ブロッホ＝ドミニシスの定理
⟨ T τ [ A 1 ( τ 1 ) A 2 ( τ 2 ) ⋯ A n ( τ n ) ] ⟩ 0 = A 1 ( τ 1 ) ∘ A 2 ( τ 2 ) ∘ A 3 ( τ 3 ) ∘ ∘ ⋯ A n ( τ n ) ∘ ∘ ∘ + A 1 ( τ 1 ) ∘ A 2 ( τ 2 ) ∘ ∘ A 3 ( τ 3 ) ∘ ⋯ A n ( τ n ) ∘ ∘ ∘ + A 1 ( τ 1 ) ∘ A 2 ( τ 2 ) ∘ ∘ A 3 ( τ 3 ) ∘ ∘ ⋯ A n ( τ n ) ∘ ∘ ∘ + ⋯ {\displaystyle {\begin{aligned}\langle T_{\tau }[A_{1}(\tau _{1})A_{2}(\tau _{2})\cdots A_{n}(\tau _{n})]\rangle _{0}=&A_{1}(\tau _{1})^{\circ }A_{2}(\tau _{2})^{\circ }A_{3}(\tau _{3})^{\circ \circ }\cdots A_{n}(\tau _{n})^{\circ \circ \circ }\\&+A_{1}(\tau _{1})^{\circ }A_{2}(\tau _{2})^{\circ \circ }A_{3}(\tau _{3})^{\circ }\cdots A_{n}(\tau _{n})^{\circ \circ \circ }+A_{1}(\tau _{1})^{\circ }A_{2}(\tau _{2})^{\circ \circ }A_{3}(\tau _{3})^{\circ \circ }\cdots A_{n}(\tau _{n})^{\circ \circ \circ }+\cdots \end{aligned}}}
と表すことができる。
⟨ T τ [ A 1 ( τ 1 ) A 2 ( τ 2 ) ⋯ A n ( τ n ) ] ⟩ 0 = ∑ P ′ ( ± 1 ) P ⟨ T τ [ A i 1 ( τ i 1 ) A i 2 ( τ i 2 ) ] ⟩ 0 ⟨ T τ [ A i 3 ( τ i 3 ) A i 4 ( τ i 4 ) ] ⟩ 0 ⋯ ⟨ T τ [ A i n − 1 ( τ i n − 1 ) A i n ( τ i n ) ] ⟩ 0 {\displaystyle \langle T_{\tau }[A_{1}(\tau _{1})A_{2}(\tau _{2})\cdots A_{n}(\tau _{n})]\rangle _{0}=\sum _{P}'(\pm 1)^{P}\langle T_{\tau }[A_{i_{1}}(\tau _{i_{1}})A_{i_{2}}(\tau _{i_{2}})]\rangle _{0}\langle T_{\tau }[A_{i_{3}}(\tau _{i_{3}})A_{i_{4}}(\tau _{i_{4}})]\rangle _{0}\cdots \langle T_{\tau }[A_{i_{n-1}}(\tau _{i_{n-1}})A_{i_{n}}(\tau _{i_{n}})]\rangle _{0}}
が成り立つ。
この場合も縮約について、
A ( τ a ) ∘ B ( τ b ) ∘ := ⟨ T τ [ A ( τ a ) B ( τ b ) ] ⟩ 0 {\displaystyle A(\tau _{a})^{\circ }B(\tau _{b})^{\circ }:=\langle T_{\tau }[A(\tau _{a})B(\tau _{b})]\rangle _{0}}
となる記法を導入すれば、
で定義されるグランドカノニカル分布での熱平均であり、H はハミルトニアン、N は数演算子、βは逆温度、μは化学ポテンシャルを表す。またA (τ)は
A ( τ ) = e + K τ A e − K τ A ( τ ) † = e + K τ A † e − K τ {\displaystyle {\begin{aligned}A(\tau )&=e^{+K\tau }Ae^{-K\tau }\\A(\tau )^{\dagger }&=e^{+K\tau }A^{\dagger }e^{-K\tau }\end{aligned}}}
で定義される虚時間τ=it についてのハイゼンベルク表示の演算子である。Tτは虚時間についての時間順序積であり、
T τ { A 1 ( τ 1 ) A 2 ( τ 2 ) } = { A 1 ( τ 1 ) A 2 ( τ 2 ) ( τ 1 > τ 2 ) ± A 2 ( τ 2 ) A 1 ( τ 1 ) ( τ 2 > τ 1 ) {\displaystyle T_{\tau }\{A_{1}(\tau _{1})A_{2}(\tau _{2})\}=\left\{{\begin{matrix}A_{1}(\tau _{1})A_{2}(\tau _{2})&\quad (\tau _{1}>\tau _{2})\\\pm A_{2}(\tau _{2})A_{1}(\tau _{1})&\quad (\tau _{2}>\tau _{1})\end{matrix}}\right.}
を意味する。
⟨ T τ [ A ( τ a ) B ( τ b ) C ( τ c ) D ( τ d ) ] ⟩ 0 = A ( τ a ) ∘ B ( τ b ) ∘ C ( τ c ) ∘ ∘ D ( τ d ) ∘ ∘ + A ( τ a ) ∘ B ( τ b ) ∘ ∘ C ( τ c ) ∘ D ( τ d ) ∘ ∘ + A ( τ a ) ∘ B ( τ b ) ∘ ∘ C ( τ c ) ∘ ∘ D ( τ d ) ∘ = ⟨ T τ [ A ( τ a ) B ( τ b ) ] ⟩ 0 ⟨ T τ [ C ( τ c ) D ( τ d ) ] ⟩ 0 ± ⟨ T τ [ A ( τ a ) C ( τ c ) ] ⟩ 0 ⟨ T τ [ B ( τ b ) D ( τ d ) ] ⟩ 0 + ⟨ T τ [ A ( τ a ) D ( τ d ) ] ⟩ 0 ⟨ T τ [ B ( τ b ) C ( τ c ) ] ⟩ 0 {\displaystyle {\begin{aligned}\langle T_{\tau }[A(\tau _{a})B(\tau _{b})C(\tau _{c})D(\tau _{d})]\rangle _{0}=&A(\tau _{a})^{\circ }B(\tau _{b})^{\circ }C(\tau _{c})^{\circ \circ }D(\tau _{d})^{\circ \circ }\\&+A(\tau _{a})^{\circ }B(\tau _{b})^{\circ \circ }C(\tau _{c})^{\circ }D(\tau _{d})^{\circ \circ }+A(\tau _{a})^{\circ }B(\tau _{b})^{\circ \circ }C(\tau _{c})^{\circ \circ }D(\tau _{d})^{\circ }\\=&\langle T_{\tau }[A(\tau _{a})B(\tau _{b})]\rangle _{0}\langle T_{\tau }[C(\tau _{c})D(\tau _{d})]\rangle _{0}\\&\pm \langle T_{\tau }[A(\tau _{a})C(\tau _{c})]\rangle _{0}\langle T_{\tau }[B(\tau _{b})D(\tau _{d})]\rangle _{0}+\langle T_{\tau }[A(\tau _{a})D(\tau _{d})]\rangle _{0}\langle T_{\tau }[B(\tau _{b})C(\tau _{c})]\rangle _{0}\end{aligned}}}
が成り立つ。