Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.Comment: 14 pages. Latex_2e fil

Singular indecomposable representations of sl(2,ℂ) and relativistic wave equations

A detailed summary is given of the structure of singular indecomposable representations of si (2,ℂ), as developed by Gel'fand and Ponomarev [Usp. Mat. Nauk 23, 3 (1968); translated in Russ. Math. Surveys 23, 1 (1968)]. A variety of four-vector operators Γμ is constructed, acting within direct sums of such representations, including some with nonsingular Γ0. Associated wave equations of Gel'fand-Yaglom type are considered that admit timelike solutions and lead to mass-spin spectra of the Majorana type. A subclass of these equations is characterized in an invariant way by obtaining basis-independent expressions for the commutator and anticommutator of Γμ and Γν. A brief discussion is given of possible applications to physics of these equations and of others in which nilpotent scalar operators appear

Parafermionic algebras, their modules and cohomologies

We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded 2-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant's theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincar\'e characteristics of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.Comment: 10 pages, talk presented at the International Workshop "Lie theory and its applications in Physics" (17-23 June 2013, Varna, Bulgaria

Temperature and filling dependence of the superconducting π\pi-phase in the Penson-Kolb-Hubbard model

We investigate in the Hartree Fock approximation the temperature and filling dependence of the superconducting $\pi$-phase for the Penson-Kolb-Hubbard model. Due to the presence of the pair-hopping term, the phase survives for repulsive values of the on-site Coulomb interaction, exhibiting an interesting filling and temperature dependence. The structure of the self-consistent equations peculiar to the $\pi$-phase of the model allows to explicitly solve them for the chemical potential. The phase diagrams are shown and discussed in dimension 2 and 3. We also show that, when a next-nearest neighbours hopping term is included, the critical temperature of the superconducting region increases, and the corresponding range of filling values is shifted away from half-filling. Comparison with known exact results is also discussed.Comment: 20 pages, REVTEX, 8 eps figure

Weak measurement of arrival time

The arrival time probability distribution is defined by analogy with the classical mechanics. The difficulty of requirement to have the values of non-commuting operators is circumvented using the concept of weak measurements. The proposed procedure is suitable to the free particles and to the particles subjected to an external potential, as well. It is shown that such an approach imposes an inherent limitation to the accuracy of the arrival time determination.Comment: 3 figure

Finite-dimensional representations of the quantum superalgebra Uq[gl(n/m)]U_q[gl(n/m)] and related q-identities

Explicit expressions for the generators of the quantum superalgebra $U_q[gl(n/m)]$ acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of $q$-number identities.Comment: 12 page

Detection model based on representation of quantum particles by classical random fields: Born's rule and beyond

Recently a new attempt to go beyond quantum mechanics (QM) was presented in the form of so called prequantum classical statistical field theory (PCSFT). Its main experimental prediction is violation of Born's rule which provides only an approximative description of real probabilities. We expect that it will be possible to design numerous experiments demonstrating violation of Born's rule. Moreover, recently the first experimental evidence of violation was found in the triple slits interference experiment, see \cite{WWW}. Although this experimental test was motivated by another prequantum model, it can be definitely considered as at least preliminary confirmation of the main prediction of PCSFT. In our approach quantum particles are just symbolic representations of "prequantum random fields," e.g., "electron-field" or "neutron-field"; photon is associated with classical random electromagnetic field. Such prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. `t Hooft's attempt to go beyond QM \cite{H1}--\cite{TH2}. In this paper we elaborate a detection model in the PCSFT-framework. In this model classical random fields (corresponding to "quantum particles") interact with detectors inducing probabilities which match with Born's rule only approximately. Thus QM arises from PCSFT as an approximative theory. New tests of violation of Born's rule are proposed.Comment: Relation with recent experiment on violation of Born's rule in the triple slit experiment is discussed; new experimental test which might confirm violation of Born's rule are presented (double stochsticity test and interference magnitude test); the problem of "double clicks" is discusse

Time-of-arrival distributions from position-momentum and energy-time joint measurements

The position-momentum quasi-distribution obtained from an Arthurs and Kelly joint measurement model is used to obtain indirectly an ``operational'' time-of-arrival (TOA) distribution following a quantization procedure proposed by Kocha\'nski and W\'odkiewicz [Phys. Rev. A 60, 2689 (1999)]. This TOA distribution is not time covariant. The procedure is generalized by using other phase-space quasi-distributions, and sufficient conditions are provided for time covariance that limit the possible phase-space quasi-distributions essentially to the Wigner function, which, however, provides a non-positive TOA quasi-distribution. These problems are remedied with a different quantization procedure which, on the other hand, does not guarantee normalization. Finally an Arthurs and Kelly measurement model for TOA and energy (valid also for arbitrary conjugate variables when one of the variables is bounded from below) is worked out. The marginal TOA distribution so obtained, a distorted version of Kijowski's distribution, is time covariant, positive, and normalized

Exact thermodynamics of an Extended Hubbard Model of single and paired carriers in competition

By exploiting the technique of Sutherland's species, introduced in \cite{DOMO-RC}, we derive the exact spectrum and partition function of a 1D extended Hubbard model. The model describes a competition between dynamics of single carriers and short-radius pairs, as a function of on-site Coulomb repulsion ($U$) and filling ($\rho$). We provide the temperature dependence of chemical potential, compressibility, local magnetic moment, and specific heat. In particular the latter turns out to exhibit two peaks, both related to `charge' degrees of freedom. Their origin and behavior are analyzed in terms of kinetic and potential energy, both across the metal-insulator transition point and in the strong coupling regime.Comment: 14 pages, 15 eps figure

Exact diagonalization of the generalized supersymmetric t-J model with boundaries

We study the generalized supersymmetric $t-J$ model with boundaries in three different gradings: FFB, BFF and FBF. Starting from the trigonometric R-matrix, and in the framework of the graded quantum inverse scattering method (QISM), we solve the eigenvalue problems for the supersymmetric $t-J$ model. A detailed calculations are presented to obtain the eigenvalues and Bethe ansatz equations of the supersymmetric $t-J$ model with boundaries in three different backgrounds.Comment: Latex file, 32 page