}_2} \left( \bar{\theta}_1 M_{11} \theta_1 \bar{\theta}_2 M_{22} \theta_2 + \bar{\theta}_1 M_{12} \theta_2 \bar{\theta}_2 M_{21} \theta_1 \right)\\
+&=& \frac{\partial}{\partial\theta_1} \frac{\partial}{\partial\bar{\theta}_1} \frac{\partial}{\partial \theta_2} \frac{\partial}{\partial\bar{\theta}_2} \left(  \bar{\theta}_2 \theta_2 \bar{\theta}_1 \theta_1 M_{11}M_{22} - \bar{\theta}_2 \theta_2 \bar{\theta}_1 \theta_1 M_{12} M_{21} \right)\\
+&=& M_{11} M_{22} - M_{12} M_{21}.
+\end{eqnarray*}
-======함께 보기=====
+반대칭 행렬 $A$에 대해 ($A_{ij}+A_{ji}=0$)
+$$\text{Pf}(A) = \frac{1}{2^n n!} \int d\theta_{2n} \ldots d\theta_1 \exp\left(\frac12 \sum_{i,j=1}^{2n} \theta_i A_{ij} \theta_j \right)$$
+이며, $\text{Pf}^2(A) = \det A$.
+======함께 보기======
   * [[물리:2차원 이징 모형]]
   * [[수학:허바드-스트라토노비치_변환|허바드-스트라토노비치 변환]]
+  * [[물리:구면_p-스핀_유리_모형|구면 $p$-스핀 유리 모형]]
+======참고문헌======
+  * Jean Jinn-Justin, //Quantum Field Theory and Critical Phenomena// (Oxford University Press, Oxford, 1989).
+  * [[https://gandhiviswanathan.wordpress.com/2018/10/16/berezin-integration-of-grassmann-variables/|Gandhi Viswanathan's Blog]]