Bosonic Algebra Research Articles

Q-Deformed quantum mechanics related to the Tamm-Dancoff oscillator algebra and some physical applications

In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.

Schottky cohomology for rank two bosonic vertex operator algebra

Using the Schottky procedure of forming a genus [Formula: see text] Riemann surface by attaching handles to the Riemann sphere, we construct coboundary operators and corresponding cohomology for the double complexes of rational functions associated to a particular example of the rank two bosonic vertex operator algebra [Formula: see text].

Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators

In this paper we use the canonical complex structure \mathbb{J}𝕁 on \mathbb{R}^{2n}ℝ2n to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra \mathfrak{su}(1,2)𝔰𝔲(1,2) which leads to the Howe dual pair (U(n),\mathfrak{su}(1,2))(U(n),𝔰𝔲(1,2)).

Bosonic Ghostbusting: The Bosonic Ghost Vertex Algebra Admits a Logarithmic Module Category with Rigid Fusion

The rank 1 bosonic ghost vertex algebra, also known as the beta gamma ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor C_2-cofinite. We identify a module category, denoted category mathscr {F}, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category mathscr {F} has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as C_2-cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors’ knowledge this is the first example of a proof of rigidity for a logarithmic non-C_2-cofinite vertex algebra.

Deformed Boson Algebra with Parity Operator and Non-classical Properties of Coherent States

In this paper we introduce the deformed boson algebra where the even number states as well as the odd number states are deformed. Based on the set of observables associated with the quantum state, we assume that photon obeys this deformed algebra. We construct the coherent states and investigate the non-classical properties of the coherent states such as photon statistics, Mandel parameter, and quantum fluctuations of quadrature.

Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd [Formula: see text] describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number [Formula: see text]. In the limit [Formula: see text], Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

Multi-dimensional q-deformed bosonic Newton oscillators and the related q-calculus, q-coherent states, and Hermite q-polynomials

The multi-dimensional q-deformed bosonic Newton oscillator algebra with SU(d)-symmetry is considered. In this framework, we first introduce some new properties concerning the q-deformed calculus related to the algebra, and we then discuss possible consequences of applying these deformed oscillators in some quantum optical issues such as in the construction of coherent states and their effects on the photon statistics. Second, we investigate the role of q-deformation on both the energy levels and the wave functions of the bosonic Newton oscillators by constructing the q-deformed Hermite polynomials. The results obtained in this work might have some implications for studies on quantum information based technologies such as in photonic quantum computing.

Q-Deformed Coherent States for q-Deformed Photon by Using the Tsallis’s q-Deformed Exponential Function in the Non-Extensive Thermodynamics

In this manuscript we consider q-deformed boson algebra by using the Tsallis’s q-deformed exponential which arises in the non-extensive thermodynamics. Using this we construct q-deformed coherent states and even and odd q-deformed coherent states. We investigate non-classical properties of these q-deformed coherent states such as photon distribution, super-/sub-Poissonian structure and bunching or anti-bunching effect.

Rotation-time symmetry in bosonic systems and the existence of exceptional points in the absence of {mathscr{PT}} symmetry

We study symmetries of open bosonic systems in the presence of laser pumping. Non-Hermitian Hamiltonians describing these systems can be parity-time ({{mathscr{PT}}}) symmetric in special cases only. Systems exhibiting this symmetry are characterised by real-valued energy spectra and can display exceptional points, where a symmetry-breaking transition occurs. We demonstrate that there is a more general type of symmetry, i.e., rotation-time ({mathscr{RT}}) symmetry. We observe that {mathscr{RT}}-symmetric non-Hermitian Hamiltonians exhibit real-valued energy spectra which can be made singular by symmetry breaking. To calculate the spectra of the studied bosonic non-diagonalisable Hamiltonians we apply diagonalisation methods based on bosonic algebra. Finally, we list a versatile set rules allowing to immediately identifying or constructing {mathscr{RT}}-symmetric Hamiltonians. We believe that our results on the {mathscr{RT}}-symmetric class of bosonic systems and their spectral singularities can lead to new applications inspired by those of the {{mathscr{PT}}}-symmetric systems.

A new boson realization of fusion polynomial algebras in non-Hermitian quantum mechanics: γ-deformed generators, -symmetry in Fock space and Higgs algebra

A γ-deformed version of algebra with non-Hermitian generators has been obtained from a bi-orthogonal system of vectors in C 2 . The related Jordan–Schwinger map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type which eventually leads to several multi-boson non-Hermitian Hamiltonians. Finally, the notion of global and partial -symmetry have been introduced in a typical Fock space setting. The possibility of -symmetry breaking is also discussed. The deformation parameter γ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration. The symmetry behavior has been conceptualized in terms of a couple of toy models.

Symmetry reduction of tensor networks in many-body theory

The ongoing progress in (nuclear) many-body theory is accompanied by an ever-rising increase in complexity of the underlying formalisms used to solve the stationary Schrödinger equation. The associated working equations at play in state-of-the-art ab initio nuclear many-body methods can be analytically reduced with respect to angular-momentum, i.e. SU(2), quantum numbers whenever they are effectively employed in a symmetry-restricted context. The corresponding procedure constitutes a tedious and error-prone but yet an integral part of the implementation of those many-body frameworks. Indeed, this symmetry reduction is a key step to advance modern simulations to higher accuracy since the use of symmetry-adapted tensors can decrease the computational complexity by orders of magnitude. While attempts have been made in the past to automate the (anti-) commutation rules linked to Fermionic and Bosonic algebras at play in the derivation of the working equations, there is no systematic account to achieve the same goal for their symmetry reduction. In this work, the first version of an automated tool performing graph-theory-based angular-momentum reduction is presented. Taking the symmetry-unrestricted expressions of a generic tensor network as an input, the code provides their angular-momentum-reduced form in an error-safe way in a matter of seconds. Several state-of-the-art many-body methods serve as examples to demonstrate the generality of the approach and to highlight the potential impact on the many-body community.

Finite W-superalgebras and quadratic spacetime supersymmetries

We consider Lie superalgebras under constraints of Hamiltonian reduction, yielding finite W-superalgebras which provide candidates for quadratic spacetime superalgebras. These have an undeformed bosonic symmetry algebra (even generators) graded by a fermionic sector (supersymmetry generators) with anticommutator brackets which are quadratic in the even generators. We analyze the reduction of several Lie superalgebras of type gl(M|N) or osp(M|2N) at the classical (Poisson bracket) level, and also establish their quantum (Lie bracket) equivalents. Purely bosonic extensions are also considered. As a special case we recover a recently identified quadratic superconformal algebra, certain of whose unitary irreducible massless representations (in four dimensions) are ‘zero-step’ multiplets, with no attendant superpartners. Other cases studied include a six dimensional quadratic superconformal algebra with vectorial odd generators, and a variant quadratic superalgebra with undeformed osp(1|2N) singleton supersymmetry, and a triplet of spinorial supercharges.

The Grassmannian VOA

We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT mathfrak{u} (M + N )k /( mathfrak{u} (M )k× mathfrak{u} (N )k ). This VOA serves as a basic building block for a large class of cosets and generalizes the {mathcal{W}}_{infty } algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the mathcal{N} = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.

Computation of cohomology of Lie conformal and Poisson vertex algebras

We develop methods for computation of Poisson vertex algebra cohomology. This cohomology is computed for the free bosonic and fermionic Poisson vertex (super)algebras, as well as for the universal affine and Virasoro Poisson vertex algebras. We establish finite dimensionality of this cohomology for conformal Poisson vertex (super)algebras that are finitely and freely generated by elements of positive conformal weight.

Quantum Markov chains: A unification approach

In this paper, we study a unified approach for quantum Markov chains (QMCs). A new quantum Markov property that generalizes the old one, is discussed. We introduce Markov states and chains on general local algebras, possessing a generic algebraic property. We stress that this kind of algebras includes both Boson and Fermi algebras. Our main results concern two reconstruction theorems for quantum Markov chains and for quantum Markov states. Namely, we illustrate the results through examples.

Blackbody radiation and Debye model based on q-deformed bosonic Newton oscillator algebra

In this paper, we consider the [Formula: see text]-mode [Formula: see text]-deformed bosonic Newton oscillators. Regarding photon and phonon as the particles obeying the multi-mode [Formula: see text]-deformed bosonic Newton algebra, we discuss the blackbody radiation and Debye model.

φ-deformed boson algebra based on φ-deformed addition and non-classical properties of φ-deformed coherent states

In this paper the general deformed boson algebra called a -deformed boson algebra is discussed based on the -addition and -subtraction. The -derivative and the realization of -deformed boson algebra are also discussed. Besides, -deformed su(2) algebra ( algebra) and supersymmetric quantum mechanics are also discussed. Seven possible -deformed boson algebras are discussed and -coherent states and non-classical properties are investigated.

Possible non-additive entropy based on the $$\alpha $$-deformed addition

We reexamine the non-additive entropy which was first proposed by Tsallis [1] where this deformed entropy (q-entropy) is related to a special map. We consider a new map giving the \(\alpha \)-additive property. Based on this, we propose a new type of non-additive entropy which we call \(\alpha \)-entropy and derive the \(\alpha \)-deformed Boltzmann factor and \(\alpha \)-deformed free energy. As an example, we discuss the black body radiation in the \(\alpha \)-deformed thermodynamics where we regard photon as a particle obeying \(\alpha \)-deformed boson algebra.

A correct way to determine the mean occupation numbers for the q-deformed boson algebras

In this paper we point out some serious problems in having derived the deformed occupation numbers for three types of q -boson algebras: Arik-Coon, Macfarlane-Biedenharn, and Tamm-Dancoff. Instead we present the correct way to determine the deformed occupation numbers for three types of q -boson algebras.

Modified multi-dimensional q-deformed Newton oscillators: Algebra, interpolating statistics and thermodynamics

In this work, we present a new algebraic modelby constructing the modified multi-dimensional q-deformed bosonic and fermionic Newton oscillator algebras. This construction leads effectively to describe interpolating statistics, which can be used to approach the properties of quasi-particle excitations occurring in many-body interacting quantum systems. It is shown that the model algebras are endowed with the SUφ(d)- and SUθ(d)-symmetries, where φ and θ are real parameters describing the special phase shifts between two different modes of the models. Particular emphasis is given to the two-dimensional case, which reveals a suitable framework for analyzing anyonic behavior of the models. Furthermore, we discuss the effects of deformation on the general thermodynamical and statistical properties of gas models of these interpolating statistics particles such as the q-deformed statistical distribution function and the equation of state in two and three dimensions. Finally, we concisely point out other possible physical applications of the present deformed (quasi)particle models.