Showing papers on "Quantum geometry published in 1985"
TL;DR: Basic features of quantum mechanics follow, such as the identification of observables with self-adjoint operators, and canonical quantization rules, which gives a new insight on the geometry of quantum theory.
Abstract: The generator aspect of observables in classical mechanics leads naturally to a generalized classical mechanics, of which quantum mechanics is shown to be a particular case. Basic features of quantum mechanics follow, such as the identification of observables with self-adjoint operators, and canonical quantization rules. This point of view also gives a new insight on the geometry of quantum theory: Planck's constant is related for instance to the curvature of the quantum-mechanical space of states, and the uniqueness of quantum mechanics can be proved. Finally, the origin of the probabilistic interpretation is discussed.
248 citations
Book•
11 Jun 1985
TL;DR: In this paper, the concepts of modern differential geometry are presented in a comprehensive study of classical mechanics, field theory, and simple quantum effects, and the concepts can be used to describe the physical world.
Abstract: This work shows how the concepts of manifold theory can be used to describe the physical world. The concepts of modern differential geometry are presented in this comprehensive study of classical mechanics, field theory, and simple quantum effects.
77 citations
64 citations
TL;DR: In this paper, quantum properties of the Yang-Mills homogeneous space model were considered in the Schrodinger representation, and the authors considered quantum mechanical properties of homogeneous spaces.
52 citations
50 citations
TL;DR: In this paper, a sum over histories formulation of quantum geometry could involve sums over different topologies as well as sum over different metrics, but it is difficult to implement such sums over manifolds in quantum gravity and motivation is found for including geometries defined on more general objects than manifolds.
Abstract: A sum over histories formulation of quantum geometry could involve sums over different topologies as well as sums over different metrics. In classical gravity a geometry is a manifold with a metric, but it is difficult to implement a sum over manifolds in quantum gravity. In this difficulty, motivation is found for including in the sum over histories, geometries defined on more general objects than manifolds-unruly topologies. In simplicial two-dimensional quantum gravity a class of simplicial complexes is found to which the gravitational action can be extended, for which sums over the class are straightforwardly defined, and for which a manifold dominates the sum in the classical limit. The situation in higher dimensions is discussed.
33 citations
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TL;DR: In this article, the authors provide a general, co-ordinate-free geometrical presentation of ideas advanced before in coordinate language with reference to specific instances, which generalizes to non-Abelian fields results of extant "geometrical quantization".
Abstract: The aim of this paper is to provide a general, co-ordinate-free geometrical presentations of ideas advanced before in co-ordinate language with reference to specific instances. This approach generalizes to non-Abelian fields results of extant “geometrical quantization»; it also permits a natural connection between quantum mechanics and information theory, as discussed elsewhere.
18 citations
TL;DR: The notion of covariance and propagation in quantum space-times constituting such fiber bundles is investigated in this paper, which gives rise in a natural manner to certain realizations of relativistic canonical commutation relations in terms of covariant derivatives involving internal as well as external degrees of freedom of space-time excitons.
Abstract: Quantum geometries whose points are stochastic and serve as seats for quantum space-time excitons are formulated as fibre bundles over base spaces of mean values with a Minkowski or general relativistic structure. The fibres contain the proper wave functions of all exciton states in a given model. The notion of covariance and propagation in quantum space-times constituting such fibre bundles is investigated. Maxwell and Yang-Mills gauge degrees of freedom are introduced by appropriately enlarging the structure group, which in all cases contains phase-space representations of the Poincare group corresponding to the exciton wave function sample space specific to a given model. It is shown that these formulations give rise in a natural manner to certain realizations of the relativistic canonical commutation relations in terms of covariant derivatives involving internal as well as external degrees of freedom of space-time excitons.
17 citations
11 citations
TL;DR: In this article, a scattering amplitude for the Polyakov quantum fermionic string is proposed, which leads to a spectrum without the usual tachionic excitation, obtained in the semi-classical limit D → −∞, where D is the space-time dimension.
7 citations
TL;DR: The recent discovery of quantized red-shifts for galaxies suggests a cosmic form of quantum mechanics which may explain the observed properties of galaxies as mentioned in this paper, and the connection between the cosmic Planck constant, (hg), and other fundamental constants of nature.
Abstract: The recent discovery of quantized red-shifts for galaxies suggests a cosmic form of quantum mechanics which may explain the observed properties of galaxies Some consequences of the new theory were presented in two papers This paper adds to the list by exploring possible connections between the cosmic Planck’s constant, (hg), and other fundamental constants of nature
TL;DR: The implications of supersymmetry for the scattering parameters of some quantum mechanical supersymmetric models in one and three dimensions are discussed in this article, where the authors also consider the implications of quantum mechanics on quantum mechanics.
TL;DR: In this paper, it was shown that in d = 11 simple supergravity, with a minisuperspace ansatz, all instantons must have a four dimensional sector.
Abstract: The quantum state of the universe is described by Hartle and Hawking's ground state which is defined by a path integral over all compact metrics. The most probable classical evolution of the universe can be considered to come from some gravitational instanton by a quantum tunneling. These arguments have been generalized to the case of Kaluza-Klein models. It is found that in d= 11 simple supergravity, with a minisuperspace ansatz, all instantons must have a four dimensional sector. It suggests that this is the main reason why space-time is four-dimensional.
TL;DR: The theory of symmetry for a two-level quantum system was developed in this paper to illustrate the main ideas of symmetry in quantum theory, which is based on the diffeomorphism of the two-dimensional sphere S.............. 2>>\s onto the space of states ℂP.............. 1>>\s and the isomorphism between the groups P ℳ(2) and SO.............. 3>>\s (ℝ).
Abstract: We develop the theory of symmetry for a two-level quantum system in oder to illustrate the main ideas of the general theory of symmetry in quantum theory. It is based on the diffeomorphism of the two-dimensional sphere S
2
onto the space of states ℂP
1
and the isomorphism between the groups Pℳ(2) and SO
3
(ℝ). In particular, rotational invariance leads to the appearance of the spin1/2 in a natural way.
TL;DR: In this article, an algorithm for the computer evaluation of Sturm solutions of the two-centre problem of quantum mechanics in the case of different values of the inter-nuclear distance R and of energies E R → 0 and R → ∞ was given.
Abstract: An algorithm is given for the computer evaluation of the Sturm solutions of the two-centre problem of quantum mechanics in the case of different values of the inter-nuclear distance R and of energies E R →0 and R →∞. The results are given graphically.