19 results
Search Results
2. Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'.
- Author
-
Znojil, Miloslav
- Subjects
QUANTUM theory ,HILBERT space ,OPEN-ended questions ,ALGEBRA - Abstract
In Alase et al (2022 J. Phys. A: Math. Theor. 55 244003), Alase et al wrote that 'the constraint of quasi-Hermiticity on observables' is not 'sufficient to extend the standard quantum theory' because 'such a system is equivalent to a standard quantum system.' Three addenda elucidating the current state of the art are found necessary. The first one concerns the project: in the related literature the original 'aim of extending standard quantum theory' has already been abandoned shortly after its formulation. The second comment concerns the method, viz., the study in 'the framework of general probabilistic theories' (GPT). It is noticed that a few other, mathematically consistent GPT-like theories are available. The authors do not mention, in particular, the progress achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras. We add that this approach already found its advanced realistic applications in the quasi-Hermitian models using the unbounded operators of observables acting in the infinite-dimensional Hilbert spaces. Thirdly, the 'intriguing open question' about 'what possible constraints, if any, could lead to such a meaningful extension' (in the future) is given an immediate tentative answer: the possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of the observables, provided only that one has in mind its recently developed non-stationary version. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. Comment on 'Twisted bialgebroids versus bialgebroids from a Drinfeld twist'.
- Author
-
Škoda, Zoran and Stojić, Martina
- Subjects
NONCOMMUTATIVE algebras ,HOPF algebras ,ALGEBRA ,MODEL theory ,MATHEMATICS - Abstract
A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where H
F and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
4. Reply to Comment on 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras'.
- Author
-
Jacobsen, Jesper Lykke
- Subjects
EIGENVALUES ,ALGEBRA ,MATHEMATICS - Abstract
The authors replies to the comment made by Yang and Zhou (2024 J. Phys. A: Math. Theor.) on his 2015 paper entitled 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras' (Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces.
- Author
-
Marquette, Ian, Zhang, Junze, and Zhang, Yao-Zhong
- Subjects
ALGEBRA - Abstract
Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics.
- Author
-
Stoilova, N I and der Jeugt, J Van
- Subjects
LIE superalgebras ,ALGEBRA ,PARASOCIAL relationships ,YANG-Baxter equation - Abstract
A Z 2 × Z 2 -graded Lie superalgebra g is a Z 2 × Z 2 -graded algebra with a bracket [ [ ⋅ , ⋅ ] ] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z 2 × Z 2 -graded Lie superalgebra o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z 2 × Z 2 -graded matrix. Since the orthosymplectic Lie superalgebra o s p (2 m + 1 | 2 n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) . Some special cases are of particular interest, even when one is dealing with parabosons only. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. Integrable boundary conditions for staggered vertex models.
- Author
-
Frahm, Holger and Gehrmann, Sascha
- Subjects
HAMILTONIAN systems ,ALGEBRA ,TRANSFER matrix - Abstract
Yang–Baxter integrable vertex models with a generic Z 2 -staggering can be expressed in terms of composite R -matrices given in terms of the elementary R -matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices K ± . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang–Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. Classification of the reducible Verma modules over the Jacobi algebra G2.
- Author
-
Aizawa, N, Dobrev, V K, and Doi, S
- Subjects
ALGEBRA ,REPRESENTATIONS of algebras ,LIE algebras ,CLASSIFICATION ,INDECOMPOSABLE modules ,JACOBI polynomials - Abstract
In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra G 2 . We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
9. Dynamical symmetry algebras of two superintegrable two-dimensional systems.
- Author
-
Marquette, I and Quesne, C
- Subjects
ALGEBRA ,SYMMETRY ,POLYNOMIALS ,INTEGRALS - Abstract
A complete classification of 2D quadratically superintegrable systems with scalar potential on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E
8 and E10 from such a classification by a new approach based on extra sets of ladder operators. They correspond in fact to two of those equivalence classes. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
10. Comment on 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras'.
- Author
-
Yang, Yi and Zhou, Shuigeng
- Subjects
ALGEBRA ,TIME complexity ,EIGENVALUES ,TRANSFER matrix ,EXTRAPOLATION ,RANDOM graphs - Abstract
We present an algorithm to compute the exact critical probability h (n) for an n × ∞ helical square lattice with random and independent site occupancy. The algorithm has time complexity O (n 2 c n) and space complexity O (c n) with c = 2.7459... and allows us to compute h (n) up to n = 24. Since the extrapolation result of h (n) is inconsistent with the current best estimation of p
c , we also compute and extend the exact critical probability p c (n) for an n × ∞ cylindrical square lattice to n = 24. Our calculation shows that the current best result of p c = 0.592 746 050 792 10 (2) by Jacobsen (2015 J. Phys. A: Math. Theor. 48 454003) is incorrect and the corrected value should be 0.592 746 050 7896 (1) . [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
11. Triangular solutions to the reflection equation for Uq(sln^).
- Author
-
Kolyaskin, Dmitry and Mangazeev, Vladimir V
- Subjects
YANG-Baxter equation ,AFFINE algebraic groups ,EQUATIONS ,ALGEBRA - Abstract
We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane.
- Author
-
Marquette, Ian and Parr, Anthony
- Subjects
SPACES of constant curvature ,FINITE integration technique ,ALGEBRA ,MATHEMATICS ,DIFFERENTIAL operators - Abstract
We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Tetrahedron equation and quantum cluster algebras.
- Author
-
Inoue, Rei, Kuniba, Atsuo, and Terashima, Yuji
- Subjects
CLUSTER algebras ,YANG-Baxter equation ,EQUATIONS ,TETRAHEDRA ,ALGEBRA - Abstract
We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y -variables in terms q -Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes.
- Author
-
Bernard, Pierre-Antoine, Mann, Zachary, Parez, Gilles, and Vinet, Luc
- Subjects
FERMIONS ,CUBES ,VERTEX operator algebras ,REPRESENTATION theory ,QUANTUM entanglement ,ALGEBRA - Abstract
This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and s o (3) − 1 . The entanglement Hamiltonian and its relation to Heun operators are also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. Integral preserving discretization of 2D Toda lattices.
- Author
-
Smirnov, Sergey V
- Subjects
GENERALIZED integrals ,LIE algebras ,CHARACTERISTIC functions ,INTEGRALS ,BANACH lattices ,ALGEBRA - Abstract
There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case (that is, they admit complete families of characteristic integrals in both directions). We prove that semi-discrete analogs of Toda lattices associated to the Cartan matrices of all simple Lie algebras are Darboux integrable. By examining the properties of Habibullin's discretization we show that if a function is a characteristic integral for a generalized Toda lattice in the continuous case, then the same function is a characteristic integral in the semi-discrete case as well. We consider characteristic algebras of such integral-preserving discretizations of Toda lattices to prove the existence of complete families of characteristic integrals in the second direction. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
16. On S -expansions and other transformations of Lie algebras.
- Author
-
Alvarez, M A and Rosales-Gómez, J
- Subjects
LIE algebras ,ALGEBRA - Abstract
The aim of this work is to study the relation between S -expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S -expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S -expanded algebras are preserved by the S -expansion procedure. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
17. Runge–Lenz vector as a 3d projection of SO(4) moment map in R4×R4 phase space.
- Author
-
Ikemori, Hitoshi, Kitakado, Shinsaku, Matsui, Yoshimitsu, and Sato, Toshiro
- Subjects
GRAPHICAL projection ,KEPLER problem ,PARTICLE motion ,SYMPLECTIC geometry ,ALGEBRA ,PHASE space - Abstract
We show, using the methods of geometric algebra, that Runge–Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, Runge–Lenz vector is a consequence of geometric symmetry of R 4 × R 4 phase space. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
18. Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series so2n+1.
- Author
-
Nazarov, Anton, Nikitin, Pavel, and Postnova, Olga
- Subjects
LIE algebras ,CENTRAL limit theorem ,PROBABILITY measures ,TENSOR products ,ALGEBRA - Abstract
We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of s o 2 n + 1 . The irreducible representations correspond to the generalized Young diagrams. With respect to this measure the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the generalized Young diagram when the tensor power N and the rank n of the algebra tend to infinity with N / n fixed. We derive an explicit formula for the limit shape and prove convergence to it in probability. We prove central limit theorem for global fluctuations around the limit shape. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
19. Regularization of δ ′ potential in general case of deformed space with minimal length.
- Author
-
Samar, M I and Tkachuk, V M
- Subjects
FUNCTION spaces ,POTENTIAL energy ,ALGEBRA ,EIGENFUNCTIONS - Abstract
In the general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the δ ′(x) potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for δ ′(x) and δ (x) â' δ ′(x) potentials in deformed space with arbitrary function of deformation. The energy spectrum for different partial cases of deformation function is analysed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.