Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical system in classical mechanics that share many features with the ModMax theory. It consists of a couple of $\sqrt{T\bar{T}}$-deformed oscillators that nevertheless preserves duality $(q \rightarrow p,p \rightarrow -q)$ and depends on a continuous parameter $γ$, as in the ModMax case. Despite its non-linear features, the system is integrable. Remarkably can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a non-linear map dependent on $γ$ that maps the oscillator in 2D to the nonlinear one, but with parameter $2γ$. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.
Source-free so-called ModMax theories of nonlinear electrodynamics in the four dimensional Minkowski spacetime vacuum are the only possible continuous deformations -- and as a function of a single real and positive parameter -- of source-free Maxwell linear electrodynamics in the same vacuum, which preserve all the same Poincaré and conformal spacetime symmetries as well as the continuous duality invariance of Maxwell's theory. Null field configurations of the latter however, including null electromagnetic knots, are singular for the Lagrangian formulation of any spacetime Poincaré and conform…
In this letter, we investigate the deformation of the ModMax theory, as a unique Lagrangian of non-linear electrodynamics preserving both conformal and electromagnetic-duality invariance, under $T\bar{T}$-like flows. We will show that the deformed theory is the generalized non-linear Born-Infeld electrodynamics. Being inspired by the invariance under the flow equation for Born-Infeld theories, we propose another $T\bar{T}$-like operator generating the ModMax and generalized Born-Infeld non-linear electrodynamic theories from the usual Maxwell and Born-Infeld theories, respectively.
We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schrödinger algebra, these equations reduce to a system of ordinary harmonic oscillators. We provide two clarifying examples of such deformed oscillators: one system invariant under SO(2,3) transformations, and another system featuring $G_{2(2)}$ symmetry. The construction of invariant actions requires adding semi-dynamical degrees of freedom; we illustrate the algorithm with the two examples mention…
We show that the crossing symmetric dispersion relation (CSDR) for 2-2 scattering leads to a fascinating connection with knot polynomials and q-deformed algebras. In particular, the dispersive kernel can be identified naturally in terms of the generating function for the Alexander polynomials corresponding to the torus knot $(2,2n+1)$ arising in knot theory. Certain linear combinations of the low energy expansion coefficients of the amplitude can be bounded in terms of knot invariants. Pion S-matrix bootstrap data respects the analytic bounds so obtained. We correlate the $q$-deformed harmonic…
We derive and spell out the structure constants of the $\mathbb{Z}_2$-graded algebra $\mathfrak{shs}[λ]\,$ by using deformed-oscillators techniques in $Aq(2;ν)\,$, the universal enveloping algebra of the Wigner-deformed Heisenberg algebra in 2 dimensions. The use of Weyl ordering of the deformed oscillators is made throughout the paper, via the symbols of the operators and the corresponding associative, non-commutative star product. The deformed oscillator construction was used by Vasiliev in order to construct the higher spin algebras in three spacetime dimensions. We derive an expression for…
We investigate the dynamical symmetry superalgebras of the one-dimensional Matrix Superconformal Quantum Mechanics with inverse-square potential. They act as spectrum-generating superalgebras for the systems with the addition of the de Alfaro-Fubini-Furlan oscillator term. The undeformed quantum oscillators are expressed by $2^n\times 2^n$ supermatrices; their corresponding spectrum-generating superalgebras are given by the $osp(2n|2)$ series. For $n=1$ the addition of a inverse-square potential does not break the $osp(2|2)$ spectrum-generating superalgebra. For $n=2$ two cases of inverse-squa…
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…
We present the deformed (for the presence of Calogero potential terms) one-dimensional quantum oscillator with the exceptional Lie superalgebra $F(4)$ as spectrum-generating superconformal algebra. The Hilbert space is given by a $16$-ple of square-integrable functions. The energy levels are $\frac{2}{3}+n$, with $n=0,1,2,\ldots$. The ground state is $7$ times degenerate. The excited states are $8$ times degenerate. The $(7,8,8,8,\ldots )$ semi-infinite tower of states is recovered from the $(7;8;1)$ supermultiplet of the ${\cal N}=8$ worldline supersymmetry. The model is unique, up to similar…
We give explicit analytical expressions for the partition function of $U(N)_{k}\times U(N+M)_{-k}$ ABJ theory at weak coupling ($k\rightarrow \infty )$ for finite and arbitrary values of $N$ and $M$ (including the ABJM case and its mass-deformed generalization). We obtain the expressions by identifying the one-matrix model formulation with a Meixner-Pollaczek ensemble and using the corresponding orthogonal polynomials, which are also eigenfunctions of a $su(1,1)$ quantum oscillator. Wilson loops in mass-deformed ABJM are also studied in the same limit and interpreted in terms of $su(1,1)$ cohe…
We consider the deformed harmonic oscillator as a discrete version of the Liouville theory and study this model in the presence of local integrable defects. From this, the time evolution of the defect degrees of freedom are determined, found in the form of the local equations of motion. We also revisit the continuous Liouville theory, deriving its local integrals of motion and comparing these with previous results from the sine-Gordon point of view.Finally, the generic Backlund type relations are presented, corresponding to the implementation of time-like and space-like impurities in the conti…
We point out a misleading treatment in a recent paper published in this Journal [Eur. Phys. J. C (2018)78:106] regarding solutions for the Schrödinger equation with a anharmonic oscillator potential embedded in the background of a cosmic string mapped into biconfluent Heun equation. This fact jeopardizes the thermodynamical properties calculated in this system.
In this paper we study the quantisation of scalar field theory in $κ$-deformed space-time. Using a quantisation scheme that use only field equations, we derive the quantisation rules for deformed scalar theory, starting from the $κ$-deformed equations of motion. This scheme allows two choices; (i)a deformed commutation relation between the field and its conjugate which leads to usual oscillator algebra, (ii) an undeformed commutation relation between field and its conjugate leading to a deformed oscillator algebra. This deformed oscillator algebra is used to derive modification to Unruh effect…
In this Letter, 2D Dirac oscillator in the quantum deformed framework generated by the $κ$-Poincaré-Hopf algebra is considered. The problem is formulated using the $κ$-deformed Dirac equation. The resulting theory reveals that the energies and wave functions of the oscillator are modified by the deformation parameter.
Representation Theory of Generalized Deformed Oscillator Algebras
发展了广义变形振子代数的表示理论。GDOA由四个运算符${1,a,a^†,N}$生成。除了$[a, a^†]_q = G(N)$之外，它们的算符和厄米性都是玻色子振子代数的算符和厄米性，其中$[a,b]_q = a b - q b a$和$G(N)$是厄米解析函数。利用Casimir算子$C$和半正算子$a^† a$得到了酉不可约表示.它们可能属于四类之一：从下有界（BFB），从上有界（BFA），有限维（FD），无界（UB）。列举了这些不同类型的联合国区域代表的一些例子。
An analogue of the Moyal star product is presented for the deformed oscillator algebra. It contains several homotopy-like additional integration parameters in the multiplication kernel generalizing the differential Moyal star-product formula $\exp[iε_{αβ}\partial^α\partial^β]$. Using Pochhammer formula, integration over these parameters is carried over a Riemann surface associated with the expression of the type $z^x (1-z)^y$ where $x$ and $y$ are arbitrary real numbers.
We study the solutions of the (2+1)-dimensional kappa-deformed Dirac oscillator in the presence of a constant transverse magnetic field. We demonstrate how the deformation parameter affects the energy eigenvalues of the system and the corresponding eigenfunctions. Our findings suggest that this system could be used to detect experimentally the effect of the deformation. We also show that the hidden supersymmetry of the non-deformed system reduces to a hidden pseudo-supersymmetry having the same algebraic structure as a result of the k-deformation.
