The extended future tube is a domain of holomorphy
The author presents an approach to prove the famous extended future tube conjecture. Let \(M^4\) be the Minkowski space, i.e. \(\mathbb{R}^4\) with Lorentz metric \(x\cdot y=x_0 y_0-x_1y_1-x_2 y_2-x_3y_3\). The future (past) light cone \(V^+(V^-)\) is defined as \[ V^+: =\{y\in M^4: y\cdot y>0,\;y_0>0\}, \quad V^-=-V^+. \] The future and past tubes \(\tau^\pm\) are defined as the corresponding tube domains \(\tau^\pm= \mathbb{R}^4+iV^\pm\) in \(\mathbb{C}^4\). The \(N\)-point future (past) tube is defined as \(\tau^\pm_N=\tau^\pm\times\cdots\times\tau^\pm\) \(N\)-times. The identity component of the complex Lorentz group is called the proper complex Lorentz group \(L_+(\mathbb{C})=SO(1,3,\mathbb{C})\cong SO(4,\mathbb{C})\). The extended \(N\)-print future tube is defined as \(\tau_N'= L_+(\mathbb{C}) \cdot\tau^+_N= \{(\Lambda z^{(1)}, \dots, \Lambda z^{(N)}): z^{(\delta)}\in\tau^+\), \(\Lambda\in L_+(\mathbb{C})\}\). The result is the following: Theorem. For every \(N\geq 1\) \(\tau_N'\) is a domain of holomorphy.
scientific article; zbMATH DE number 1187592