Yosuke Morimoto
Using the known duality between events and states, we establish the fact that there is a duality between quantization of events and measurement of states. With this duality, we will show a generalized system of imprimitivity and a covariant measurement are in a dual relation. We also propose a novel quantization scheme as a dual of a measurement model.
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We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our a…
Quantization of the damped harmonic oscillator is taken as leitmotiv to gently introduce elements of quantum probability theory for physicists. To this end, we take (graduate) students in physics as entry level and explain the physical intuition and motivation behind the, sometimes overwhelming, math machinery of quantum probability theory. The main text starts with the quantization of the (undamped) harmonic oscillator from the Heisenberg and Schroedinger point of view. We show how both treatments are special instances of a quantum probabilistic quantization procedure: the second quantization…
The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula ari…
An algebraic quantization procedure for discretized spacetime models is suggested based on the duality between finitary substitutes and their incidence algebras. The provided limiting procedure that yields conventional manifold characteristics of spacetime structures is interpreted in the algebraic quantum framework as a correspondence principle.
Quantization of classical systems using the star-product of symbols of observables is discussed. In the star-product scheme an analysis of dual structures is performed and a physical interpretation is proposed. At the Lie algebra level duality is shown to be connected to double Lie algebras. The analysis is specified to quantum tomography. The classical tomographic Poisson bracket is found.
The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjoint regions of the space. By wave, to the contrary, is understood a distributed item, occupying in some cases two or more disjoint regions of the space. The quantum formalism did not explain until today the so called "collapse" of the wave function, i.e. the shrinking of the wave function to one small region of the space, when a macroscopic object is encountered. This seems to hap…
The Newton--Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton's characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling e…
The geometric quantization problem is considered from the point of view of the Davies and Lewis approach to quantum mechanics. The influence of the measuring device is accounted in the classical and quantum case and it is shown that the conditions of the measurement define the type of quantization (Weyl, normal, antinormal, etc.). The quantum states and quantum operators are obtained by means of the projection, defined from the system of generalized coherent states.
The Mach-Zehnder interferometric setup quantitatively characterizing the wave-particle duality implements in fact a joint measurement of two unsharp observables. We present a necessary and sufficient condition for such a pair of unsharp observables to be jointly measurable. The condition is shown to be equivalent to a duality inequality, which for the optimal strategy of extracting the which-path information is more stringent than the Jaeger-Shimony-Vaidman-Englert inequality.
By making the second quantization for the Cini Model of quantum measurement without wave function collapse [M. Cini, Nuovo Cimento, B73 27(1983)], the second order quantum decoherence (SOQD) is studied with a two mode boson system interacting with an idealized apparatus composed by two quantum oscillators. In the classical limit that the apparatus is prepared in a Fock state with a very large quantum number, or in a coherent state with average quantum numbers large enough, the SOQD phenomenon appears similar to the first order case of quantum decoherence.
On the Quantization of the Electromagnetic Field with Magnetic Monopoles
本文基于Cabibbo-Ferrari双势形式论，在Feynman-'t Hooft规范中利用Gupta-Bleuler程序实现了含磁单极子的电磁场协变量子化，预测了与磁荷相关的对偶光子的存在。
Power-Duality in Path Integral Formulation of Quantum Mechanics
本文研究费曼路径积分表述下的幂对偶性，发现修改后的promotor在经典角动量下形式不变，角动量量子化后对偶性破坏，进而提出准幂对偶性概念并给出由已知格林函数确定对偶系统格林函数的公式。
This paper presents a philosophically realistic analysis of quantization, field-particle duality, superposition, entanglement, nonlocality, and measurement. These are logically related: Realistically understanding measurement depends on realistically understanding superposition, entanglement, and nonlocality; understanding these three depends on understanding field-particle duality and quantization. This paper resolves all six, based on a realistic view of standard quantum physics. It concludes that, for these issues, standard quantum physics is consistent with scientific practice since Copern…
In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable $f:\R^2\to \R$ is associated with a unique positive operator measure (POM) $E^f$, which is not necessarily projection valued. The motivation for such a scheme comes from the well-known fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly,…
Duality and measurement: the Copenhagen reconciliation
本文通过双方面解读重构量子理论，解决测量问题及相关的知识-信息和宏观-微观二元性，回归哥本哈根诠释的核心精神。
The only evidence we have for a discrete reality comes from quantum measurements; without invoking these measurements, quantum theory describes continuous entities. This seeming contradiction can be resolved via analysis that treats measurements as boundary constraints. It is well-known that boundaries can induce apparently-discrete behavior in continuous systems, and strong analogies can be drawn to the case of quantum measurement. If quantum discreteness arises in this manner, this would not only indicate an analog reality, but would also offer a solution to the so-called "measurement proble…
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