Concepts of coarse graining in quantum mechanics

Various procedures of coarse graining in quantum mechanics and their relationships are reviewed. A recently developed notion of relative coarse graining is described which is based on a certain type of classical embedding of quantum states. The ensuing structure of the set of quantum observables is studied. As an application of the new concept an operational classical limit procedure is sketched out.     

Description of physical systems

A general “logical” scheme, containing both classical and quantum mechanics, is developed on the basis of plausible axioms. We introduce the division of states and yes-no measurements into sharp and diffuse ones, and prove that sharp states possess their carriers. Owing to this result, the existence of lattice joins and meets is proved for a wide class of elements of the logic. This “semi-lattice” structure gives the familiar lattice picture for special cases of classical and quantum mechanics. The notion of quantum superposition is introduced in this general scheme. It is proved that if in a theory appear nontrivial quantum superpositions, then this theory is “undeterministic” and vise versa. Further analysis of the pure state space leads to the construction of the canonical embedding of the general logic into an orthomodular complete ortho-lattice. After defining the probability of transition between pure states, the pure state space appears to be a generalization of Mielnik's “probability space” of quantum mechanics.     

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.     

BBGKY hierarchy underlying many-particle quantum mechanics

Recently, the one-particle quantum mechanics has been obtained in the framework of an entirely classical subquantum kinetics. In the present Letter we argue that, within the same scheme and without any additional assumption, it is possible to obtain also the n-particle non-relativistic quantum mechanics. The main goal of the present effort is to show that the classical BBGKY hierarchical equation, for the n-particle reduced distribution function, is the ancestor of the n-particle Schrödinger equation. On the other hand we show that within the scenario of the subquantum structure of quantum particle, the Fisher information measure emerges naturally in quantum mechanics.     

Nonrelativistic Quantum Mechanics with Spin in the Framework of a Classical Subquantum Kinetics

Recently it has been shown that the spinless one-particle quantum mechanics can be obtained in the framework of entirely classical subquantum kinetics. In the present paper we argue that, within the same scheme and without any extra assumption, it is possible to obtain both the nonrelativistic quantum mechanics with spin, in the presence of an arbitrary external electromagnetic field, as well as the nonlinear quantum mechanics.     

Strong Analog Classical Simulation of Coherent Quantum Dynamics

A strong analog classical simulation of general quantum evolution is proposed, which serves as a novel scheme in quantum computation and simulation. The scheme employs the approach of geometric quantum mechanics and quantum informational technique of quantum tomography, which applies broadly to cases of mixed states, nonunitary evolution, and infinite dimensional systems. The simulation provides an intriguing classical picture to probe quantum phenomena, namely, a coherent quantum dynamics can be viewed as a globally constrained classical Hamiltonian dynamics of a collection of coupled particles or strings. Efficiency analysis reveals a fundamental difference between the locality in real space and locality in Hilbert space, the latter enables efficient strong analog classical simulations. Examples are also studied to highlight the differences and gaps among various simulation methods.     

A Quantum Proxy Blind Signature Scheme Based on Genuine Five-Qubit Entangled State

In this paper, a quantum proxy blind signature scheme based on controlled quantum teleportation is proposed. This scheme uses a genuine five-qubit entangled state as quantum channel and adopts the classical Vernam algorithm to blind message. We use the physical characteristics of quantum mechanics to implement delegation, signature and verification. Security analysis shows that our scheme is valid and satisfy the properties of a proxy blind signature, such as blindness, verifiability, unforgeability, undeniability.     

Nonlinear quantum mechanics is a classical theory

Quantum mechanics with nonlinear operators is shown to be an essentially classical theory. A general scheme of delinearization of a quantum theory is described.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Quantum Mechanics and the Metrics of General Relativity

A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics, will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges.     

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach I

Schwinger's quantization scheme is extended in order to solve the problem of the formulation of quantum mechanics on a space with a group structure. The importance of Killing vectors in the quantization scheme is shown. Usage of these vectors makes the algebraic properties of the operators consistent with the geometrical structure of the manifold. The procedure of the definition of the quantum Lagrangian of a free particle and the norm of the velocity (momentum) operators is given. These constructions are invariant under a general coordinate transformation. The unified procedure for constructing the quantum theory on a space with a group structure is developed. Using this, quantum mechanics on a Riemannian manifold with a simply transitive group acting on it is investigated. Received: 27 June 2000 / Revised version: 10 May 2001 / Published online: 19 July 2001     

An Improved Proxy Blind Signature Scheme

In this paper, we propose an improved proxy blind signature scheme based on controlled quantum teleportation. Five-qubit entangled state functions as quantum channel. We use physical characteristics of quantum mechanics to implement signature, delegation and verification. Furthermore, quantum key distribution (QKD) protocol and one-time pad are adopted in this scheme. Like classical signature protocols, our scheme can be used in many application scenarios, such as e-government and e-business.

     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

Quantum Physics and Classical Physics—In the Light of Quantum Logic

In contrast to the Copenhagen interpretation we consider quantum mechanics as universally valid and query whether classical physics is really intuitive and plausible. We discuss these problems within the quantum logic approach to quantum mechanics where the classical ontology is relaxed by reducing metaphysical hypotheses. On the basis of this weak ontology a formal logic of quantum physics can be established which is given by an orthomodular lattice. By means of the Solèr condition and Piron's result one obtains the classical Hilbert spaces. However, this approach is not fully convincing. There is no plausible justification of Solèr's law and the quantum ontology is partly too weak and partly too strong. We propose to replace this ontology by an ontology of unsharp properties and conclude that quantum mechanics is more intuitive than classical mechanics and that classical mechanics is not the macroscopic limit of quantum mechanics.     

Quantization postulate in quantum mechanics

The procedure of transition from classical observables to quantum (operator) observables in quantum mechanics is discussed. By an example it is shown that, even in simple cases, the method of self-adjoint extensions of formal differential expressions for defining physical observables as operators is not equivalent to the procedure of forming operator functions corresponding to these observables. This inequivalence is not a formal one but has physical consequences connected with the compatibility of observables.