Ｔｈｅ ｐｈｉｌｏｓｏｐｈｙ ｏｆ ｍｅａｓｕｒｅｍｅｎｔ ｔｈｅｏｒｙ Ｍｅａｓｕｒｅｍｅｎｔｓ（Ａｘｉｏｍ １） Ｔｈｅ ｒｅｌａｔｉｏｎ ａｍｏｎｇ ｓｙｓｔｅｍｓ（Ａｘｉｏｍ ２） Ｂｏｌｔｚｍａｎｎ’ｓ ｅｑｕｉｌｉｂｒｉｕｍ ｓｔａｔｉｓｔｉｃａｌ ｍｅｃｈａｎｉｃｓ Ｆｉｓｈｅｒ’ｓ ｓｔａｔｉｓｔｉｃｓ Ｉ（ｕｎｄｅｒ Ａｘｉｏｍ １） Ｆｉｓｈｅｒ’ｓ ｓｔａｔｉｓｔｉｃｓ ＩＩ（ｒｅｌａｔｅｄ ｔｏ Ａｘｉｏｍｓ １ ａｎｄ ２） Ｐｒａｃｔｉｃａｌ ｌｏｇｉｃ Ｓｔａｔｉｓｔｉｃａｌ ｍｅａｓｕｒｅｍｅｎｔｓ ｉｎ Ｃ＊‐ａｌｇｅｂｒａｉｃ ｆｏｒｍｕｌａｔｉｏｎ Ｓｔａｔｉｓｔｉｃａｌ ｍｅａｓｕｒｅｍｅｎｔｓ ｉｎ Ｗ＊‐ａｌｇｅｂｒａｉｃ ｆｏｒｍｕｌａｔｉｏｎ Ｎｅｗｔｏｎｉａｎ ｍｅｃｈａｎｉｃｓ ｉｎ ｍｅａｓｕｒｅｍｅｎｔ Ｔｈｅｏｒｙ Ｍｅａｓｕｒｅｍｅｎｔ ｅｒｒｏｒ Ｈｅｉｓｅｎｂｅｒｇ’ｓ ｕｎｃｅｒｔａｉｎｔｙ ｒｅｌａｔｉｏｎ
We investigate here the concept of generalized complementarity introduced by Bohr. It is shown that it is not possible to ascertain the microscopic state of a system without, for the purpose of measurement, imposing upon it physical conditions so stringent that they preclude the application of the statistical description which was appropriate before the measurement. One may conclude that the statistical averages which appear in the formation of an ensemble should in quantum statistics be given an interpretation somewhat different from that of classical statistics. These results lead naturally to a critical survey of the foundations of quantum statistical mechanics. We discuss the physical interpretation of statistical matrices which are the quantum equivalent of the classical ensemble. By a straightforward generalization of classical methods one obtains a statistical representation of a system if a set of measurements are given. Let $\ensuremath{\rho}$ designate the statistical matrix which represents the system; the method consists in making the diagonal sum of $\ensuremath{\rho} log \ensuremath{\rho}$ a minimum, while the results of the performed measurements appear as conditions of the minimum. As an application we treat the composition of two separate systems and derive from this the general proof of the
H theorem which was first given by Klein.