Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

On quantum iterated function systems

A Quantum Iterated Function System on a complex projective space is defined through a family of linear operators on a complex Hilbert space. The operators define both the maps and their probabilities by one algebraic formula. Examples with conformal maps (relativistic boosts) on the Bloch sphere are discussed.     

Application of the group foliation method to the complex Monge-Ampère equation

We apply the method of group foliation to the complex Monge-Ampère equation (CMA ₂) to establish a regular framework for finding its non-invariant solutions. We employ an infinite symmetry subgroup ofCMA ₂ to produce a foliation of the solution space into orbits of solutions with respect to this group and a corresponding splitting ofCMA ₂ into an automorphic system and a resolvent system. We propose a new approach to group foliation which is based on the commutator algebra of operators of invariant differentiation. This algebra together with its Jacobi identities provides the commutator representation of the resolvent system. Presented by M.B. Sheftel at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

On Some Second Hochschild Cohomology Spaces for Algebras of Functions on a Manifold

We show that the second Hochschild cohomology space for the space of smooth functions on a manifold corresponding to cochains defined by continuous operators is the same as the one corresponding to differentiable operators, i.e. is given by the space of skewsymmetric contravariant 2-tensors on the manifold. We do this using a coboundary construction due to Omori, Maeda and Yoshioka.     

The Operator Formulation of Classical Mechanics and Semiclassical Limit

The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h ₀. For the later of these two extreme values, introduced operator algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operator framework.     

Generalized q-Hermite Polynomials

We consider two operators A and A ⁺ in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ₀ and that A and A ⁺ are ladder operators for the sequence . The sequence (ψ _n /ψ₀) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001     

Lagrangian submanifolds and an application to the reduced Schrödinger equation in central force problems

In this Letter, a Lagrangian foliation of the zero energy level is constructed for a family of planar central force problems. The dynamics on the leaves are explicitly computed and these dynamics are given a simple interpretation in terms of the dynamics near the singularity of the potential. Lagrangian submanifolds also arise when seeking asymptotic solutions to certain partial differential equations with a large parameter. In determining such solutions, an operator between half densities on the Lagrangian submanifold and half densities on the configuration space is computed. This operator is derived for the given example, and the corresponding first order asymptotic solution to the reduced Schrödinger equation is given.     

How different are the supermanifolds of Rogers and DeWitt?

A DeWitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold, whereas a Rogers supermanifold is not so restricted. Corresponding to the vector space fibers of the DeWitt supermanifold, a Rogers supermanifold has a foliation by submanifolds, or leaves, parametrized by soul coordinates only. We show that the universal covering space of any leaf always admits a flat metric. If the covering space is complete in this metric, it must in fact be a vector space. We combine this result with known theorems about foliations to give conditions under which a compact Rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace. We also show that a supermanifold defined by a polynomial equation in flat superspace is always of the DeWitt type. Finally, we exhibit new supermanifold structures forR ² and the 2-torus which show that the foliation of a Rogers supermanifold can be quite exotic.Enrico Fermi Fellow. Research supported by the NSF: PHY 83-01221, and the Department of Energy: DE AC02-82-ER-40073     

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.     

Riemann-Cartan-Weyl quantum geometry,I: Laplacians and supersymmetric systems

In this first article of a series dealing with the geometry of quantum mechanics, we introduce the Riemann-Cartan-Weyl (RCW) geometries of quantum mechanics for spin-0 systems as well as for systems of nonzero spin. The central structure is given by a family of Laplacian (or D'Alembertian) operators on forms of arbitrary degree associated to the RCW geometries. We show that they are conformally equivalent with the Laplacian operators introduced by Witten in topological quantum field theories. We show that the Laplacian RCW operators yield a supersymmetric system, in the sense of Witten, and study the relation between the RCW geometries and the symplectic structure of loop space. The RCW family of Laplacians are the infinitesimal generators of diffusion processes on nondegenerate space-times of systems of arbitrary spin.     

High order Cartan's homotopy formulae and the algebraic approach to anomalies

The homotopy operators in high-dimensional parameter space are introduced and the high order Chern-Simons-like characteristic polynomials are precisely defined. These quantities are closely related to the algebraic approach to anomalies recently proposed by Faddeev.     

Influence of Constraint in Parameter Space on Quantum Games

We study the influence of the constraint in the parameter space on quantum games. Decomposing SU(2) operator into product of three rotation operators and controlling one kind of them, we impose a constraint on the parameter space of the players' operator. We find that the constraint can provide a tuner to make the bilateral payoffs equal, so that the mismatch of the players' action at multi-equilibrium could be avoided. We also find that the game exhibits an intriguing structure as a function of the parameter of the controlled operators, which is useful for making game models.     

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

Representation of hysteresis operators for vector-valued inputs by functions on strings

In 1996, Brokate and Sprekels have shown that scalar-valued hysteresis operators for scalar-valued continuous input functions being piecewise monotone can be uniquely represented by functionals defined on the set of all finite alternating strings of real numbers.In this work, it is shown that a similar result can also be derived for hysteresis operators dealing with inputs in a general normed vector space. Considering hysteresis operators defined for continuous inputs that are piecewise monotaffine, it will be shown that these operators can be uniquely represented by functionals acting on an appropriate set of finite strings of elements of this space.     

Exact one-point function of N=1 super-Liouville theory with boundary

In this paper, exact one-point functions of N=1 super-Liouville field theory in two-dimensional space–time with appropriate boundary conditions are presented. Exact results are derived both for the theory defined on a pseudosphere with discrete (NS) boundary conditions and for the theory with explicit boundary actions which preserves super conformal symmetries. We provide various consistency checks. We also show that these one-point functions can be related to a generalized Cardy conditions along with corresponding modular S-matrices. Using this result, we conjecture the dependence of the boundary two-point functions of the (NS) boundary operators on the boundary parameter.     

Second order connections and stochastic horizontal lifts

In this paper we develop a theory of second order connections with a view towards the associated stochastic calculus. Connections in principal fiber bundles are defined as sections of the tangent space of second order differential operators. We prove existence and uniqueness of stochastic horizontal lifts for semimartingales with respect to these connections. Finally, the parallel transport along semimartingales on the base space is studied.     

On the calculation of the Berry phase in a solvable model

We study fermions defined on a one-dimensional interval, for which the interaction is given by a four-parameter family of boundary conditions. We compare the full solution to the adiabatic approximation and determine the Berry phase for a number of typical orbits in parameter space. We observe the occurrence of a non-trivial fundamental group and discuss the possibilities of avoided crossings and apparent crossings.     

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