New Application of Noncanonical Maps in Quantum Mechanics

One invertible and one unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. The original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified equations of motion which do not preserve the properties of the original first-order set for annihilation and creation operators. Their relation with commutation relations is also studied.     

Thermal QCD sum rules in the \rho^0 channel revisited

From the hypothesis that at zero temperature the square root of the spectral continuum threshold is linearly related to the QCD scale we derive in the chiral limit and for temperatures considerably smaller than scaling relations for the vacuum parts of the Gibbs averaged scalar operators contributing to the thermal operator product expansion of the current-current correlator. The scaling with being the T-dependent perturbative QCD continuum threshold in the spectral integral, is simple for renormalization group invariant operators, and becomes nontrivial for a set of operators which mix and scale anomalously under a change of the renormalization point. In contrast to previous works on thermal QCD sum rules with this approach the gluon condensate exhibits a sizable T-dependence. The -meson mass is found to rise slowly with temperature which coincides with the result found by means of a PCAC and current algebra analysis of the correlator. Received: 16 November 1999 / Revised version: 20 May 2000 / Published online: 23 October 2000     

Momentum and Hamiltonian operators in generalized coordinates

The self-adjointness of momentum operators in generalized coordinates, questioned by Domingos and Caldeira is shown. The momentum operators of a particle and the kinetic part of its Hamiltonian operator constructed from them are characterized as self-adjoint operators and geometrical objects in coordinate-free form. Local coordinates of ann-dimensional Riemannian manifold are taken as the generalized coordinates of the particle. As an example the curvilinear coordinates of Euclidean space are treated. The coefficients of connection and curvature are given on the manifold for which the assumed momentum operators exist. It is found that if our momentum operators form a complete set of mutually commuting observables, the manifold is locally Euclidean, i.e., there exists a local coordinate system such that we obtain the usual Schrödinger correspondence rule.     

Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice. Received: 27 June 2000 / Published online: 9 August 2000     

Quantum Theory of Path-Dependent Field Operator Based on Characteristics of Displacement Operators

The path-dependent operator formalism of quantum electrodynamics proposed by Mandelstam is reformulated through quantum field theory based on characteristics of displacement operators in Minkowski space. It is shown that total energy- and total angular-momentum operators can generate inhomogeneous Lorentz transformations on any local operator including path-independent bilinear forms constructed of path-dependent electron operator Ψ(x, P), but that generators for Ψ(x, P) itself are only their Ψ(x, P)-dependent parts. Such an unfamiliar feature is characteristic of the path-dependent operator formalism. The present approach possesses unique merits in making the logic of the formalism transparent as described in the following: i) Quantum electrodynamics can be formulated but for the help of potential operator even as a tool for calculation up to a final step. ii) Some restriction, which can be used to discuss propriety of gauge conditions, can be figured out. iii) By introducing a path-rearrangement operator, we can keep infinite variety of space-like pathes with the same end point throughout our formulation as they stand. iv) Several points which must be modified in the presence of magnetic monopole are closed up.     

An interesting representation of lie algebras of linear groups

I have presented a means of getting a representation space of a general linear group ofn dimensions in terms of homogeneous functions ofn,n-dimensional vectors. Except in particular cases, the representation is of the Lie algebra, rather than the group. A general formalism is set up to evaluate the Casimir operators of the Lie algebra of the group in terms of the degrees of homogeneity of the functions (which are eigenfunctions of the Casimir operators) in then variables. It is noticed that the Casimir operators exhibit certain symmetries in these degrees of homogeneity which relate different representations having the same eigenvalues for the Casimir operators. Contour integral formulas that enable one to pass from one such representation to another are presented. An expression for the eigenvalues of a general Casimir operator in terms of the degree of homogeneity is presented.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

Tensor operators for quantum algebras

Tensor operators are discussed for Hopf algebras and, in particular, for a quantum (q-deformed) algebraUq(g), whereg is any simple finite-dimensional or affine Lie algebra. These operators are defined via an adjoint action in a Hopf algebra. There are two types of the tensor operators which correspond to two coproducts in the Hopf algebra. In the case of tensor products of two tensor operators one can obtain 8 types of the tensor operators and so on. We prove the relations which can be a basis for a proof of the Wigner-Eckart theorem for the Hopf algebras. It is also shown that in the case ofUq(g) a scalar operator can be differed from an invariant operator but atq=1 these operators coincide. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163, and by INTAS-00-00055.     

Dirac Operators on Taub-NUT Space: Relationship and Discrete Transformations

It is shown that the N = 4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant constant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N = 4 supersymmetry.     

Note on the Paper “The Norm Convergence¶ of the Trotter–Kato Product Formula with Error Bound” by Ichinose and Tamura

The norm convergence of the Trotter–Kato product formula is established with ultimate optimal error bound for the selfadjoint semigroup generated by the operator sum of two selfadjoint operators. A generalization is also given to the operator sum of several selfadjoint operators. Received: 5 October 2000 / Accepted: 12 March 2001     

Centre and representations of small quantum superalgebras at roots of unity

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU _q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU _q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU _q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

Supersymmetric Pair of q-Deformed Nonlocal Operators

A simple version of the q-deformed calculus is used to generate a pair ofq-nonlocal, second-order difference operators by means of deformed counterpartsof Darboux intertwining operators for the Schrödinger—Hermite oscillators atzero factorization energy. These deformed nonlocal operators may be consideredas supersymmetric partners and their structure contains contributions originatingin both the Hermite operator and the quantum harmonic oscillator operator. Thereare also extra ±x contributions. The undeformed limit, in which allq-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanicalharmonic oscillator Hamiltonians. The more general case of negative factorizationenergy is briefly discussed as well.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Ruijsenaars' Commuting Difference Operators as Commuting Transfer Matrices

For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL=LLR, taking the trace of the L-operator gives a set of commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars' operators. The relationship between the difference L-operator and Krichever's Lax matrix is given, and an explicit formula for elliptic commuting differential operators is derived. We also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space. Received: 27 December 1995 / Accepted: 11 November 1996     

Generalized Lamé Operators

We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero–Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lamé operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh–Veselov conjecture for the elliptic Calogero–Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to the B₂ case, another one is a certain deformation of the A₂ case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald–Ruijsenaars type.On leave of absence from: Advanced Education and Science Centre, Moscow State University, Moscow 119899, Russia     

Topological order in an exactly solvable 3D spin model

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest-neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on Ω(R²) qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.     

Neutrino oscillations with three flavors in matter of varying density

In this paper, we discuss the evolution operator and the transition probabilities expressed as functions of the vacuum mass squared differences, the vacuum mixing angles, and the matter density parameter for three flavor neutrino oscillations in matter of varying density in the plane wave approximation. The applications of this to neutrino oscillations in a model of the earth's matter density profile, step function matter density profiles, constant matter density profiles, linear matter density profiles, and finally in a model of the sun's matter density profile are discussed. We show that for matter density profiles which do not fluctuate too much, the total evolution operator consisting of n operators can be replaced by one single evolution operator in the semi-classical approximation. Received: 23 March 2001 / Published online: 8 June 2001     

Calculations in external fields in quantum chromodynamics. Technical review

We review the technique of calculation of operator expansion coefficients. The main emphasis is put on gluon operators which appear in expansion of n-point functions induced by colourless quark currents. Two convenient schemes are discussed in detail: the abstract operator method and the method based on the Fock-Schwinger gauge for the vacuum gluon field. We consider a large number of instructive examples important from the point of view of physical applications.