THE DIFFEOMORPHISM GROUP APPROACH TO NONLINEAR QUANTUM SYSTEMS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1905-1916 ◽  
Author(s):  
GERALD A. GOLDIN
Keyword(s):  
Nonlinear Effects ◽  
Quantum Systems ◽  
Incompressible Fluids ◽  
Diffeomorphism Groups ◽  
Group Approach ◽  
Quantum Vortex ◽  
Nonlinear Quantum ◽  
New Applications ◽  
Fundamental Physical ◽  
Natural Description

Unitary representations of diffeomorphism groups predict some unusual possibilities in quantum theory, including non-standard statistics and certain nonlinear effects. Many of the fundamental physical properties of “anyons” were first derived from their study by R. Menikoff, D.H. Sharp, and the author. This paper surveys new applications in two other domains: first (with Menikoff and Sharp) some surprising conclusions about the nature of quantum vortex configurations in ideal, incompressible fluids; second (with H.-D. Doebner) a natural description of dissipative quantum mechanics by means of a nonlinear Schrödinger equation different from the sort usually studied. Our equation follows from including a diffusion current in the equation of continuity.

THE DIFFEOMORPHISM GROUP APPROACH TO ANYONS

1991 ◽  
Vol 05 (16n17) ◽  
pp. 2625-2640 ◽  
Author(s):  
GERALD A. GOLDIN ◽  
DAVID H. SHARP
Keyword(s):  
Quantum Mechanics ◽  
Quantum Systems ◽  
Diffeomorphism Group ◽  
Diffeomorphism Groups ◽  
Group Approach ◽  
Fundamental Properties ◽  
Local Current ◽  
Unified Description ◽  
Particle Statistics ◽  
Insight Into

We explain an approach to quantum mechanics, based on diffeomorphism groups and local current algebras, that predicts many of the fundamental properties of anyons. Our formulation also yields further insight into the meaning of particle statistics, and permits the unified description of a wide variety of quantum systems.

scholarly journals Structure of gauge theories

2021 ◽  
Vol 136 (3) ◽  
Author(s):  
Víctor Aldaya
Keyword(s):  
Canonical Quantization ◽  
Experimental Parameter ◽  
Diffeomorphism Groups ◽  
Gauge Groups ◽  
Yang Mills ◽  
Vector Potentials ◽  
Generalized Symmetry ◽  
Weinberg Angle ◽  
Stueckelberg Formalism ◽  
Fundamental Physical

AbstractElementary interactions are formulated according to the principle of minimal interaction although paying special attention to symmetries. In fact, we aim at rewriting any field theory on the framework of Lie groups, so that, any basic and fundamental physical theory can be quantized on the grounds of a group approach to quantization. In this way, connection theory, although here presented in detail, can be replaced by “jet-gauge groups” and “jet-diffeomorphism groups.” In other words, objects like vector potentials or vierbeins can be given the character of group parameters in extended gauge groups or diffeomorphism groups. As a natural consequence of vector potentials in electroweak interactions being group variables, a typically experimental parameter like the Weinberg angle $$\vartheta _W$$ ϑ W is algebraically fixed. But more general remarkable examples of success of the present framework could be the possibility of properly quantizing massive Yang–Mills theories, on the basis of a generalized Non-Abelian Stueckelberg formalism where gauge symmetry is preserved, in contrast to the canonical quantization approach, which only provides either renormalizability or unitarity, but not both. It proves also remarkable the actual fixing of the Einstein Lagrangian in the vacuum by generalized symmetry requirements, in contrast to the standard gauge (diffeomorphism) symmetry, which only fixes the arguments of the possible Lagrangians.

Simplified P-representation operator correspondence applied to quantum systems with generalized Kerr nonlinearity

2019 ◽  
Vol 33 (28) ◽  
pp. 1950340 ◽  
Author(s):  
S. Chiangga ◽  
S. Pitakwongsaporn ◽  
Till D. Frank
Keyword(s):  
Energy Levels ◽  
Kerr Nonlinearity ◽  
Quantum Systems ◽  
Matter Wave ◽  
Quantum Oscillator ◽  
Text Representation ◽  
Nonlinear Quantum ◽  
Quantum Optical ◽  
Field Oscillation ◽  
Quadratic And Cubic Nonlinearities

A simplified operator correspondence scheme is derived to address nonlinear quantum systems within the framework of the [Formula: see text]-representation. The simplified method is applied to a general nonlinear quantum oscillator model that has been used in the literature to describe nonlinear quantum optical and matter wave systems. The [Formula: see text]-representation evolution equation for the model is derived for arbitrary nonlinearity exponents. It is shown that in the high temperature case, the [Formula: see text]-representation is sufficient to describe the model and its associated systems such that there is no need to use alternative, mathematically more involved representations. Systems with quadratic and cubic nonlinearities are considered in more detail. Distributions for the energy levels and photon and particle numbers are obtained within the framework of the [Formula: see text]-representation. Moreover, the electrical field oscillation frequency dependency is studied numerically when interpreting the model as model for quantum optical nonlinear oscillators.

scholarly journals Non-hermitian topology as a unifying framework for the Andreev versus Majorana states controversy

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
J. Avila ◽  
F. Peñaranda ◽  
E. Prada ◽  
P. San-Jose ◽  
R. Aguado
Keyword(s):  
Quantum Systems ◽  
Topological Classification ◽  
Exceptional Point ◽  
Complex Spectrum ◽  
Zero Energy ◽  
Zero Modes ◽  
Large Subset ◽  
General Systems ◽  
Natural Description ◽  
Surprising Fact

Abstract Zero-energy Andreev levels in hybrid semiconductor-superconductor nanowires mimic all expected Majorana phenomenology, including $$2{e}^{2}/h$$ 2 e 2 ∕ h conductance quantisation, even where band topology predicts trivial phases. This surprising fact has been used to challenge the interpretation of various transport experiments in terms of Majorana zero modes. Here we show that the Andreev versus Majorana controversy is clarified when framed in the language of non-Hermitian topology, the natural description for quantum systems open to the environment. This change of paradigm allows one to understand topological transitions and the emergence of zero modes in more general systems than can be described by band topology. This is achieved by studying exceptional point bifurcations in the complex spectrum of the system’s non-Hermitian Hamiltonian. Within this broader topological classification, Majoranas from both conventional band topology and a large subset of Andreev levels at zero energy are in fact topologically equivalent, which explains why they cannot be distinguished.

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