A New Boson realization of Fusion Polynomial Algebras in Non-Hermitian Quantum Mechanics : $γ$-deformed $su(2)$ generators, Partial $\mathcal{PT}$-symmetry and Higgs algebra
A $γ$-deformed version of $su(2)$ algebra with non-hermitian generators has been obtained from a bi-orthogonal system of vectors in $\bf{C^2}$. The related Jordan-Schwinger(J-S) 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. Finally the notion of partial $\mathcal{PT}$ symmetry has been introduced as characteristic feature of some operators as well as their eigenfunctions. The possibility of partial $\mathcal{PT}$-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.
A non-Hermitian generalized oscillator model, generally known as the Swanson model, has been studied in the framework of R-deformed Heisenberg algebra. The non-Hermitian Hamiltonian is diagonalized by generalized Bogoliubov transformation. A set of deformed creation annihilation operators is introduced whose algebra shows that the transformed Hamiltonian has conformal symmetry. The spectrum is obtained using algebraic technique. The superconformal structure of the system is also worked out in detail. An anomaly related to the spectrum of the Hermitian counterpart of the non-Hermitian Hamiltoni…
This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and $W$-representation. We apply it to prove the formula for the Jack functions averages - appropriate analogue of characters for the $β$-deformed Hermitian Gaussian matrix model. We also sketch the construction of $W$-operators from Calogero-Ruijsenaars H…
We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins-Szèkelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau.
Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.
In this paper, we provide a necessary and sufficient condition for the solvability of the supercritical deformed Hermitian-Yang-Mills equation using integrals on subvarieties. This result confirms the mirror version of the Thomas-Yau conjecture about special Lagrangian submanifolds on Calabi-Yau manifolds.
We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hatθ \in (\fracπ{2},\frac{3π}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Ampère equation with mixed sign coefficients.
Based on the usual Fedosov construction of star products for a symplectic manifold M we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle this gives a Morita equivalence bimodule where the relation between the characteristic classes of the Morita equivalent star products can be found very easily in this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian …
We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analog of this construction for the most general Polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.
Eigenvalues and eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The deformed spectrum is annular-shaped, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that correspond to the two-arc and wings observed by Hatano and Nelson in deformed Hermitian models, and are explained in terms of localization of eigenstates via a spectral duality and the Argument principle.
Nonrelativistic quantum mechanics and conformal quantum mechanics are deformed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the so-called "unfolded formalism" discussed in previous works. A Hopf algebra structure, compatible with the physical interpretation of the coproduct, is introduced for the Universal Enveloping Algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained amon…
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog and we extend Daskaloyannis' construction in obtained in context of quadratic algebras and we obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite dimensional unitary irreductible representations of quartic algebras and obtain algebraically …
We review deformed quantum phase spaces and their realizations in terms of undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed quantum phase spaces and Snyder type spaces are considered. Examples of linear realizations of the $κ$-Minkowski spacetime are elaborated. Finally, some new results on quadratic deformations of quantum phase spaces and a generalization of Yang and triply special relativity models are presented.
General realizations, star products and plane waves for kappa-Minkowski spacetime are considered. Systematic construction of general hermitian realization is presented, with special emphasis on noncommutative plane waves and hermitian star product. Few examples are elaborated and possible physical applications are mentioned.
We investigate the observational consequences of the light-like deformations of the Poincaré algebra induced by the jordanian and the extended jordanian classes of Drinfel'd twists. Twist-deformed generators belonging to a Universal Enveloping Algebra close nonlinear algebras. In some cases the nonlinear algebra is responsible for the existence of bounded domains of the deformed generators. The Hopf algebra coproduct implies associative nonlinear additivity of the multi-particle states. A subalgebra of twist-deformed observables is recovered whenever the twist-deformed generators are either he…
We found hermitian realizations of the position vector $\vec{r}$, the angular momentum $\vecΛ$ and the linear momentum $\vec{p}$, all behaving like vectors under the $su_q(2)$ algebra, generated by $L_0$ and $L_\pm$. They are used to introduce a $q$-deformed Schr\" odinger equation. Its solutions for the particular cases of the Coulomb and the harmonic oscillator potentials are given and briefly discussed.
Symmetric ordering and Weyl realizations for non commutative quantum Minkowski spaces are reviewed. Weyl realizations of Lie deformed spaces and corresponding star products, as well as twist corresponding to Weyl realization and coproduct of momenta are presented. Drinfeld twists understood in Hopf algebroid sense are also discussed. A few examples of corresponding Weyl realizations are given. We show that for the original Snyder space there exists symmetric ordering, but no Weyl realization. Quadratic deformations of Minkowski space are considered and it is demonstrated that symmetric orderin…
In this paper we work out the RTT-realization for the Yangian algebra of the Hubbard model and AdS/CFT correspondence. We find that this Yangian algebra is of a non-standard type in which the levels of the Yangian mix. The crucial feature that allows this is a braiding factor that deforms the coproduct and generates the central extensions of the underlying sl(2|2) Lie algebra. In our RTT-realization we have also been able to incorporate the so-called secret symmetry and we were able to extend it to higher Yangian levels. Finally, we discuss the center of the Yangian and the automorphisms relat…
The Gardiner's phonon presented for a particle-number conserving approximation method to describe the dynamics of Bose-Einstein Condesation (BEC) (C.W. Gardiner, Phys. Rev. A 56, 1414 (1997)) is shown to be a physical realization of the $q$-deformed boson, which was abstractly developed in quantum group theory. On this observation, the coherent output of BEC atoms driven by a radio frequency (r.f) field is analyzed in the viewpoint of a $q$-deformed Fock space. It is illustrated that the $q$-deformation of bosonic commutation relation corresponds to the non-ideal BEC with the finite particle n…
In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite II polynomials recently introduced in [13]. Furthermore, we construct the wave functions and we determine the q-coherent states.
In this parer, q-deformed oscillator for pseudo-Hermitian systems is investigated and pseudo-Hermitian appropriate coherent and squeezed states are studied. Also, some basic properties of these states is surveyed. The over-completeness property of the para-Grassmannian pseudo-Hermitian coherent states (PGPHCSs) examined, and also the stability of coherent and squeezed states discussed. The pseudo-Hermitian supercoherent states as the product of a pseudo-Hermitian bosonic coherent state and a para-Grassmannian pseudo-Hermitian coherent state introduced, and the method also developed to define p…
For the two-parameter $p,q$-deformed Heisenberg algebra introduced recently and in which, instead of usual commutator of $X$ and $P$ in the l.h.s. of basic relation $[X,P] = {\rm i}\hbar$, one uses the $p,q$-commutator, we established interesting properties. Most important is the realizability of the $p,q$-deformed Heisenberg algebra by means of the appropriate deformed oscillator algebra. Another uncovered property is special extension of the usual mutual Hermitian conjugation of the creation and annihilation operators, namely the so-called $η(N)$-pseudo-Hermitian conjugation rule, along with…
Deformed Heisenberg algebra and its Hilbert space representations
African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究所）；Universite Thomas Sankara（托马斯·桑卡拉大学）；Université Felix Houphouët Boigny-Abidjan（费利克斯·乌弗埃-博瓦尼大学）
Composite bosons, here called {\it quasibosons} (e.g. mesons, excitons, etc.), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the "fermion+fermion" composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (de…
Darboux-deformations of short range one-dimensional potentials are constructed by means of Gamow-Siegert functions (resonance states). Results include both Hermitian and non-Hermitian short range potentials which are exactly solvable. As illustration, the method is applied to square wells and barriers for which the transmission coefficient is a superposition of Fock-Breit-Wigner distributions. Resonance levels are calculated in the long lifetime limit by means of analytical and numerical approaches. The new complex potentials behave as an optical device which both refracts and absorbs light wa…
We investigate here all the possible invariant metric functions under the action of various kinds of semi-direct product Poincaré subgroups and their deformed partners. The investigation exhausts the possible theoretical frameworks for the spacetime realization of Cohen-Glashow's very special relativity and the deformation very special relativity approach by Gibbons-Gomis-Pope. Within Finsler-Minkowski type of spacetime, we find that the spacetime emerge a Finsler type of geometry in most cases both for undermed Poincaré subgroup and for deformed one. We give an explanation that the rotation o…
A realization of a deformed Lorentz algebra is considered and its irreducible representations are found; in the limit $q\to 1$, these are precisely the irreducible representations of the classical Lorentz group.
Starting from a faithful five-dimensional matrix representation of the group of two independent oscillators and applying the R-matrix method we generate some classes of deformed fermionic-bosonic quantum Hopf algebras. The corresponding Lie deformed superalgebras of type I--II, obtained by duality, are computed and a realization of generators of these deformed superalgebras are given in terms of the usual fermionic and bosonic creation and annihilation operators associated to the supersymmetric harmonic oscillator. Then, a generalized deformed annihilator is construted and their eigenstates ar…
Motivated by deformation quantization, we consider in this paper $^*$-algebras $\mathcal A$ over rings $\ring C = \ring{R}(i)$, where $\ring R$ is an ordered ring and $i^2 = -1$, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) $\mathcal A$-valued inner product. For $A=C^\infty(M)$, M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star-product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that…
We unify kappa-Minkowki spacetime and Lorentz algebra in unique Lie algebra. Introducing commutative momenta, a family of kappa-deformed Heisenberg algebras and kappa-deformed Poincare algebras are defined. They are specified by the matrix depending on momenta. We construct all such matrices. Realizations and star product are defined and analyzed in general and specially, their relation to coproduct of momenta is pointed out. Hopf algebra of the Poincare algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are intr…
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed S…
The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the past decades. In general, non-Hermitian Hamiltonians are constructed by an ad hoc manner. Here, we study the (2+1) Dirac oscillator and show that in the context of the $κ$-deformed Poincaré-Hopf algebra its Hamiltonian is non-Hermitian but has real eigenvalues. The non-Hermiticity stems from the $κ$-deformed algebra. From the mapping in Bermudez et al., Phys. Rev. A 76, 041801(R) (2007), we propose the $κ$-Jaynes-Cummings and $κ$-anti-Jaynes-Cummings models, which describe…
The phenomenon of degeneracy of an $N-$plet of bound states is studied in the framework of quantum theory of closed (i.e., unitary) systems. For an underlying Hamiltonian $H=H(λ)$ the degeneracy occurs at a Kato's exceptional point $λ^{(EPN)}$ of order $N$ and of the spectral geometric multiplicity $K<N$. In spite of the phenomenological appeal of the concept (tractable as a quantum phase transition, or as a unitary processes of the loss of the observability of the system), the dedicated literature deals, predominantly, just with the models where $N=2$ and $K=1$. In our paper it is shown that …
With a view towards future applications in nuclear physics, the fermion realization of the compact symplectic sp(4) algebra and its q-deformed versions are investigated. Three important reduction chains of the sp(4) algebra are explored in both the classical and deformed cases. The deformed realizations are based on distinct deformations of the fermion creation and annihilation operators. For the primary reduction, the su(2) sub-structure can be interpreted as either the spin, isospin or angular momentum algebra, whereas for the other two reductions su(2) can be associated with pairing between…
n-ary algebras have played important roles in mathematics and mathematical physics. The purpose of this paper is to construct a deformation of Virasoro-Witt n-algebra based on an oscillator realization with two independent parameters (p, q) and investigate its n-Lie subalgebra.
Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphi…
Virasoro constraint is the operator algebra version of one-loop equation for a Hermitian one-matrix model, and it plays an important role in solving the model. We construct the realization of the Virasoro constraint from the Conformal Field Theory (CFT) method. From multi-loop equations of the one-matrix model, we get a more general constraint. It can be expressed in terms of the operator algebras, which is the Virasoro subalgebra with extra parameters. In this sense, we named as generalized Virasoro constraint. We enlarge this algebra with central extension, this is a new kind of algebra, and…
A three-block higher-rank deformed Hermitian--Yang--Mills correspondence on $\mathbb P^1\times\mathbb P^1$
C. N. Yang Institute for Theoretical Physics, Stony Brook University（纽约州立大学石溪分校杨振宁理论物理研究所）；Weinberg Institute, The University of Texas at Austin（德克萨斯大学奥斯汀分校温伯格研究所）；SpaceXAI
The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.
We note that, though nonanticommutative (NAC) deformations of Minkowski supersymmetric theories do not respect the reality condition and seem to lead to non-Hermitian Hamiltonians H, the latter belong to the class of ``cryptoreal'' Hamiltonians considered recently by Bender and collaborators. They can be made manifestly Hermitian via the similarity transformation H -> exp{R} H exp{-R} with a properly chosen R. The deformed model enjoys the same supersymmetry algebra as the undeformed one, though being realized differently on the involved canonical variables. Besides quantum-mechanical models, …
We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg's uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Poeschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the g…
We discuss non-Hermitian field theories where the spectrum of the Hamiltonian involves only real energies. We make three observations. (i) The theories obtained from supersymmetric theories by nonanticommutative deformations belong in many cases to this class. (ii) When the deformation parameter is small, the deformed theory enjoys the same supersymmetry algebra as the undeformed one. Half of the supersymmetries are manifest and the existence of another half can be deduced from the structure of the spectrum. (iii) Generically, the conventionally defined S--matrix is not unitary for such theori…
