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

2001 ◽  
Vol 21 (3) ◽  
pp. 587-595 ◽  
Author(s):  
N.M. Chepilko ◽  
A.V. Romanenko
2017 ◽  
Vol 29 (03) ◽  
pp. 1750006
Author(s):  
Partha Mukhopadhyay

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space [Formula: see text] corresponding to a Riemannian manifold [Formula: see text] around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of [Formula: see text] around the diagonal submanifold, where [Formula: see text] is the Cartesian product of [Formula: see text] copies of [Formula: see text] with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to [Formula: see text] can be obtained by taking the limit [Formula: see text]. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-[Formula: see text] limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of [Formula: see text] which, for [Formula: see text], is the tangent bundle [Formula: see text].


2021 ◽  
pp. 2150187
Author(s):  
F. M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort ◽  
G. Marmo ◽  
L. Schiavone ◽  
...  

A novel derivation of Feynman’s sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac–Feynman–Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function [Formula: see text] on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian [Formula: see text] allows us to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman’s original derivation of the propagator for a point particle described by a classical Lagrangian L.


2019 ◽  
Author(s):  
Yang Zhang

AbstractFrom Synthesis perspective, whether Logic Synthesis, Physical Synthesis, Chemical Synthesis, or Biological Synthesis, Physical Geometry such as Universal Geometry and Quantum Geometry, and Biological Geometry like Conformal Geometry supported by Tensors and Manifolds, are the outcome of physical laws and biological laws in modeling non-linear physical and biological dynamics as opposed to traditional partial differential/difference equation way. We discover that Multiversal SpaceTime instead of Neural Network, governing physical and biological world at macroscopic and microscopic level, is the ultimate source of intelligence. With that we propose Multiversal Synthesis-based Artificial Design Automation (ADA), a bio-physical inspired model based on Multiverse in Darwin Dynamics, Generalized Quantum Mechanics, and Extended General Relativity, for Artificial Super Intelligence (ASI) implementation. Based on Schrodinger Equation of Quantum Mechanics, we generalize the 4-Dimensional Hilbert Space based Discrete Quantum SpaceTime to N-Dimensional (1 ≪ N < M, with M is limited by Planck Length) Hilbert Space based Discrete MSpaceTime as part of MSpaceTime, in modeling both Micro-Environment Intelligence and Micro-Agent Intelligence of ASI; likewise based on Einstein Equations of General Relativity, we make a T-Symmetry extension first, and then extend the 4-Dimensional Pseudo-Riemannian Manifold based Continuous Curved SpaceTime as part of MSpaceTime to N-Dimensional (1 ≪ N < ∞) Pseudo-Riemannian Manifold based Continuous MSpaceTime extension, in modeling both Macro-Environment Intelligence and Macro-Agent Intelligence of ASI. Our discovery only solves the black box puzzle of AI, but also paves the way in achieving ASI through ADA. Of course, our Multiverse Endeavor will never stop from there.


1991 ◽  
Vol 85 (6) ◽  
pp. 1189-1201 ◽  
Author(s):  
N. Ogawa ◽  
K. Fujii ◽  
N. Chepilko ◽  
A. Kobushkin

1990 ◽  
Vol 83 (5) ◽  
pp. 894-905 ◽  
Author(s):  
N. Ogawa ◽  
K. Fujii ◽  
A. Kobushukin

Sign in / Sign up

Export Citation Format

Share Document