Recently a new kind of approximation to continuum topological spaces has been introduced, the app... more Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approxi...
A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution i... more A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected Principal Fibre Bundles, and the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint orbits associated with Lie groups. It is shown that this framework generalises in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of Null Phase Curves and Pancharatnam lifts from pure to mixed states are also presented.
The possibility of deforming the (associative or Lie) product to obtain alternative descriptions ... more The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by chang-ing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical conguration space Q that can be considered as \adapted " to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in dierent nonequivalent ways, \evading, " so to speak, the von Neumann uniqueness theorem.
We show how it is possible to include the constraint of single site-occupancy in the large U limi... more We show how it is possible to include the constraint of single site-occupancy in the large U limit of the Hubbard model at half-filling by means of a local chemical potential. The formalism which is in principle exact allows for finite temperature mean field results satisfying the constraint within 10%. The method is here applied to the dimerized phase and the commensurate flux phase which are still found as local saddle points of the action. The symmetry properties of these actions have been studied. One loop static corrections have been added to obtain an effective free energy and an estimate of the specific heat. In the limit of zero temperature our results reproduce those of the conventional mean field theory of these phases, because quantum fluctuations have not been included. We find that the temperature destabilizes the flux phase more than it does with the dimer phase, especially due to the constraint, which does not suppress phase fluctuating modes, provided the flux is con...
Page 1. Lecture Notes in Physics - - Vol. 38 HUBBARD MODEL AND ANYON AP Balachandran E. Ercolessi... more Page 1. Lecture Notes in Physics - - Vol. 38 HUBBARD MODEL AND ANYON AP Balachandran E. Ercolessi G. Morandi AM Srivastava World Scientific Page 2. Page 3. HUBBARD MODEL AND ANYON SUPERCONDUCTIVITY This One 97PJ-YXT-JW59 Page 4. Page 5. ...
We discuss how the existence of a regular Lagrangian description on the tangent bundle $TQ$ of so... more We discuss how the existence of a regular Lagrangian description on the tangent bundle $TQ$ of some configuration space $Q$ allows for the construction of a linear structure on $TQ$ that can be considered as "adapted" to the given dynamical system. The fact then that many dynamical systems admit alternative Lagrangian descriptions opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.
A complete solution to the problem of setting up Wigner distribution for N-level quantum systems ... more A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.
Recently a new kind of approximation to continuum topological spaces has been introduced, the app... more Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approxi...
A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution i... more A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected Principal Fibre Bundles, and the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint orbits associated with Lie groups. It is shown that this framework generalises in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of Null Phase Curves and Pancharatnam lifts from pure to mixed states are also presented.
The possibility of deforming the (associative or Lie) product to obtain alternative descriptions ... more The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by chang-ing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical conguration space Q that can be considered as \adapted " to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in dierent nonequivalent ways, \evading, " so to speak, the von Neumann uniqueness theorem.
We show how it is possible to include the constraint of single site-occupancy in the large U limi... more We show how it is possible to include the constraint of single site-occupancy in the large U limit of the Hubbard model at half-filling by means of a local chemical potential. The formalism which is in principle exact allows for finite temperature mean field results satisfying the constraint within 10%. The method is here applied to the dimerized phase and the commensurate flux phase which are still found as local saddle points of the action. The symmetry properties of these actions have been studied. One loop static corrections have been added to obtain an effective free energy and an estimate of the specific heat. In the limit of zero temperature our results reproduce those of the conventional mean field theory of these phases, because quantum fluctuations have not been included. We find that the temperature destabilizes the flux phase more than it does with the dimer phase, especially due to the constraint, which does not suppress phase fluctuating modes, provided the flux is con...
Page 1. Lecture Notes in Physics - - Vol. 38 HUBBARD MODEL AND ANYON AP Balachandran E. Ercolessi... more Page 1. Lecture Notes in Physics - - Vol. 38 HUBBARD MODEL AND ANYON AP Balachandran E. Ercolessi G. Morandi AM Srivastava World Scientific Page 2. Page 3. HUBBARD MODEL AND ANYON SUPERCONDUCTIVITY This One 97PJ-YXT-JW59 Page 4. Page 5. ...
We discuss how the existence of a regular Lagrangian description on the tangent bundle $TQ$ of so... more We discuss how the existence of a regular Lagrangian description on the tangent bundle $TQ$ of some configuration space $Q$ allows for the construction of a linear structure on $TQ$ that can be considered as "adapted" to the given dynamical system. The fact then that many dynamical systems admit alternative Lagrangian descriptions opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.
A complete solution to the problem of setting up Wigner distribution for N-level quantum systems ... more A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.
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