scholarly journals A geometric Hamiltonian description of composite quantum systems and quantum entanglement

2015 ◽  
Vol 12 (07) ◽  
pp. 1550069 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Quantum Entanglement ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Quantum Systems ◽  
Point Of View ◽  
Entanglement Measure ◽  
Hamiltonian Description

Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kähler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be P(H1 ⊗ H2) and not simply P(H1) × P(H2) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.

Geometric Hamiltonian quantum mechanics and applications

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630017 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Star Product ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Point Of View ◽  
Quantum Theories ◽  
Geometric Point ◽  
Set Up

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

scholarly journals Paths of unitary access to exceptional points

2021 ◽  
Vol 2038 (1) ◽  
pp. 012026
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Phase Transitions ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
Point Of View ◽  
Explicit Description ◽  
Direct Access ◽  
Exceptional Point ◽  
Exceptional Points ◽  
Innovative Idea

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

EQUIVALENCE OF THE LAGRANGIAN AND HAMILTONIAN BRST CHARGES FOR THE BOSONIC STRING IN THE HARMONIC AND CONFORMAL GAUGES

1988 ◽  
Vol 03 (05) ◽  
pp. 1081-1101 ◽  
Author(s):  
V. DEL DUCA ◽  
L. MAGNEA ◽  
P. VAN NIEUWENHUIZEN
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Point Of View ◽  
Bosonic String ◽  
Lagrangian Formalism ◽  
Brst Charge ◽  
Gauge Fixing ◽  
Conformal Gauge ◽  
Brst Invariance

We consider the BRST formalism for the bosonic string in arbitrary gauges, both from the Hamiltonian and from the Lagrangian point of view. In the Hamiltonian formulation we construct the BRST charge Q(H) following the Batalin-Fradkin-Fradkina-Vilkovisky (BFFV) formalism in phase space. In the Lagrangian formalism, we use the Noether procedure to construct the BRST charge Q(L) in configuration space. We then discuss how to go from configuration to phase space and demonstrate that the dependence of Q(L) on the gauge fixing disappears and that both charges become equal. We work through two gauges in detail: the conformal gauge and the de Donder (harmonic) gauge. In the conformal gauge one must use equations of motion, and a simple canonical transformation is found which exhibits the equivalence. In the de Donder gauge, nontrivial canonical transformations are needed. Our results overlap with work by Beaulieu, Siegel and Zwiebach on the de Donder gauge, but since we only require BRST invariance and not anti-BRST invariance, we need simpler field redefinitions; moreover, we stay off-shell.

scholarly journals HAMILTONIAN DESCRIPTION OF HIGHER ORDER LAGRANGIANS

1996 ◽  
Vol 11 (25) ◽  
pp. 4551-4559 ◽  
Author(s):  
M. S. RASHID ◽  
S. S. KHALIL
Keyword(s):  
Phase Space ◽  
Symplectic Structure ◽  
Hamiltonian Formulation ◽  
Higher Order ◽  
Hamiltonian Description

Using Dirac’s analysis of second class constraints, a Hamiltonian formulation of higher order Lagrangians is reached. The new Hamiltonian description recovers the conventional description of such systems due to Ostrogradsky, but unlike the latter, it provides a simpler structure for the phase space and consequently the underlying symplectic structure.

PASSIVITY OF GROUND STATES OF QUANTUM SYSTEMS

2005 ◽  
Vol 17 (01) ◽  
pp. 1-14 ◽  
Author(s):  
WALTER F. WRESZINSKI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Ground State ◽  
Quantum Systems ◽  
Point Of View ◽  
Cyclic Vector ◽  
Vector State ◽  
Unitary Evolution ◽  
C Algebra ◽  
Local Perturbations

We consider a quantum system described by a concrete C*-algebra acting on a Hilbert space ℋ with a vector state ω induced by a cyclic vector Ω and a unitary evolution Ut such that UtΩ = Ω, ∀t ∈ ℝ. It is proved that this vector state is a ground state if and only if it is non-faithful and completely passive. This version of a result of Pusz and Woronowicz is reviewed, emphasizing other related aspects: passivity from the point of view of moving observers and stability with respect to local perturbations of the dynamics.

Quantization and Group Representations

1977 ◽  
Vol 29 (6) ◽  
pp. 1264-1276 ◽  
Author(s):  
R. Cressman
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Mechanical System ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Group Representations ◽  
Additional Requirement ◽  
Adjoint Operators

A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.

scholarly journals QSES's and the Quantum Jump

2001 ◽  
Vol 56 (1-2) ◽  
pp. 228-229
Author(s):  
David Salgado ◽  
José Luis Sánchez-Gómez
Keyword(s):  
Hilbert Space ◽  
Markov Property ◽  
Quantum Systems ◽  
Point Of View ◽  
Stochastic Methods ◽  
Stochastic Evolution ◽  
Quantum Jumps ◽  
Practical Point ◽  
Quantum Jump ◽  
Evolution Scheme

Abstract The stochastic methods in Hilbert space have been used both from a fundamental and a practical point of view. The result reported here concerns the application of these methods to model the evolution of quantum systems and does not enter into the question of their fundamental or practical status. It can easily be stated as follows: Once a quantum stochastic evolution scheme is assumed, the incompatibility of the Markov property with the notion of quantum jumps is easily established

NONLOCAL LAGRANGIANS AND HAMILTONIAN FORMALISM

1994 ◽  
Vol 03 (01) ◽  
pp. 211-214
Author(s):  
LLOSA J. ◽  
VIVES J.
Keyword(s):  
Phase Space ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Lagrangian Systems ◽  
Infinite Dimensional ◽  
First Order ◽  
Dimensional Phase Space ◽  
Set Up

A presymplectic formalism is set up for nonlocal Lagrangian systems. The method is based on an ‘equivalent’ first order Lagrangian that is processed by standard ways of classical mechanics. The Hamiltonian formalism for the latter is then pulled back onto the infinite dimensional phase space of the nonlocal system.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

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