scholarly journals GEOMETRIC PHASES IN THE QUANTISATION OF BOSONS AND FERMIONS

2011 ◽  
Vol 90 (2) ◽  
pp. 221-235 ◽  
Author(s):  
SIYE WU
Keyword(s):  
Hilbert Spaces ◽  
Maslov Index ◽  
Harmonic Oscillators ◽  
Complex Structures ◽  
Flat Connection ◽  
Geometric Phases ◽  
Berry's Phase ◽  
Geometric Quantisation ◽  
Projectively Flat ◽  
Fermionic Systems

AbstractAfter reviewing geometric quantisation of linear bosonic and fermionic systems, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the Maslov index and its various generalisations. We also consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the above holonomy.

scholarly journals Complex structures on twisted Hilbert spaces

2017 ◽  
Vol 222 (2) ◽  
pp. 787-814 ◽  
Author(s):  
Jesús M. F. Castillo ◽  
Wilson Cuellar ◽  
Valentin Ferenczi ◽  
Yolanda Moreno
Keyword(s):  
Hilbert Spaces ◽  
Complex Structures

scholarly journals Distilling Entanglement from Fermions

2009 ◽  
Vol 16 (02n03) ◽  
pp. 243-258
Author(s):  
Michael Keyl
Keyword(s):  
Hilbert Spaces ◽  
Point Of View ◽  
Entangled States ◽  
Algebraic Point ◽  
Gaussian States ◽  
Free Fermions ◽  
One Dimensional ◽  
Fermionic Systems ◽  
Maximally Entangled States ◽  
Local Observables

Since fermions are based on anti-commutation relations, their entanglement cannot be studied in the usual way, such that the available theory has to be modified appropriately. Recent publications consider in particular the structure of separable and of maximally entangled states. In this paper we want to discuss local operations and entanglement distillation from bipartite fermionic systems. To this end we apply an algebraic point of view where algebras of local observables, rather than tensor product Hilbert spaces play the central role. We apply our scheme in particular to fermionic Gaussian states, where the whole discussion can be reduced to properties of the covariance matrix. Finally, the results are demonstrated with free fermions on an infinite, one-dimensional lattice.

Linear symplectic geometry

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Symplectic Geometry ◽  
Vector Bundles ◽  
Maslov Index ◽  
Vector Spaces ◽  
Complex Structures ◽  
Homotopy Equivalence ◽  
Linear Complex ◽  
Symplectic Forms ◽  
Lagrangian Subspaces ◽  
First Chern Class

The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the Maslov index. In the section on linear complex structures particular emphasis is placed on the homotopy equivalence between the space of symplectic forms and the space of linear complex structures. The chapter includes sections on symplectic vector bundles and the first Chern class.

FOCK BUNDLES AND QUANTIZATION OF SYSTEMS WITH FIRST CLASS CONSTRAINTS

1994 ◽  
Vol 09 (37) ◽  
pp. 3467-3479 ◽  
Author(s):  
A.D. POPOV
Keyword(s):  
Phase Space ◽  
Symmetry Group ◽  
Hilbert Spaces ◽  
Complex Structures ◽  
Constrained System

The flat phase space (R2n, ω) with a symmetry group G generated by quadratic first class constraints is considered. We analyze the quantization of this constrained system in terms of the Fock bundle [Formula: see text], where S=Sp(2n, R)/U(n) is the space of positive translationally invariant complex structures J on (R2n, ω) and fibers HJ are the Hilbert spaces of quantization associated with the space (R2n, ω, J).

scholarly journals Majorana fermions and orthogonal complex structures

2018 ◽  
Vol 33 (14) ◽  
pp. 1840001 ◽  
Author(s):  
J. S. Calderón-García ◽  
A. F. Reyes-Lega
Keyword(s):  
Hilbert Space ◽  
Ground States ◽  
Majorana Fermions ◽  
Complex Structures ◽  
Quadratic Hamiltonians ◽  
Fermionic Systems

Ground states of quadratic Hamiltonians for fermionic systems can be characterized in terms of orthogonal complex structures. The standard way in which such Hamiltonians are diagonalized makes use of a certain “doubling” of the Hilbert space. In this work, we show that this redundancy in the Hilbert space can be completely lifted if the relevant orthogonal structure is taken into account. Such an approach allows for a treatment of Majorana fermions which is both physically and mathematically transparent. Furthermore, an explicit connection between orthogonal complex structures and the topological [Formula: see text]-invariant is given.

scholarly journals COHERENT STATES AND GEOMETRIC PHASES IN THE CALOGERO–SUTHERLAND MODEL

2002 ◽  
Vol 17 (02) ◽  
pp. 259-267 ◽  
Author(s):  
DAE-YUP SONG ◽  
JEONGHYEONG PARK
Keyword(s):  
Coherent State ◽  
Coherent States ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Periodic Systems ◽  
Classical Solutions ◽  
Geometric Phases ◽  
Sutherland Model ◽  
Inverse Square Potentials ◽  
Time Periodic

Exact coherent states in the Calogero–Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the AN-1 Calogero–Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.

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