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extended_quantum_action_functional_._finite [2014/03/12 00:14] nikolaj |
extended_quantum_action_functional_._finite [2015/01/30 12:55] nikolaj |
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| ===== Extended quantum action functional . finite ===== | ===== Extended quantum action functional . finite ===== | ||
| - | ==== partial function ==== | + | ==== Partial function ==== |
| - | | @#88DDEE: $ \mathbb K = \mathbb C \lor \mathbb R $ | | + | | @#55CCEE: context | @#55CCEE: $ \mathbb K = \mathbb C \lor \mathbb R $ | |
| - | | @#88DDEE: $ m\in\mathbb N $ | | + | | @#55CCEE: context | @#55CCEE: $ m\in\mathbb N $ | |
| - | | @#88DDEE: $ D $ .... self-adjoint operator in $\mathbb K^m$ with well behaved inverse at least for $D+i\,\varepsilon\,\mathrm 1$ | | + | | @#55CCEE: context | @#55CCEE: $ D $ .... self-adjoint operator in $\mathbb K^m$ with well behaved inverse at least for $D+i\,\varepsilon\,\mathrm 1$ | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $Z:(\mathbb K^2\to\mathbb R)\to \mathbb K^4\to \mathbb K $ | | |
| - | | @#FFBB00: $Z:(\mathbb K^2\to\mathbb R)\to \mathbb K^4\to \mathbb K $ | | + | | @#FFBB00: definiendum | @#FFBB00: $Z_{\mathcal L_\mathrm{int}}(J,K,\phi,\psi):=\mathrm{e}^{i\hbar^{-1}\sum_{i=1}^m\mathcal L_\mathrm{int}\left(-i\,\hbar\frac{\partial}{\partial J_i},-i\,\hbar\frac{\partial}{\partial K_i}\right)} \left( \mathrm{e}^{i\,\hbar^{-1} \left\langle J\left|\,\mathcal{R}_\varepsilon\,\right|K\right\rangle}\cdot\mathrm{e}^{Z_\text{source}(J,K,\phi,\psi)}\right)$ | |
| - | | @#FFBB00: $Z_{\mathcal L_\mathrm{int}}(J,K,\phi,\psi):=\mathrm{e}^{i\hbar^{-1}\sum_{i=1}^m\mathcal L_\mathrm{int}\left(-i\,\hbar\frac{\partial}{\partial J_i},-i\,\hbar\frac{\partial}{\partial K_i}\right)} \left( \mathrm{e}^{i\,\hbar^{-1} \left\langle J\left|\,\mathcal{R}_\varepsilon\,\right|K\right\rangle}\cdot\mathrm{e}^{Z_\text{source}(J,K,\phi,\psi)}\right)$ | | + | |
| | @#BBDDEE: $\mathcal{R}_\varepsilon\equiv-\left(D+i\,\varepsilon\,\mathrm{1}\right)^{-1}$ | | | @#BBDDEE: $\mathcal{R}_\varepsilon\equiv-\left(D+i\,\varepsilon\,\mathrm{1}\right)^{-1}$ | | ||
| | @#BBDDEE: $Z_\text{source}(J,K,\phi,\psi):=i\,\hbar^{-1}\left(\left\langle J\left|\right. \phi\right\rangle+\left\langle K\left|\right. \psi\right\rangle\right)$ | | | @#BBDDEE: $Z_\text{source}(J,K,\phi,\psi):=i\,\hbar^{-1}\left(\left\langle J\left|\right. \phi\right\rangle+\left\langle K\left|\right. \psi\right\rangle\right)$ | | ||
| - | ==== Discussion ==== | + | ----- |
| + | === Discussion === | ||
| The operator $\mathcal{R}_\varepsilon$ is denoted the response function because it relates the classical free field to the source, $(D+i\,\varepsilon)\phi=-J$. The brackets are the standard inner product on $\mathbb C^n$ or $\mathbb R^n$. The finite quantum action functional presented here can be generalized by | The operator $\mathcal{R}_\varepsilon$ is denoted the response function because it relates the classical free field to the source, $(D+i\,\varepsilon)\phi=-J$. The brackets are the standard inner product on $\mathbb C^n$ or $\mathbb R^n$. The finite quantum action functional presented here can be generalized by | ||
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| {{qed_two-three-four_vertex-diagrams.png}} | {{qed_two-three-four_vertex-diagrams.png}} | ||
| - | ==== Parents ==== | + | ----- |
| === Context === | === Context === | ||
| [[Square matrix]], [[Self-adjoint operator]] | [[Square matrix]], [[Self-adjoint operator]] | ||
