It is shown that Uhlmann's parallel transport of purifications along a path of mixed states repre... more It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well-known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just Pancharatnam transport of quaternionic spinors.
By using the modified Newton-Sabatier inversion method, a model independent local potential is pr... more By using the modified Newton-Sabatier inversion method, a model independent local potential is presented which describes elastic scattering of two 12C nuclei at c.m. energy of E-14.1 MeV. Only the physical partial waves of even angular momenta are involved in the inversion procedure. The potential obtained is in general smooth and real in the outer (Coulombic) and inner (fusion) regions, between which it becomes complex showing characteristic oscillations responsible for reactions and nonlocal effects.
The modified Newton-Sabatier method is applied to invert electron-atom scattering phase-shifts in... more The modified Newton-Sabatier method is applied to invert electron-atom scattering phase-shifts into effective potentials. The long range (–/2r4) tail of the polarization interaction requires a special treatment. Examples involving inversion of synthetic as well as experimental phase-shifts are presented.
The quantization of the chaotic geodesic motion on Riemann surfaces Σ g,κ of constant negative cu... more The quantization of the chaotic geodesic motion on Riemann surfaces Σ g,κ of constant negative curvature with genus g and a finite number of points κ infinitely far away (cusps) describing scattering channels is investigated. It is shown that terms in Selberg's trace formula describing scattering states can be expressed in terms of a regularized time delay. Poles in these quantities give rise to resonances reflecting the chaos of the underlying classical dynamics. Illustrative examples for a class of Σ g,2 surfaces are given.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1988
The Kohn and Schwinger variational methods are applied to electron scattering by local spherical ... more The Kohn and Schwinger variational methods are applied to electron scattering by local spherical potentials and the numerical results are compared with those of Ladanyi's method (1970) based on the standard form of the Schrodinger equation. Both the stability and the convergence of the results are carefully analysed and, in the case of Ladanyi's method, the measure of the error of the approximate solution is investigated as well. The analysis implies that Ladanyi's method is competitive in comparison with the standard variational procedures.
Journal of Physics A: Mathematical and General, 1994
Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group... more Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately Sd (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.
Journal of Physics A: Mathematical and General, 1995
Using a first-order Casimir operator calculated in a non-standard realization for the so(3,1) alg... more Using a first-order Casimir operator calculated in a non-standard realization for the so(3,1) algebra, we obtain a one-dimensional scattering problem with LS-type interaction terms. It is shown that for this realization the square of this operator can be expressed in terms of the usual quadratic Casimir. Due to this constraint the scattering states are completely specified by restricting the possible set of eigenvalues accordingly. The results show that the use of extra Casimir operators can provide additional insight into the group theoretical structure of the scattering problem. A generalization for the so(2n-1,1), n>2 case is also given. The underlying supersymmetry of the resulting Schrodinger equations is pointed out. The supersymmetric charge operators are related to our first-order Casimir operators.
International Journal of Geometric Methods in Modern Physics, 2010
It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic t... more It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic to PG(5, 2). Since the GQ(2, 4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2, 2)s, the 63 points of PG(5, 2) split into two families; 27 being represented by perp-sets and 36 by GQ(2, 2)s. The 651 lines of PG(5, 2) are found to fall into four distinct classes: in particular, 45 of them feature only perp-sets, 216 comprise two perp-sets and one GQ(2, 2), 270 consist of one perp-set and two GQ(2, 2)s and the remaining 120 are composed solely of GQ(2, 2)s, according to the intersection of two distinct hyperplanes determining the (Veldkamp) line is, respectively, a line, an ovoid, a perp-set and a grid (i.e. GQ(2, 1)) of a copy of GQ(2, 2). A direct "by-hand" derivation of the above-listed properties is followed by their heuristic justification based on the properties of an elliptic quadric of PG(5, 2) and complemented by a proof employing ...
Employing the fact that the geometry of the N-qubit (N≥2) Pauli group is embodied in the structur... more Employing the fact that the geometry of the N-qubit (N≥2) Pauli group is embodied in the structure of the symplectic polar space W(2N−1,2) and using properties of the Lagrangian Grassmannian LGr(N,2N) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N-qubit Pauli group and a certain subset of elements of the 2N−1-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N=3 (also addressed recently by Lévay, Planat and Saniga (JHEP 09 (2013) 037)) and N=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2N−1,2) of the 2N−1-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋅⋅⋅×SL(2,2)⋊SN, to decompose π––(LGr(N,2N)) into non-equivalent orbits. This leads to a partition of LGr(N,2N) into distinguished classes that can be labeled by elements of the above-mentione...
The complete quantum metric of a parametrized quantum system has a real part (usually known as th... more The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the SU(1, 1) group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the SU(1, 1) group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.
We study the thermal state of a two dimensional conformal field theory which is dual to the stati... more We study the thermal state of a two dimensional conformal field theory which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into N subsystems we show that there is an underlying CN−1 cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed N is the cyclohedron CN−1. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov Y-system of CN−1 type. The boundary condition for such an Y-system is featuring the entropy of the BTZ black hole.
A geometrical description of three qubit entanglement is given. A part of the transformations cor... more A geometrical description of three qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric Q a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of Q. An invariant vanishing for the GHZ class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.
Abstract: It is shown that the E 6(6) symmetric entropy formula describing black holes and black ... more Abstract: It is shown that the E 6(6) symmetric entropy formula describing black holes and black strings in D = 5 is intimately tied to the geometry of the generalized quadrangle GQ(2, 4) with automorphism group the Weyl group W(E6). The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ(2, 4). Different truncations with 15, 11 and 9 charges are represented by three distinguished subconfigurations of GQ(2, 4), well-known to finite geometers; these are the “doily ” (i. e. GQ(2, 2)) with 15, the “perp-set ” of a point with 11, and the “grid ” (i. e. GQ(2, 1)) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non-commutative labelling for the points of GQ(2, 4). For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares — objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ...
It is shown that Uhlmann's parallel transport of purifications along a path of mixed states repre... more It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well-known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just Pancharatnam transport of quaternionic spinors.
By using the modified Newton-Sabatier inversion method, a model independent local potential is pr... more By using the modified Newton-Sabatier inversion method, a model independent local potential is presented which describes elastic scattering of two 12C nuclei at c.m. energy of E-14.1 MeV. Only the physical partial waves of even angular momenta are involved in the inversion procedure. The potential obtained is in general smooth and real in the outer (Coulombic) and inner (fusion) regions, between which it becomes complex showing characteristic oscillations responsible for reactions and nonlocal effects.
The modified Newton-Sabatier method is applied to invert electron-atom scattering phase-shifts in... more The modified Newton-Sabatier method is applied to invert electron-atom scattering phase-shifts into effective potentials. The long range (–/2r4) tail of the polarization interaction requires a special treatment. Examples involving inversion of synthetic as well as experimental phase-shifts are presented.
The quantization of the chaotic geodesic motion on Riemann surfaces Σ g,κ of constant negative cu... more The quantization of the chaotic geodesic motion on Riemann surfaces Σ g,κ of constant negative curvature with genus g and a finite number of points κ infinitely far away (cusps) describing scattering channels is investigated. It is shown that terms in Selberg's trace formula describing scattering states can be expressed in terms of a regularized time delay. Poles in these quantities give rise to resonances reflecting the chaos of the underlying classical dynamics. Illustrative examples for a class of Σ g,2 surfaces are given.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1988
The Kohn and Schwinger variational methods are applied to electron scattering by local spherical ... more The Kohn and Schwinger variational methods are applied to electron scattering by local spherical potentials and the numerical results are compared with those of Ladanyi's method (1970) based on the standard form of the Schrodinger equation. Both the stability and the convergence of the results are carefully analysed and, in the case of Ladanyi's method, the measure of the error of the approximate solution is investigated as well. The analysis implies that Ladanyi's method is competitive in comparison with the standard variational procedures.
Journal of Physics A: Mathematical and General, 1994
Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group... more Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately Sd (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.
Journal of Physics A: Mathematical and General, 1995
Using a first-order Casimir operator calculated in a non-standard realization for the so(3,1) alg... more Using a first-order Casimir operator calculated in a non-standard realization for the so(3,1) algebra, we obtain a one-dimensional scattering problem with LS-type interaction terms. It is shown that for this realization the square of this operator can be expressed in terms of the usual quadratic Casimir. Due to this constraint the scattering states are completely specified by restricting the possible set of eigenvalues accordingly. The results show that the use of extra Casimir operators can provide additional insight into the group theoretical structure of the scattering problem. A generalization for the so(2n-1,1), n>2 case is also given. The underlying supersymmetry of the resulting Schrodinger equations is pointed out. The supersymmetric charge operators are related to our first-order Casimir operators.
International Journal of Geometric Methods in Modern Physics, 2010
It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic t... more It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic to PG(5, 2). Since the GQ(2, 4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2, 2)s, the 63 points of PG(5, 2) split into two families; 27 being represented by perp-sets and 36 by GQ(2, 2)s. The 651 lines of PG(5, 2) are found to fall into four distinct classes: in particular, 45 of them feature only perp-sets, 216 comprise two perp-sets and one GQ(2, 2), 270 consist of one perp-set and two GQ(2, 2)s and the remaining 120 are composed solely of GQ(2, 2)s, according to the intersection of two distinct hyperplanes determining the (Veldkamp) line is, respectively, a line, an ovoid, a perp-set and a grid (i.e. GQ(2, 1)) of a copy of GQ(2, 2). A direct "by-hand" derivation of the above-listed properties is followed by their heuristic justification based on the properties of an elliptic quadric of PG(5, 2) and complemented by a proof employing ...
Employing the fact that the geometry of the N-qubit (N≥2) Pauli group is embodied in the structur... more Employing the fact that the geometry of the N-qubit (N≥2) Pauli group is embodied in the structure of the symplectic polar space W(2N−1,2) and using properties of the Lagrangian Grassmannian LGr(N,2N) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N-qubit Pauli group and a certain subset of elements of the 2N−1-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N=3 (also addressed recently by Lévay, Planat and Saniga (JHEP 09 (2013) 037)) and N=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2N−1,2) of the 2N−1-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋅⋅⋅×SL(2,2)⋊SN, to decompose π––(LGr(N,2N)) into non-equivalent orbits. This leads to a partition of LGr(N,2N) into distinguished classes that can be labeled by elements of the above-mentione...
The complete quantum metric of a parametrized quantum system has a real part (usually known as th... more The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the SU(1, 1) group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the SU(1, 1) group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.
We study the thermal state of a two dimensional conformal field theory which is dual to the stati... more We study the thermal state of a two dimensional conformal field theory which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into N subsystems we show that there is an underlying CN−1 cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed N is the cyclohedron CN−1. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov Y-system of CN−1 type. The boundary condition for such an Y-system is featuring the entropy of the BTZ black hole.
A geometrical description of three qubit entanglement is given. A part of the transformations cor... more A geometrical description of three qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric Q a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of Q. An invariant vanishing for the GHZ class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.
Abstract: It is shown that the E 6(6) symmetric entropy formula describing black holes and black ... more Abstract: It is shown that the E 6(6) symmetric entropy formula describing black holes and black strings in D = 5 is intimately tied to the geometry of the generalized quadrangle GQ(2, 4) with automorphism group the Weyl group W(E6). The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ(2, 4). Different truncations with 15, 11 and 9 charges are represented by three distinguished subconfigurations of GQ(2, 4), well-known to finite geometers; these are the “doily ” (i. e. GQ(2, 2)) with 15, the “perp-set ” of a point with 11, and the “grid ” (i. e. GQ(2, 1)) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non-commutative labelling for the points of GQ(2, 4). For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares — objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ...
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Papers by Péter Lévay