Classical limit of fermions in phase space

2001 ◽  
Vol 42 (9) ◽  
pp. 4020-4030 ◽  
Author(s):  
A. O. Bolivar
Keyword(s):  
Phase Space ◽  
Classical Limit

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.

scholarly journals The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as an emergent phenomenon

2021 ◽  
Author(s):  
Valter Moretti ◽  
Christiaan J. F. van de Ven
Keyword(s):  
Phase Space ◽  
Probability Measure ◽  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Classical Limit ◽  
Schrödinger Operators ◽  
Algebraic Approach ◽  
Schrodinger Operators ◽  
Berezin Quantization ◽  
Emergent Phenomenon

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off [Formula: see text]. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.

scholarly journals Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model

2020 ◽  
Vol 32 (10) ◽  
pp. 2050031 ◽  
Author(s):  
Klaas Landsman ◽  
Valter Moretti ◽  
Christiaan J. F. van de Ven
Keyword(s):  
Phase Space ◽  
State Space ◽  
Deformation Quantization ◽  
Classical Limit ◽  
Poisson Manifold ◽  
Quantum Spin ◽  
The State ◽  
Symmetric Convex ◽  
Algebraic Vector ◽  
Classical Phase Space

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

scholarly journals CLASSICAL FERMI FLUID AND GEOMETRIC ACTION FOR c=1

1993 ◽  
Vol 08 (02) ◽  
pp. 325-349 ◽  
Author(s):  
AVINASH DHAR ◽  
GAUTAM MANDAL ◽  
SPENTA R. WADIA
Keyword(s):  
Phase Space ◽  
Matrix Model ◽  
Classical Limit ◽  
Phase Plane ◽  
Coadjoint Orbit ◽  
Gauge Fields ◽  
Quantum Fluid ◽  
Field Action ◽  
Classical Phase Space ◽  
Geometric Action

We formulate the c=1 matrix model as a quantum fluid and discuss its classical limit in detail, emphasizing the ħ corrections. We view the fermi fluid profiles as elements of w∞-coadjoint orbit and write down a geometric action for the classical phase space. In the specific representation of fluid profiles as “strings” the action is written in a four-dimensional form in terms of gauge fields built out of the embedding of the “string” in the phase plane. We show that the collective field action can be derived from the above action provided one restricts to quadratic fluid profiles and ignores the dynamics of their “turning points”.

Generalized Moyal structures in phase-space, master equations and their classical limit

1998 ◽  
Vol 256 (1-2) ◽  
pp. 112-128 ◽  
Author(s):  
Constantinos Tzanakis ◽  
Alkis P. Grecos ◽  
Polyxeni Hatjimanolaki
Keyword(s):  
Phase Space ◽  
Classical Limit ◽  
Master Equations

Semi-classical mechanics in phase space: A study of Wigner’s function

1977 ◽  
Vol 287 (1343) ◽  
pp. 237-271 ◽  
Keyword(s):  
Phase Space ◽  
Classical Limit ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Delta Function ◽  
Full Detail ◽  
Integrable Case ◽  
Classical Motion ◽  
Classical Path ◽  
Path Structure

We explore the semi-classical structure of the Wigner functions Ψ( q,p ) representing bound energy eigenstates | Ψ 〉 for systems with f degrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f -dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2 f -dimensional phase space. In the semi-classical limit of Ψ ( ℏ small but not zero) the delta function softens to a peak of order Ψ−  f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲ ,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.

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