arXiv subjects

N. Aizawa

Publications and source records attributed to N. Aizawa.

At least 37 records · Page 2Linked to original sources

arXiv2020

$\mathbb{Z}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras

It is shown that the ${\cal N}=1$ supersymmetric quantum mechanics (SQM) can be extended to a $\mathbb{Z}_2^n$-graded superalgebra. This is done by presenting quantum mechanical models which realize, with the aid of Clifford gamma matrices, the $\mathbb{Z}_2^n$-graded Poincaré algebra in one-dimensional spacetime. Reflecting the fact that the $\mathbb{Z}_2^n$-graded Poincaré algebra has a number of central elements, a sequence of models defining the $\mathbb{Z}_2^n$-graded version of SQM are provided for a given value of $n.$ In a model of the sequence, the central elements having the same $\mathbb{Z}_2^n$-degree are realized as dependent or independent operators. It is observed that as use the Clifford algebra of lager dimension, more central elements are realized as independent operators.

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arXiv2019

Conformal mechanical treatment of Calogero-Moser model and infinite dimensional Lie algebra of conformal Galilei type

We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module. More precisely, it is a correspondence of excited states of the model and singular vectors in Verma modules over the algebra. This is found by a free field realization of the time evolution operator of the model. We investigate the Verma modules and some explicit example of singular vectors are given.

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arXiv2019

On the spectrum-generating superalgebras of the deformed one-dimensional quantum oscillators

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-square potential deformations arise. The first one produces Klein deformed quantum oscillators; the corresponding spectrum-generating superalgebras are given by the $D(2,1;α)$ class, with $α$ determining the inverse-square potential coupling constants. The second $n=2$ case corresponds to deformed quantum oscillators of non-Klein type. In this case the $osp(4|2)$ spectrum-generating superalgebra of the undeformed theory is broken to $osp(2|2)$. The choice of the Hilbert spaces corresponding to the admissible range of the inverse-square potential coupling constants and the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented.

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arXiv2018

$ \mathbb{Z}_2 \times \mathbb{Z}_2 $ generalizations of ${\cal N} = 1$ superconformal Galilei algebras and their representations

We introduce two classes of novel color superalgebras of $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ grading. This is done by realizing members of each in the universal enveloping algebra of the ${\cal N}=1$ supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ graded setting.

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arXiv2018

Generalization of Superalgebras to Color Superalgebras and Their Representations

For a given Lie superalgebra, two ways of constructing color superalgebras are presented. One of them is based on the color superalgebraic nature of the Clifford algebras. The method is applicable to any Lie superalgebras and results in color superalgebra of $ {\mathbb Z}_2^{\otimes N} $ grading. The other is discussed with an example, a superalgebra of boson and fermion operators. By treating the boson operators as "second" fermionic sector we obtain a color superalgebra of ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ grading. A vector field representation of the color superalgebra obtaind from the boson-fermion system is also presented.

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arXiv2018

Verma Modules over a ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ Graded Superalgebra and Invariant Differential Equations

Lowest weight representations of the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra introduced by Rittenberg and Wyler are investigated. We give a explicit construction of Verma modules over the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra and show their reducibility by using singular vectors. The explicit formula of singular vectors are given and are used to derive partial differential equations invariant under the color supergroup generated by the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra.

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arXiv2017

The quasi-nonassociative exceptional $F(4)$ deformed quantum oscillator

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 similarity transformations, and admits an octonionic-covariant formulation which manifests itself as "quasi-nonassociativity". This means, in particular, that the Calogero coupling constants are expressed in terms of the octonionic structure constants. The associated $F(4)$ superconformal quantum mechanics is also presented.

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arXiv2017

${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalizations of ${\cal N} = 2$ super Schrödinger algebras and their representations

We generalize the real and chiral $ {\cal N} =2 $ super Schrödinger algebras to ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded Lie superalgebras. This is done by $D$-module presentation and as a consequence, the $D$-module presentations of ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded superalgebras are identical to the ones of super Schrödinger algebras. We then generalize the calculus over Grassmann number to ${\mathbb Z}_2 \times {\mathbb Z}_2 $ setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded superalgebras. A vector field realization of the ${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalization of ${\cal N} = 1 $ super Schrödinger algebra is also presented.

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arXiv2016

Aspects of infinite dimensional $\ell$-super Galilean conformal algebra

In this work we construct a infinite dimensional $\ell$-super Galilean conformal algebra, which is a generalization of the $\ell=1$ algebra found in the literature. We give a classification of central extensions, the vector field representation, the coadjoint representation and the operator product expansion of the infinite dimensional $\ell$-super Galilean conformal algebra, keeping possible applications in physics and mathematics in mind.

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arXiv2016

${\mathbb Z}_2\times {\mathbb Z}_2$-graded Lie Symmetries of the Lévy-Leblond Equations

The first-order differential Lévy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of ($1+d$)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schrödinger equations for free particles or in the presence of a harmonic potential), admit a natural ${\mathbb Z}_2$-graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric PDE's, a natural ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the $(1+1)$-dimensional Lévy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie superalgebra, realized by first and second-order differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the ${\mathbb Z}_2$- and ${\mathbb Z}_2\times{\mathbb Z}_2$- graded extensions are no longer allowed. The construction of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded Lie symmetry of the ($1+2$)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra.

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arXiv2016

Generalized supersymmetry and Lévy-Leblond equation

The symmetries of the Lévy-Leblond equation are investigated beyond the standard Lie framework. It is shown that the equation has two remarkable symmetries. One is given by the super Schrödinger algebra and the other one by a $\ZZ$ graded Lie algebra. The $\ZZ$ graded Lie algebra is achieved by transforming bosonic into fermionic operators in the super Schrödinger algebra and introducing second order differential operators as generators of symmetry.

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arXiv2016

Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions

We investigate the general class of second-order PDEs, invariant under the $d=1$ $\ell=\frac{1}{2}+{\mathbb N}_0$ centrally extended Conformal Galilei Algebras, pointing out that they are deformations of decoupled systems. For $\ell=\frac{3}{2}$ the unique deformation parameter $γ$ belongs to the fundamental domain $γ\in ]0,+\infty[$. We show that, for any $γ\neq 0$, invariant PDEs with discrete spectrum (either bounded or unbounded) induce cryptohermitian operators possessing the same spectrum as two decoupled oscillators, provided that their frequencies are in the special ratio $r=\frac{ω_2}{ω_1}=\pm\frac{1}{3},\pm 3$ (the negative energy solutions correspond to a special case of Pais-Uhlenbeck oscillator), where $ω_1,ω_2$ are two different parameters of the invariant PDEs. We also consider the $γ=0$ decoupled system for any value $r$ of the ratio. It possesses enhanced symmetry at the critical values $r=\pm \frac{1}{3}, \pm 1,\pm 3$. Two inequivalent $12$-generator symmetry algebras are found at $r =\pm\frac{1}{3},\pm 3$ and $r=\pm 1$, respectively. The $\ell=\frac{3}{2}$ Conformal Galilei Algebra is not a subalgebra of the decoupled symmetry algebra. Its $γ\rightarrow 0$ contraction corresponds to a $8$-generator subalgebra of the decoupled $r=\pm\frac{1}{3},\pm 3$ symmetry algebra. The features of the $\ell\geq \frac{5}{2}$ invariant PDEs are briefly discussed.

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arXiv2015

Invariant PDEs with Two-dimensional Exotic Centrally Extended Conformal Galilei Symmetry

Conformal Galilei Algebras labeled by $d,\ell$ (where $d$ is the number of space dimensions and $\ell$ denotes a spin-${\ell}$ representation w.r.t. the $\mathfrak{sl}(2)$ subalgebra) admit two types of central extensions, the ordinary one (for any $d$ and half-integer $\ell$) and the exotic central extension which only exists for $d=2$ and ${\ell}\in\mathbb{N}$. For both types of central extensions invariant second-order PDEs with continuous spectrum were constructed in [1]. It was later proved in [2] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing \textcolor{black}{a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The ${\ell}=1$ case (which is the prototype of this class of extensions, just like the $\ell=\frac{1}{2}$ Schrödinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.

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arXiv2014

${\ell}$-oscillators from second-order invariant PDEs of the centrally extended Conformal Galilei Algebras

We construct, for any given ${\ell}=\frac{1}{2}+{\mathbb{N}}_0$, the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra. \par At the given ${\ell}$, two invariant equations in one time and ${\ell}+\frac{1}{2}$ space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrödinger equation (recovered for ${\ell}=\frac{1}{2}$) in $1+1$ dimension. The second equation (the "$\ell$-oscillator") possesses a discrete, positive spectrum. It generalizes the $1+1$-dimensional harmonic oscillator (recovered for $\ell=\frac{1}{2}$). The spectrum of the ${\ell}$-oscillator, derived from a specific $osp(1|2\ell+1)$ h.w.r., is explicitly presented.\par The two sets of invariant PDEs are determined by imposing (representation-dependent) {\it on-shell invariant conditions} both for {\it degree} $1$ operators (those with continuum spectrum) and for {\it degree } $0$ operators (those with discrete spectrum).\par The on-shell condition is better understood by enlarging the Conformal Galilei Algebras with the addition of certain second-order differential operators. Two compatible structures (the algebra/superalgebra duality) are defined for the enlarged set of operators.

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arXiv2014

Intertwining Operator Realization of anti de Sitter Holography

We give a group-theoretic interpretation of relativistic holography as equivalence between representations of the anti de Sitter algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators for arbitrary integer spin the framework of representation theory. Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the de Sitter case, we show that each bulk field has two boundary (shadow) fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary.

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arXiv2014

Galilean conformal algebras in two spatial dimension

A class of infinite dimensional Galilean conformal algebra in (2+1) dimensional spacetime is studied. Each member of the class, denoted by \alg_{\ell}, is labelled by the parameter \ell. The parameter \ell takes a spin value, i.e., 1/2, 1, 3/2, .... We give a classification of all possible central extensions of \alg_{\ell}. Then we consider the highest weight Verma modules over \alg_{\ell} with the central extensions. For integer \ell we give an explicit formula of Kac determinant. It results immediately that the Verma modules are irreducible for nonvanishing highest weights. It is also shown that the Verma modules are reducible for vanishing highest weights. For half-integer \ell it is shown that all the Verma module is reducible. These results are independent of the central charges.

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arXiv2013

Intertwining operators for l-conformal Galilei algebras and hierarchy of invariant equations

l-Conformal Galilei algebra, denoted by g{l}{d}, is a non-semisimple Lie algebra specified by a pair of parameters (d,l). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by g{l}{d} with central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g{l}{d} and vector field representations for d = 1, 2.

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arXiv2013

Chiral and Real N=2 supersymmetric l-conformal Galilei algebras

Inequivalent N=2 supersymmetrizations of the l-conformal Galilei algebra in d-spatial dimensions are constructed from the chiral (2,2) and the real (1,2,1) basic supermultiplets of the N=2 supersymmetry. For non-negative integer and half-integer l both superalgebras admit a consistent truncation with a (different) finite number of generators. The real N=2 case coincides with the superalgebra introduced by Masterov, while the chiral N=2 case is a new superalgebra. We present D-module representations of both superalgebras. Then we investigate the new superalgebra derived from the chiral supermultiplet. It is shown that it admits two types of central extensions, one is found for any d and half-integer l and the other only for d=2 and integer l. For each central extension the centrally extended l-superconformal Galilei algebra is realized in terms of its super-Heisenberg subalgebra generators.

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