Generalized Heisenberg Algebra Research Articles

Minimum measurable length inspired thermodynamic characteristics of the Schwarzschild black hole surrounded by the cloud of strings and quintessence

We explore the thermodynamic properties of the Schwarzschild black hole surrounded by quintessence matter and a cloud of strings within the framework of the generalized Heisenberg algebra. We provide precise computations of temperature, heat capacity, and entropy associated with this black hole and investigate the impacts of various deformations of the Heisenberg algebra on these thermodynamic parameters. The findings are compared with those for the usual Schwarzschild black hole. Additionally, the behavior of the isotherm related to this black hole is studied. It is anticipated that remnants will hold the key to solving the information loss problem. It is observed that different deformations of the Heisenberg algebra, which introduce minimum measurable length, maximum measurable momenta, or both simultaneously, could lead to the existence of a black hole remnant. Furthermore, it is noted that both exotic fields, cloud of strings and quintessence contribute to the quantum gravity-corrected entropy of the composite system.

Abstract ladder operators for non self-adjoint Hamiltonians, with applications

Ladder operators are useful, if not essential, in the analysis of some given physical system since they can be used to find easily eigenvalues and eigenvectors of its Hamiltonian. In this paper we extend our previous results on abstract ladder operators considering in many details what happens if the Hamiltonian of the system is not self-adjoint. Among other results, we give an existence criterion for coherent states constructed as eigenstates of our lowering operators. In the second part of the paper we discuss two different examples of our framework: pseudo-quons and a deformed generalized Heisenberg algebra. Incidentally, and interestingly enough, we show that pseudo-quons can be used to diagonalize an oscillator-like Hamiltonian written in terms of (non self-adjoint) position and momentum operators which obey a deformed commutation rule of the kind often considered in minimal length quantum mechanics.

Turning graphene into a lab for noncommutativity

It was recently shown that, taking into account the granular structure of graphene lattice, the Dirac-like dynamics of its quasiparticles resists beyond the lowest energy approximation. This can be described in terms of new phase-space variables, (X→,P→), that enjoy generalized Heisenberg algebras. In this letter, we add to that picture the important case of noncommuting X→, for which [Xi,Xj]=iθij and we find that θij=ℓ2ϵij, with ℓ the lattice spacing. We close by giving both the general recipe and a possible specific kinematic setup for the practical implementation of this approach to test noncommutative theories in tabletop analog experiments on graphene.

Quantum Heisenberg enveloping algebras at roots of unity

In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, has been studied at roots of unity. In this context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed.

Shadows of new physics on Dirac materials, analog GUPs and other amusements

We discuss here how, when higher-order effects in the parameter , related to the lattice spacing ℓ, are considered, pristine graphene, and other Dirac materials, can be used as tabletop systems where generalized commutation relations are naturally realized. Such generalized algebras of quantization, which lead to generalized versions of the Heisenberg uncertainty principle, are under intense scrutiny these days, as they could manifest a fundamental length scale of spacetime. Despite the efforts and the many intriguing results, there are no experimental signatures of any generalized uncertainty principle (GUP). Therefore, our results here, which tell how to use tabletop physical systems to test certain GUPs in analog experiments, should be of interest to practitioners of quantum gravity. We identify three different energy regimes that we call “layers”, where the physics is still of a Dirac type but within precisely described limits. The higher the energy, the more sensitive the Dirac system becomes to the effects of the lattice. Here such lattice plays the role of a discrete space where the Dirac quasi-particles live. With the goals just illustrated, we had to identify the mapping between the high-energy coordinates, X i , and the low-energy ones, x i , i.e., those measured in the lab. We then obtained three generalized Heisenberg algebras. For two of them we have the noticeable result that X i = x i , and for the third one we obtained an improvement with respect to an earlier work: the generalized coordinates expressed in terms of the standard phase space variables, X i (x, p), and higher order terms.

Three “layers” of graphene monolayer and their analog generalized uncertainty principles

We show that graphene, in its simplest form and settings, is a practical tabletop realization of the analog of exotic quantum gravity scenarios, which are speculated to lead to certain generalized Heisenberg algebras. In particular, we identify three different energy regimes (the ``layers'') where the physics is still of a pseudorelativistic (Dirac) type but more and more sensitive to the effects of the lattice. This plays here a role analog to that of a discrete space, where the Dirac quasiparticles live. This work improves and pushes further earlier results, where the physical meaning of the high energy momenta was clear, but the conjugate coordinates only had a purely abstract description. Here we find the physical meaning of the latter by identifying the mapping between the high energy coordinates and low energy ones, i.e., those measured in the lab. We then obtain two generalized Heisenberg algebras that were not noticed earlier. In these two cases, we have the striking result that the high energy coordinates just coincide with the standard ones, measured in the lab. A third generalized Heisenberg algebra is obtained, and it is an improvement of the results obtained earlier in two respects: we now have an expression of the generalized coordinates in terms of the standard phase-space variables, and we obtain higher order terms. All mentioned results clearly open the doors to tabletop experimental verifications of many generalized uncertainty principle--corrected predictions of the quantum gravity phenomenology.

Generalized Heisenberg Algebra, Realizations of the gI(N) Algebra And Applications

We introduce the generalized Heisenberg algebra appropriate for realizations of the $\mathfrak{gl}(n)$ algebra. Linear realizations of the $\mathfrak{gl}(n)$ algebra are presented and the corresponding star product, coproduct of momenta and twist are constructed. The dual realization and dual $\mathfrak{gl}(n)$ algebra are considered. Finally, we present a general realization of the $\mathfrak{gl}(n)$ algebra, the corresponding coproduct of momenta and two classes of twists. These results can be applied to physical theories on noncommutative spaces of the $\mathfrak{gl}(n)$ type.

Generalized uncertainty principle: from the harmonic oscillator to a QFT toy model

Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg uncertainty principle into the generalized uncertainty principle. In this work, we study the implications of a polynomial generalized uncertainty principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the generalized uncertainty principle.

Abstract ladder operators and their applications

We consider a rather general version of ladder operator Z used by some authors in few recent papers, [H 0, Z] = λZ for some , , and we show that several interesting results can be deduced from this formula. Then we extend it in two ways: first we replace the original equality with formula [H 0, Z] = λZ[Z †, Z], and secondly we consider [H, Z] = λZ for some , H ≠ H †. In both cases many applications are discussed. In particular we consider factorizable Hamiltonians and Hamiltonians written in terms of operators satisfying the generalized Heisenberg algebra or the pseudo-bosonic commutation relations.

Quantum generalized Heisenberg algebras and their representations

We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras (qGHA), including both the so-called generalized Heisenberg algebras and the generalized down-up algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and provide a common framework for several previously studied classes of algebras. In particular, our class includes the enveloping algebras of the Lie algebra and of the 3-dimensional Heisenberg Lie algebra, as well as its q-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the classification of finite-dimensional irreducible representations of qGHA, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain a Lie-theoretic flavor. We work over a field of arbitrary characteristic and our results are presented in a characteristic-free fashion.

Structure and isomorphisms of quantum generalized Heisenberg algebras

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter [Formula: see text] and two polynomials [Formula: see text]. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties.

Capability of Nilpotent Lie Algebras of Small Dimension

Given a nilpotent Lie algebra $L$ of dimension $\le 6$ on an arbitrary field of characteristic $\neq 2$, we show a direct method which allows us to detect the capability of $L$ via computations on the size of its nonabelian exterior square $L \wedge L$. For dimensions higher than $ 6$, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of $L \wedge L$ via the size of the Schur multiplier $M(L/Z^\wedge(L))$ of $L/Z^\wedge(L)$, where $Z^\wedge(L)$ denotes the exterior center of $L$.

Bipartite entanglement of generalized Barut–Girardello nonlinear coherent states

We investigate the entanglement of bipartite nonlinear quantum systems in the context of generalized Heisenberg algebra and generalized su(1, 1) algebra. Particularly, we examine the entanglement properties of states prepared by Barut–Girardello nonlinear coherent states of the square-well potential. Furthermore, we show that the entanglement amount of these nonlinear coherent states depends on the corresponding algebraic structure and on the characteristic function of the associated algebras. Moreover, a large class of entangled and maximally entangled states can be recovered and constructed.

Quantum credit loans

Quantum models based on the mathematics of quantum mechanics (QM) have been developed in cognitive sciences, game theory and econophysics. In this work a generalization of credit loans is introduced by using the vector space formalism of QM. Operators for the debt, amortization, interest and periodic installments are defined and its mean values in an arbitrary orthonormal basis of the vectorial space give the corresponding values at each period of the loan. Endowing the vector space of dimension M, where M is the loan duration, with a SO(M) symmetry, it is possible to rotate the eigenbasis to obtain better schedule periodic payments for the borrower, by using the rotation angles of the SO(M) transformation. Given that a rotation preserves the length of the vectors, the total amortization, debt and periodic installments are not changed. For a general description of the formalism introduced, the loan operator relations are given in terms of a generalized Heisenberg algebra, where finite dimensional representations are considered and commutative operators are defined for the specific loan types. The results obtained are an improvement of the usual financial instrument of credit because introduce several degrees of freedom through the rotation angles, which allows to select superposition states of the corresponding commutative operators that enables the borrower to tune the periodic installments in order to obtain better benefits without changing what the lender earns.

Measuring space deformation via graphene under constraints

We describe the lattice deformation in graphene under strain effect by considering the spacial-momenta coordinates do not commute. This later can be realized by introducing the star product to end up with a generalized Heisenberg algebra. Within such framework, we build a new model describing Dirac fermions interacting with an external source that is noncommutative parameter κ dependent. The solutions of energy spectrum are showing Landau levels in similar way to the case of a real magnetic field applied to graphene. We show that some strain configurations can be used to explicitly evaluate κ and then offer a piste toward its measurement.

Phase-space behavior of physical nonlinear coherent states

In this paper, we construct the generalized su(1, 1) nonlinear coherent states for particular physical systems. We study the behavior of the Wigner function for coherent states built from differential operators that satisfy either a generalized Heisenberg algebra or a generalized su(1, 1) nonlinear algebra. The systems studied are the harmonic oscillator and those ones associated with Pöschl-Teller potentials. The results show that the Wigner function for the associated nonlinear coherent states is positive for small values of their convergence radius. Furthermore, the Wigner function for these coherent states depends on the corresponding algebraic structure.

Generalized su(1,1) algebra and the construction of nonlinear coherent states for Pöschl-Teller potential

We introduce a generalization structure of the su(1,1) algebra which depends on a function of one generator of the algebra, f(H). Following the same ideas developed to the generalized Heisenberg algebra (GHA) and to the generalized su(2), we show that a symmetry is present in the sequence of eigenvalues of one generator of the algebra. Then, we construct the Barut-Girardello coherent states associated with the generalized su(1,1) algebra for a particle in a Pöschl-Teller potential. Furthermore, we compare the time evolution of the uncertainty relation of the constructed coherent states with that of GHA coherent states. The generalized su(1,1) coherent states are very localized.

Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz, and Poincaré algebras and their dual extensions

We introduce the generalized Heisenberg algebra Hn and construct realizations of the orthogonal and Lorentz algebras by a formal power series in a semicompletion of Hn. The obtained realizations are given in terms of the generating function for the Bernoulli numbers. We also introduce an extension of the orthogonal and Lorentz algebras by quantum angles and study realizations of the extended algebras in Hn. Furthermore, we show that by extending the generalized Heisenberg algebra Hn, one can also obtain realizations of the Poincaré algebra and its extension by quantum angles.

Generalized Heisenberg Algebras: periodicity and finite representation

We consider the Generalized Heisenberg Algebra (GHA) and the deformed GHA with finite representations. We provide the necessary restriction on the characteristic function of the periodic GHA and that of the periodic deformed GHA and study particular examples. Then, we give a deformation of GHA raising operator of the Morse system and conclude that this system can be described by a periodic deformed GHA as it can be deduced from a nilpotent GHA.

Coherent States of the Generalized Heisenberg Algebra with $$\varvec{k-1}$$ Isolated States

In this paper we construct the generalized Heisenberg algebra with \( k-1 \) isolated states. We use this algebra to construct three types of coherent states. Finally we discuss the non-classical properties related to the coherent states.