A unified scheme of measurement and amplification processes

We present a unified scheme of quantum measurements containing the aspects of amplification processes, or translation to macro-levels on the basis of [1], a joint work with Prof. I. Ojima. This formulation is based on micro-macro duality [2] as a mathematical expression of the general idea of quantum-classical correspondence. An essential difference between classical physics and quantum physics can be seen in their algebraic structures describing the physical systems. Quantum systems are generally described by non-commutative algebras (C*algebras, von-Neumann algebras, and so on) consists of observables. The concept of amplification or translation to macro is nothing but how the information of (quantum) non-commutative observables is translated in the words of macroscopic worlds, which are described by (classical) commutative ones. This point of view is essential for understanding quantum measurements. In mathematical words, the transformation from quantum internal degrees of freedom to classical ones is constructed by Kac-Takesaki operator or multiplicative unitary, which plays a fundamental roles in harmonic analysis, and the dual structure between q-numbers and c-numbers can be understood via the Helgason duality controlling the Radon transform (in somehow generalized meanings). A concrete observation for the case of Stern-Gerlach experiment can be seen in [1], and recently, we obtained more general formulation which is applicative to various situation. In addition, this framework suggests us when ideal measurements (with no perturbation effects) go on; the so-called adiabaticity conditions can be said as sufficient for meaningful measurement results.