The hidden geometry of the quantum Euclidean space

We briefly describe how to introduce the basic notions of noncommutative differential geometry on the 3-dim quantum space covariant under the quantum group of rotations SO _q(3).     

Structure of the three-dimensional quantum euclidean space

As an example of a noncommutative space we discuss the quantum 3-dimensional Euclidean space together with its symmetry structure in great detail. The algebraic structure and the representation theory are clarified and discrete spectra for the coordinates are found. The q-deformed Legendre functions play a special role. A completeness relation is derived for these functions. Received: 13 April 2000 / Published online: 18 May 2000     

Phase space propagators for quantum operators

The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e., the Wigner function. The various propagation kernels are different representations of the super-operators which act on the space of operators of a closed quantum system. We here present the mixed semiclassical propagator, that takes translation chords to reflection centres, or vice versa. In contrast to the centre-centre propagator that directly evolves Wigner functions, they are guaranteed to be caustic free, having a simple WKB-like universal form for a finite time, whatever the number of degrees of freedom. Special attention is given to the near-classical region of small chords, since this dominates the averages of observables evaluated through the Wigner function.     

Braided differential operators on quantum algebras

We propose a general scheme of constructing braided differential algebras via algebras of “quantum exponentiated vector fields” and those of “quantum functions”. We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group GL(m)$GL\left(m\right)$. In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that “quantum adjoint vector fields” can be restricted to the so-called “braided orbits” which are counterparts of generic GL(m)$GL\left(m\right)$-orbits in gl^∗(m)$g{l}^{\ast }\left(m\right)$. Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.     

Covariant differential calculus on the quantum exterior vector space

We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying

Hidden measurement model for pure and mixed states of quantum physics in Euclidean space

We propose a representation of quantum mechanics where all pure and mixed states of a n-dimensional quantum entity are represented as points of a subset of an ²-dimensional real space. We introduce the general measurements of quantum mechanics on this entity, determined by sets of mutual orthogonal points of the representation space. Within this framework we construct a hidden measurement model for an arbitrary finite dimensional quantum entity.     

Realization of quantum permutation algorithm in high dimensional Hilbert space

Quantum algorithms provide a more efficient way to solve computational tasks than classical algorithms. We experimentally realize quantum permutation algorithm using light's orbital angular momentum degree of freedom. By exploiting the spatial mode of photons, our scheme provides a more elegant way to understand the principle of quantum permutation algorithm and shows that the high dimension characteristic of light's orbital angular momentum may be useful in quantum algorithms. Our scheme can be extended to higher dimension by introducing more spatial modes and it paves the way to trace the source of quantum speedup.     

Conformally invariant differential operators on Minkowski space and their curved analogues

This article describes the construction of a natural family of conformally invariant differential operators on a four-dimensional (pseudo-)Riemannian manifold. Included in this family are the usual massless field equations for arbitrary helicity but there are many more besides. The article begins by classifying the invariant operators on flat space. This is a fairly straightforward task in representation theory best solved through the theory of Verma modules. The method generates conformally invariant operators in the curved case by means of Penrose's local twistor transport.S.E.R.C. Advanced Fellow and Flinders University Visiting Research Fellow     

Construction of quantum operators

On the basis of an analysis of the existing methods of constructing quantum operators, requirements are formulated for a correspondence principle that is free from the defects of the familar principles. It is shown that such a principle exists. Its explicit form is determined and the properties of the resulting operators are investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 102–106, November, 1971.The author wishes to thank Professor Ya. P. Terletskii and the participants in the theoretical seminar at the University of International Friendship for their valuable comments.     

Deformed oscillator algebra for quantum superintegrable systems in two-dimensional Euclidean space and on a complex two-sphere

In this work,we study superintegrable quantum systems in two-dimensional Euclidean space and on a complex twosphere with second-order constants of motion.We show that these constants of motion satisfy the deformed oscillator algebra.Then,we easily calculate the energy eigenvalues in an algebraic way by solving of a system of two equations satisfied by its structure function.The results are in agreement to the ones obtained from the solution of the relevant Schrdinger equation.     

Covariant differential calculus on quantum Minkowski space and theq-analogue of Dirac equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices and quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects aq-analogue of Dirac equation follows directly.     

The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.     

Realization of quantum gates by Lyapunov control

We propose a Lyapunov control design to achieve specific (or a family of) unitary time-evolution operators, i.e., quantum gates in the Schrödinger picture by tracking control. Two examples are presented. In the first, we illustrate how to realize the Hadamard gate in a single-qubit system, while in the second, the controlled-NOT (CNOT) gate is implemented in two-qubit systems with the Ising and Heisenberg interactions. Furthermore, we demonstrate that the control can drive the time-evolution operator into the local equivalence class of the CNOT gate and the operator keeps in this class forever with the existence of Ising coupling.     

Realization of the N(odd)-Dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space R_q^N is a kind of noncommutative space that is obtained from ordinary Euclidean space R^N by deformation with parameter q. When N is odd, the structure of this space is similar to R_q³. Motivated by realization of R_q³ by differential operators in R³, we give such realization for R_q⁵ and R_q⁷ cases and generalize our results to R_q^N (N odd) in this paper, that is, we show that the algebra of R_q^N can be realized by differential operators acting on C^∝ functions on undeformed space R^N.     

Euclidean formulation of quantum field theory without positivity

General properties of local quantum field theories (QFT) without positivity are discussed in connection with their euclidean formulation. Modified euclidean axioms for local QFT's without positivity are presented, which allow us to recover by analytic continuation Wightman functions satisfying the modified Wightman axioms for indefinite metric QFT's.     

Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces

Covariant differential calculi on the quantum space for the quantum group SL _q (2) are classified. Our main assumptions are thatq is not a root of unity and that the differentials de _j of the generators of form a free right module basis for the first-order forms. Our result says, in particular, that apart from the two casesc =c(3), there exists a unique differential calculus with the above properties on the space which corresponds to Podles' quantum sphereS _qc ^/2 .     

New realizations of Lie algebra kappa-deformed Euclidean space

We study Lie algebra κ-deformed Euclidean space with undeformed rotation algebra SO_a(n) and commuting vectorlike derivatives. Infinitely many realizations in terms of commuting coordinates are constructed and a corresponding star product is found for each of them. The κ-deformed noncommutative space of the Lie algebra type with undeformed Poincaré algebra and with the corresponding deformed coalgebra is constructed in a unified way.