Semiclassical quantum mechanics

We consider the 0 limit of the quantum dynamics generated by the HamiltonianH()=–(²/2m)+V. We prove that the evolution of certain Gaussian states is determined asymptotically as 0 by classical mechanics. For suitable potentialsV inn3 dimensions, our estimates are uniform in time and our results hold for scattering theory.Supported in part by the National Science Foundation under Grant PHY 78-08066     

Attempt of an axiomatic foundation of quantum mechanics and more general theories. III

Starting from axioms as physical as possible [1, 2, 3] about effects and ensembles, we shall investigate further consequences.Concerning part I and II [4, 5] the axioms can be so formulated as to be surveyed more easily.Besides, it is possible to prove some important theorems more simply.New structures of the lattice of decision effects are pointed out, leading in two subsequent papers at last to the final aim, the structure of Hilbert-space.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories. IV

This contribution continues the series of papers on the same subject which has been treated byLudwig in [1–3]. Using the system of axioms as given in [3], we shall succeed in constructing an orthomodular lattice of linear operators on the real vector space generated by the physical decision effects. There results an isomorphism between the orthomodular lattice of all physical decision effects and the lattice to be constructed.     

N = 4 supersymmetric quantum mechanics and its large order behaviour

A model of N = 4 supersymmetric quantum mechanics is shown to have a perturbative expansion for which the large order behaviour is - (3/2)ⁿn!(9/π + 5/n + …). In the space of the coupling constants the supersymmetric case is found to be the dividing point between two different types of large order behaviour. In this way the large order behaviour of supersymmetric theories is special.     

Finite-dimensional quantum mechanics of a particle. III. The Weylian quantum mechanics of confined quarks

A finite-dimensional analog of Weyl's formulation of quantum kinematics of a physical system through irreducible Abelian groups of unitary ray rotations in system space offers many possibilities for the quantum mechanics of confined particles. This paper is devoted to the expansion of the recently developed framework of such Weylian finite-dimensional quantum mechanics which may provide a new way of thinking about the characteristics of quark physics.     

The large numbers hypothesis and quantum mechanics

In this paper, the suggested similarity between micro and macrocosmos is extended to quantum behavior, postulating that quantum mechanics, like general relativity and classical electrodynamics, is invariant under discrete scale transformations. This hypothesis leads to a large scale quantization of angular momenta. Using the scale factor Λ ~ 10³⁸, the corresponding quantum of action, obtained by scaling the Planck constant, is close to the Kerr limit for the spin of the universe - when this is considered as a huge rotating black hole - and to the spin of Gödel’s universe, solution of Einstein equations of gravitation. Besides, we suggest the existence of another, intermediate, scale invariance, with scale factor λ ~ 10¹⁹. With this factor we obtain, from Fermi’s scale, the values for the gravitational radius and for the collapse proper time of a typical black hole, besides the Kerr limit value for its spin. It is shown that the mass-spin relations implied by the two referred scale transformations are in accordance with Muradian’s Regge-like relations for galaxy clusters and stars. Impressive results are derived when we use a λ-scaled quantum approach to calculate the mean radii of planetary orbits in solar system. Finally, a possible explanation for the observed quantization of galactic redshifts is suggested, based on the large scale quantization conjecture.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories VI

This contribution continues the series of papers [2, 4, 5, 12] treated by Ludwig and collaborators. It is based on the generalized frame given in [6]; there Ludwig has set up an infinite axiomatic scheme as extension of the finite system [4, 5]. The results of [12] are then proved for a locally finite case; they lead to an extended representation theorem.This paper was supported by the Deutsche Forschungsgemeinschaft.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories V

We continue here the series of papers treated byLudwig in [1–5]. Using some results ofDähn in [6], we point out that each irreducible solution of the axiomatic scheme set up in [5] is represented by a system of positive-semi-definite operator pairs of a finite-dimensional Hilbert-space over the real, complex or quaternionic numbers.This paper is an abridged version of the author's thesis presented to the Marburg University and written under the direction of Prof.G. Ludwig.     

Semiclassical states in loop quantum gravity: an introduction

Unifying general relativity and quantum mechanics is a great challenge left to us by Einstein. To face this challenge, considerable progress has been made in non-perturbative canonical (loop) quantum gravity during the past 20 years. The kinematical Hilbert space of the quantum theory is constructed rigorously. However, the semiclassical analysis of the theory is still a crucial and open issue. In this review, we first introduce our work on constructing a semiclassical weave state, using the [ω] operator on the kinematical Hilbert space of loop quantum gravity. Then we give an introduction to the two different approaches currently investigated for constructing coherent states in the kinematical Hilbert space. The current status of semiclassical analysis in loop quantum gravity is then summarized.     

Canonical transformations to action and angle variables and their representations in quantum mechanics: III. The general problem

The representation in quantum mechanics of canonical transformations to action and angle variables is discussed for a general class of Hamiltonians including some that have both bounded and unbounded orbits where, in the latter case, the definition of the transformation is suitably extended. Again, as in the particular examples discussed in the previous papers of this series, the canonical transformations are nonlinear and nonbijective. We can recover though the bijectiveness, i.e., the one-to-one onto mapping, either by introducing a sheet structure in the original phase space or using the concepts of ambiguity group and ambiguity spin. With the help of these concepts we obtain an expression for the representation in quantum mechanics of the canonical transformation and recover the latter when we pass to the classical limit with the help of the WKB approximation. Furthermore, we establish in this paper a one-to-one correspondance between the arbitrariness in the phase of the representation and in the choice of the variable conjugate to the action.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories,II

The consequences of an axiomatic formulation of physical probability fields established in a first paper [1] are investigated in case of a finite dimensional ensemble-space.It will be shown that the stated number of axioms can be diminuished essentially. Further the structure of an ortho-complemented orthomodular lattice for the decision effects (also often called properties or still more misunderstandingly propositions) and the orthoadditivity of the probability measures upon this lattice, both, can be essentially inferred from the axioms 3 and 4,only. This seems to give a better comprehension of the lattice structure defined by the decision effects.Particularly, it is pointed out that no assumption (axiom) concerning the commensurability of two decision effectsE ₁ E ₂ withE ₁E ₂ must be made but that this commensurability is a theorem of the theory.     

Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

Nonstationary quantum mechanics. III. Quantum mechanics does not incorporate classical physics

It is shown that disagreement between the prediction of classical and conventional quantum mechanics about momentum probabilities exists in the case of a quasiclassical motion. The discussion is based on the detailed consideration of two specific potentials:U(x)=x and the oscillatory potentialU(x)=m ² x ²/2. The results of the present Part III represent a further development of the idea in Todorov (1980) about the possible inefficiency of conventional theory in the case of potentials swiftly varying with time.     

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and -KMS states for all inverse temperatures 0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.     

Asymptotic localization and separation of states in quantum mechanics

This paper introduces the concepts of asymptotic localization and separation, and shows that quantum mechanical states exist which are asymptotically localizable and separable; these are states with finite and disjoint momentum ranges respectively. All this applies whether the quantum particle concerned is free or is in a potential.     

The logic of quantum mechanics derived from classical general relativity

For the first time it is shown that the logic of quantum mechanics can be derived from classical physics. An orthomodular lattice of propositions characteristic of quantum logic, is constructed for manifolds in Einstein’s theory of general relativity. A particle is modelled by a topologically non-trivial 4-manifold with closed timelike curves—a 4-geon, rather than as an evolving 3-manifold. It is then possible for both the state preparationand measurement apparatus to constrain the results of experiments. It is shown that propositions about the results of measurements can satisfy a non-distributive logic rather than the Boolean logic of classical systems. Reasonable assumptions about the role of the measurement apparatus leads to an orthomodular lattice of propositions characteristic of quantum logic.