[en] The problem of quantizing dissipative systems is shown to be reducible, through embedding, to the quantization of volume prerserving dynamics. A Heisenberg picture quantization scheme for these dynamics is developed and applied to two simple examples

[en] We consider two classifications of real Kummer quartics. They use the Heisenberg invariance of Kummer quartics. The first divides the whole variety of real Kummer quartics into four classes according to the Heisenberg-invariance type and then subdivides each class into subclasses to obtain a deformation classification. This subdivision into subclasses is performed by means of the topological classification of the real parts of real Kummer quartics. The second classification deals with the set of real Kummer quartics with a fixed Heisenberg group. Such a set consists of a continuous part and a discrete part. We describe the deformation classes of the continuous part and describe its discrete part. (paper)

[en] The problem of describing all N(N-1)/2 states in two-magnon sector of 1D periodic S = 1/2 chain with the Hamiltonian H = -1/2Σ^N_j≠lρ(j - l)(σ_jσl-1)/2 of spin interaction via elliptic Weierstrass ρ function is investigated. It is proved that the set of eigenvectors having Hermite-like form is complete. (author). 9 refs

[en] A new approach is proposed to solve the quantum evolution problem for a system with an arbitrary number of coupled optical parametric processes. Our method is based on the canonical transformations which define the evolution of the system in the Heisenberg picture. This theory overcomes the difficulties arising in the Wei–Norman method. The application of the approach developed is illustrated with the example of generation of a three-mode entangled light field. (paper)

[en] In this note we present some natural extensions of the commutator that can be used for a generalization of Heisenberg's equations. An example of extension of the commutator xy-yx is p(x,y) = axbyc - aybxc, where a,b,c are noncommuting coefficients inserted between the variables. If tr denotes trace, then for this extension of the commutator the following inequalities are generally valid: trp(x,y) not equal to 0, trxp(x,y) not equal to 0, tryp(x,y) not equal to 0, trxp(y,z) not equal to trp(x,y)z; whereas for the commutator, p(x,y) = xy-yx, all these inequalities become identities. If a,b,c,x,y are elements of an associative algebra A, then the algebra A*, whose elements are the same as those of A but with the new composition law x*y = axbyc - aybxc, is not a Lie algebra; whereas if the new composition law * is the commutator, then the algebra A* is a Lie algebra. This note is a first attempt to write some extensions of the commutator that preserve the above mentioned properties of the commutator: trace zero, orthogonal to its arguments, associative and Lie-admissible

[en] In this paper we define d-polynomials which are generalized polynomials with noncommuting coefficients inserted between the variables. We show that some d-polynomials can be used as an extension of the brackets of Nambu. As a result, we obtain that n mutually commutative Heisenberg pairs of canonical variables, sigma-Pauli matrices, and lambda-Gell-Mann matrices can be considered in a unified way as canonical lists of variables

[en] Relevant algebraic structures for the description of quantum mechanics in the Heisenberg picture are replaced by tensor fields on the space of states. This replacement introduces a differential geometric point of view which allows for a covariant formulation of quantum mechanics under the full diffeomorphism group. (paper)

[en] We renormalize the model of multiple Dirac delta potentials in two and three dimensions by regularizing it through the minimal extension of Heisenberg algebra. We show that the results are consistent with the other regularization schemes given in the literature. (paper)

[en] The position-momentum Shannon and Renyi uncertainty products of general quantum systems are shown to be bounded not only from below (through the known uncertainty relations), but also from above in terms of the Heisenberg-Kennard product < r²>< p²>. Moreover, the Cramer-Rao, Fisher-Shannon, and Lopez-Ruiz, Mancini, and Calbet shape measures of complexity (whose lower bounds have been recently found) are also bounded from above. The improvement of these bounds for systems subject to spherically symmetric potentials is also explicitly given. Finally, applications to hydrogenic and oscillator-like systems are done.

[en] In this paper we discuss the norms of the Bethe states for the spin $\frac{1}{2}$ Heisenberg chain in the critical regime. Our analysis is based on the ODE/IQFT correspondence. Together with numerical work, this has lead us to formulate a set of conjectures concerning the scaling behavior of the norms. Also, we clarify the rôle of the different Hermitian structures associated with the integrable structure studied in the series of works of Bazhanov, Lukyanov and Zamolodchikov in the mid nineties.