Showing papers on "Quantum geometry published in 1973"

Quantum theory as an indication of a new order in physics. B. Implicate and explicate order in physical law

Abstract: In this paper, we inquire further into the question of the emergence of new orders in physics, first raised in an earlier paper. In this inquiry, we are led to suggest that the quantum theory indicates the need for yet another new order, which we call “enfolded” or “implicate.” One of the most striking examples of the implicate order is to be seen by considering the function of the hologram, which clearly reveals how a total content (in principle extending over the whole of space and time) is “enfolded” in the movement of waves (electromagnetic and other kinds) in any given region. We then come to the notion that the quantum theory indicates that this implicate order is not merely a dependent or fortuitous feature of the content, but rather, that it should be considered as the independent ground of existence of things, while the ordinary explicate order is what should be considered as dependent. Finally, in the appendix we point out how the implicate order is expressed naturally in terms of an algebra similar to that of the quantum theory, which is, however, subject to generalizations going beyond the limits of what has meaning in this theory. Various new directions of further research are indicated, which will be explained in later papers.

State automorphisms in axiomatic quantum mechanics

Abstract: A new metric which we call the ‘intrinsic metric’ is introduced on the statesℐ of the generalized logic of quantum mechanics It is shown that every automorphism onℐ is an isometry A norm can be defined on the linear spanE ofℐ which reduces to the intrinsic metric onℐ IfX is the completion ofE then every automorphism, onℐ has a unique extension to a linear isometry onX A comparison is made between these results and those of Kroniff

Infinite quantum systems

Abstract: A somewhat deeper understanding of the relation between spin and statistics is found. Also symmetry between charge conjugate sectors is shown. It can be demonstrated that the statistics parameters determine the metric of the scattering states and thereby enter into expressions for cross sections. Quantum statistical mechanics and quantum field theory are mentioned as possible applications. (JFP)

The particle search in a quantum field model

Abstract: The goal of quantum field theory is a description of elementary particles. When successful, this description should be both a mathematical theory and a law of nature. In our approach, we emphasize the construction of specific field theory models. The first motive for this emphasis is the mathematical consistency of quantum field theory. In addition, we follow closely the formal ideas of physics in order to provide a more solid foundation for the main task of quantum field theory: the study of elementary particles. It is clear by now that the first motive has been attained in two spacetime dimensions. In two space-time dimensions, quantum fields with nonlinear interactions have been constructed [1], [2]. For the polynomial boson (^(0)2) interactions with small coupling constant, all the Wightman axioms have been verified [5]. We announce two new results. The first is an estimate which is a major technical step toward establishing the above results in three space-time dimensions. The second result concerns the physical interpretation of our two dimensional models (the second motive of our study). In ( )2 models with a small coupling constant, we establish the existence of particles and we verify the Haag-Ruelle axioms for scattering. Thus we conclude that the ^((/>)2 model has the qualitative structure required for the description of (many particle) scattering experiments. In our first result we consider the $3 interaction: a $ 4 coupling in the interaction Hamiltonian for boson fields in three space-time dimensions. We introduce a space cut-off g e CQ(R), 0 ^ 0 ^ 1, in the interaction Hamiltonian density, and we let the resulting total Hamiltonian (with infinite counterterms from second and third order perturbation theory) be denoted H(g). Let A(g) denote the set of points within distance 1 of supp g.

Quantum Effect in D. C. and Hall Conductivities An Application of Wigner Representation

Abstract: The Wigner representation formalism is applied to investigating the effect of the momentum-coordinate commutation relation ~n the d.c. and Hall conductivities of a system of noninteracting electrons moving in a· potential :field of randomly distributed impurities. The conductivities are expanded in powers of A and the secondand fourth-order terms are shown to vanish within the Born approximation, as far as the expansion is reasonable. This situation is discussed in comparison with the result of the kinetic theory.