In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jaynes-Cummings-type models of quantum optics are presented as special cases of the method. Their spectra, eigen-spinors and Wigner functions as well as their constants of motion are also obtained within the autonomous framework of deformation quantization.
The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.
We define a q-deformation of the Dirac operator, inspired by the one dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows to construct q-deformed Schroedinger equations in higher dimensions. The equivalence of these Schroedinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford-Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal…
It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Cl_q(n|m) such that the latter is a homomorphic image of the quantum superalgebra U_q[osp(2n+1|2m)]. The Fock space F(n|m) of Cl_q(n|m) is constructed. At q being a root of unity (q=\exp ({iπl/k})) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace F_{l/k}(n|m) of F(n|m). Each F_{l/k}(n|m) is turned through the above mentioned homomorphism into an irreducible (root of unity) U_q[osp…
In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these tr…
One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on that space preserving the non-commutative structure constitute a quantum group. The non-commutativity of these variables provide an example of non-commutative geometry. In some recent work we have shown that quantum Clifford algebras --i.e. Clifford algebras of an arbitrary bilinear form-- are closely related to deformed structures as q-spin groups, Hecke a…
As a substantial generalization of the technique for constructing canonical and the related nonlinear and q-deformed coherent states, we present here a method for constructing vector coherent states in the same spirit. These vector coherent states may have a finite or an infinite number of components. As examples we first apply the technique to construct vector coherent states using the Plancherel isometry for groups and vector coherent states associated to Clifford algebras, in particular quaternions. As physical examples, we discuss vector coherent states for a quantum optical model and fina…
This paper presents a thoughful review of: (a) the Clifford algebra Cl(H_{V}) of multivecfors which is naturally associated with a hyperbolic space H_{V}; (b) the study of the properties of the duality product of multivectors and multiforms; (c) the theory of k multivector and l multiform variables multivector extensors over V and (d) the use of the above mentioned structures to present a theory of the parallelism structure on an arbitrary smooth manifold introducing the concepts of covariant derivarives, deformed covariant derivatives and relative covariant derivatives of multivector, multifo…
Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields is thus discussed. Hence, by endowing the underlying spacetime with an arbitrary bilinear form with a antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are therefore compared to the classical (non quant…
A set of two-parameter bi-orthogonal eigen-spinors has been constructed from a deformed pseudo- Hermitian extension of Pauli Hamiltonian and its Hermitian conjugate. The Hamiltonians thus obtained are iso-spectral to the original Pauli Hamiltonian. A pair of spin-projection operators has been constructed as an essential ingredient of a possible bi-orthogonal quantum mechanics. An analogue of Kramers theorem in pseudo-Hermitian setting has also been inferred in a conjectural sense. The properties of time reversal and bi-orthogonality have been elaborated in the frame work of Clifford algebra Cl…
In this work we apply the Drinfel'd twist of Hopf algebras to the study of deformed quantum theories. We prove that, by carefully considering the role of the central extension, it is indeed possible to construct the universal enveloping algebra of the Heisenberg algebra and deform it by means of a Drinfel'd twist, which yields a noncommutative theory. Furthermore, we show that in the second-quantization formalism the Hopf algebra structure of the Heisenberg algebra (both undeformed and deformed) can be obtained from the Hopf algebra of the Schrodinger fields and oscillators, as long as they ar…
Clifford algebras are naturally associated with quadratic forms. These algebras are Z_2-graded by construction. However, only a Z_n-gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) <-> \bigwedge V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Z_n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a…
In this paper we discuss a simplified approach to the symplectic Clifford algebra, the symplectic Clifford group and the symplectic spinor by first extending the Heisenberg algebra. We do this by adding a new idempotent element to the algebra. It turns out that this element is the projection operator onto the Dirac standard ket. When this algebra is transformed into the Fock algebra, the corresponding idempotent is the projector onto the vacuum. These additional elements give a very simple way to write down expressions for symplectic spinors as elements of the symplectic Clifford algebra. When…
Both the ${\cal N}=7$ superconformal quantum mechanics possessing the exceptional $G(3)$ Lie superalgebra as dynamical symmetry and its associated deformed oscillator with $G(3)$ as spectrum-generating superalgebra are presented. This superconformal quantum mechanics, uniquely defined up to similarity transformations, is obtained via the octonionically-induced "quasi-nonassociative" method employed to derive the exceptional ${\cal N}=8$ $F(4)$ model. To construct the $G(3)$ theories, the covariant embedding of the $7$-dimensional representation of the Lie algebra $g_2$ within the $8\times 8$ m…
In this paper, using the notion of nonlinear coherent states, we define a deformed bosonic dephasing channel modelling the impact of a Kerr medium on a quantum state, as it occurs, for instance, in quantum communication based on optical fibers. We show that, in certain regimes, the Kerr nonlinearity is able to compensate the dephasing. In addition, our studies reveal that the quantum capacity of the deformed bosonic dephasing channel can be greater than that of the undeformed, standard bosonic dephasing channel for certain nonlinearity parameters.
The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficient part of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle c…
Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in th…
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 construct a new family of q-deformed coherent states $|z>_q$, where $0 < q < 1$. These states are normalizable on the whole complex plane and continuous in their label $z$. They allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem and expressed in terms of one of the two Jacksons's $q$-exponentials. They also permit exact evaluation of matrix elements of physically-relevant operators. We use this to show that the photon number statistics for the states is sub-…
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…
This work addresses the study of the oscillator algebra, defined by four parameters $p$, $q$, $α$, and $ν$. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum…
We study quantum computation relations on unital finite-dimensional CAR $C^{*}$-algebras. We prove an entropy power inequality (EPI) in a fermionic setting, which presumably will permit understanding the capacities in fermionic linear optics. Similar relations to the bosonic case are shown, and alternative proofs of known facts are given. Clifford algebras and the Grassmann representation can thus be used to obtain mathematical results regarding coherent fermion states.
This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to the condensed matter physics, which obeys a $f$-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan $P$-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Mo…
This is my PhD thesis. In this thesis we study the gauge theories on noncommutative Moyal space. We find new static solitons and instantons in terms of the so called generalized Bose operators. Generalized Bose operators are constructed to describe reducible representation of the oscillator algebra. They create/annihilate $k$-quanta, $k$ being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of generalized Bose operators. The Nielsen-Olesen vortex solutions found in terms o…
