Wigner distributions for finite dimensional quantum systems: An algebraic approach

We discuss questions pertaining to the definition of ‘momentum’, ‘momentum space’, ‘phase space’ and ‘Wigner distributions’; for finite dimensional quantum systems. For such systems, where traditional concepts of ‘momenta’ established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail     

Lorentz cobordism. II

It is shown that ‘changes of topology’ (of spacelike sections) in the spacetime of classical general relativity are consistent with the following requirements: (i) stable causality, (ii) future causal geodesic completeness, and (iii) finite, positive energy density. This amounts to showing that the framework of classical general relativity encompasses ‘changes of topology’.     

Modification of Lytle’s theory: Interpretation of the extended fine structure associated with LIII absorption discontinuity of some ytterbium systems

In Lytle’s theory for the extended fine structure in x-ray absorption spectra, the potential at the boundary of the ‘equivalent sphere’ around the absorbing atom, having volume equal to that of the Wigner-Seitz cell is considered to be infinite. It has been observed that Lytle’s theory is applicable only in the case of metals and metallic systems. In the present paper the extended fine structure associated with the L_III absorption spectra of some systems of ytterbium is interpreted on the basis of Lytle’s model, modified by using a finite potential instead of an infinite one at the boundary of the equivalent sphere. The values of this potential are estimated for eight systems of ytterbium. It has been shown that there exists a correlation between the potentials and covalency of the compounds.     

Infinity and Newton’s Three Laws of Motion

It is shown that the following three common understandings of Newton’s laws of motion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisible continuous body with finite mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible.     

Gravitational scattering in the ADD model revisited

It is argued that the assumption that the standard model particles live on a finite brane in the ADD model does in itself imply a finite propagator for virtual Kaluza–Klein mode exchange. The part of the propagator relevant for large distance scattering is cut-off-independent for scattering at distances large compared to the brane width. The matrix element corresponding to this part can also, at least for an odd number of extra dimensions, be Fourier transformed to position space, giving back the extra-dimensional version of Newton’s law. For an even number of extra dimensions a corresponding result is found by requiring that Newton’s law should be recovered. PACS 04.50.+h; 11.10.Kk; 11.25.Mj     

Massless Sine-Gordon and Massive Thirring Models: Proof of Coleman’s Equivalence

We prove Coleman’s conjecture on the equivalence between the massless Sine-Gordon model with finite volume interaction and the Thirring model with a finite volume mass term.     

Conformational analysis of allyl halides from the calculation of indirect spin-spin coupling

A theoretical study on the conformation of allyl halides from the calculation of nuclear spin-spin coupling constants by adopting the finite perturbation theory (FPT), is carried out in terms of the self-consistent, semi-empirical INDO (intermediate neglect of differential overlap) approximation of molecular orbital theory. Results of the calculations performed using ‘s’ and ‘p’ valence orbitals alone (‘sp’ basis) at INDO level approximation seem to replicate the experimental trend quite satisfactorily. Despite the overall agreement of the theoretical values with the experimental ones, the uncertainties in the INDO parametrization scheme lead to overestimation of certain coupling constants. The calculations also show that the orientation of the coupled protons with respect to the substituent halogen atom is an important factor to be considered.     

On the instability of a homogeneous state of a weakly interacting Bose gas under external cooling

The behavior of a weakly interacting Bose gas with a finite particle lifetime has been studied in the framework of hydrodynamic equations under the conditions of a constant mass and energy inflow in the presence of external cooling. A spatially homogeneous state of such a gas is shown to be unstable with respect to the formation of an inhomogeneous density structure. A possible connection of the present results with experiments [3, 4] is discussed. Original Russian Text ? R.B. Saptsov, 2007, published in Pis’ma v Zhurnal éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2007, Vol. 86, No. 10, pp. 779–784.     

Gallavotti–Cohen Theorem,Chaotic Hypothesis and the Zero-Noise Limit

The Fluctuation Relation for a stationary state, kept at constant energy by a deterministic thermostat—the Gallavotti–Cohen Theorem— relies on the ergodic properties of the system considered. We show that when perturbed by an energy-conserving random noise, the relation follows trivially for any system at finite noise amplitude. The time needed to achieve stationarity may stay finite as the noise tends to zero, or it may diverge. In the former case the Gallavotti–Cohen result is recovered, while in the latter case, the crossover time may be computed from the action of ‘instanton’ orbits that bridge attractors and repellors. We suggest that the ‘Chaotic Hypothesis’ of Gallavotti, Cohen and Ruelle can thus be reformulated as a matter of stochastic stability of the measure in trajectory space. In this form this hypothesis may be directly tested.     

Spin dynamics of Rashba-Dresselhaus two-dimensional electron systems with electron-electron interactions

Spin dynamics of Rashba-Dresselhaus two-dimensional electron systems is studied by taking account of electron-electron interactions under the D’yakonov-Perel’ mechanism. The diffusion equations for charge and spin densities are obtained through decoupling of the interactions using the auxiliary Bose field. We show that the electron-electron interaction has no effect on the infinite spin lifetime when the Rashba and Dresselhaus coupling constants satisfy the condition α = ±β. If the general condition α≠±β is satisfied, the spin lifetime is finite and enhanced by the electron-electron interaction with the increment of the temperature in the ballistic regime. The increasing amplitude of the spin lifetime depends on the ratio of the temperature to the Fermi temperature.     

Quasiparticles in the theory of fermion condensation

It is shown that Landau’s quasiparticle formalism continues to work in systems with a fermion condensate. In the case of a finite system this formalism is suitable for describing the restructuring of states at the Fermi surface. It also works in an infinite system, and the idea of quasiparticles at low temperature as well-defined excitations at the Fermi surface remains valid. The quasiparticle lifetime is directly proportional to the temperature, and the density of states is inversely proportional to the temperature. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 9, 719–723 (10 May 1996)     

Double-Delta Potentials: One Dimensional Scattering The Casimir Effect and Kink Fluctuations

The path is explored between one-dimensional scattering through Dirac-δ walls and one-dimensional quantum field theories defined on a finite length interval with Dirichlet boundary conditions. It is found that two δ’s are related to the Casimir effect whereas two δ’s plus the first transparent P?sch-Teller well arise in the context of the sine-Gordon kink fluctuations, both phenomena subjected to Dirichlet boundary conditions. One or two delta wells will be also explored in order to describe absorbent plates, even though the wells lead to non unitary Quantum Field Theories.     

On the Convergence to the Continuum of Finite Range Lattice Covariances

In (J. Stat. Phys. 115:415–449, 2004) Brydges, Guadagni and Mitter proved the existence of multiscale expansions of a class of lattice Green’s functions as sums of positive definite finite range functions (called fluctuation covariances). The lattice Green’s functions in the class considered are integral kernels of inverses of second order positive self-adjoint elliptic operators with constant coefficients and fractional powers thereof. The rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing L ⁻ⁿ and satisfies uniform bounds. Our main result in this note is that the sequence of these terms converges in appropriate norms at a rate L ^−n/2 to a smooth, positive definite, finite range continuum function.     

Existence and Uniqueness Theorems for Massless Fields on a Class of Spacetimes with Closed Timelike Curves

We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC’s), in which all future-directed CTC’s traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in L ₂ on ℐ ⁻ , we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite L ₂ -norm) and have the given initial data on ℐ ⁻ . A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position. For a complementary class of spacetimes in which CTC’s are confined to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC’s. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally, we discuss a conjecture whose meaning is essentially that the Cauchy problem for free fields is well defined in the presence of CTC’s whenever the problem is well-posed in a geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region. Received: 28 November 1994/Accepted: 20 May 1996     

Heisenberg’s uncertainty relation may be violated in a single measurement

The well-known Heisenberg’s uncertainty relation is an inequality between uncertainties of canonically conjugate observables in a given state. In this interpretation, the Heisenberg’s uncertainty relation is a rigorous mathematical theorem and is, therefore, always valid. However, the same inequality is often applied in the situation of measurement, where it is illustrated in a quite different way. The uncertainty relation is then an inequality connecting the precision (resolution) of the measurement of one observable and the uncertainty of the conjugate observable in the state arising after the measurement. It turns out that in such an interpretation the Heisenberg’s inequality may be violated for some measurement readouts that emerge with small but finite probabilities. Making use of the uncertainties averaged in a special way over all possible measurement readouts, one may formulate an inequality of the type of Heisenberg’s inequality but valid for any measurement. Paper submitted by the author in English on 28 April 2006.     

Stability of quark gluon plasma to nielsen-olesen mode

Nielsen and Olesen showed that perturbative vacuum with uniform chromomagnetic field in one space and one color direction is unstable. This instability is called Nielsen-Olesen instability (NOI), and leads to formation of a ‘spaghetti of flux tubes’ as a model for non-perturbative vacuum and confinement. We re-examine this instability in presence of color sources, quarks and gluons, at a finite temperature and find that at sufficiently high temperature NOI is stabilized due to an ‘effective mass’ of gluons arising through plasma effects. This explains how a QGP with no confinement effects may exist at high temperature. As the temperature is lowered, NOI reappears at a valueT=T _c, which is very close to confinement-deconfinement transition from hadrons to QGP..     

Non-trivial Berry phase for an asymmetric one-dimensional potential in the free electron limit

We find a non-trivial Berry phase for a finite potential, even in the delocalized-electron limit. We obtain this result for a one-dimensional, periodic sawtooth potential, though our analysis is applicable to other systems. We show that this result is due to both the periodicity of the time-independent wave function and the effects of interband degeneracies. We also find, for the sawtooth potential, that each band is uniquely characterized by its Berry phase’s functional dependence on the potential’s parameters. Thus, each band may be identified by the behavior of its Berry phase.     

Evidence of quantum interference in transport properties of a triple quantum dot T-shape system

We consider the transport and the noise characteristic in the case of a triple quantum dots T-shape system where two of the dots form a two-level system and the other works in a detector-like setup. Our theoretical results are obtained using the equation of motion method for the case of zero and finite on-site Coulomb interaction in the detector dot. We present analytic results for the electronic Green’s functions in the system’s component quantum dots, and we used numerical calculations to evaluate the system’s transport properties. The transport trough the T-shaped system can be controlled by varying the coupling between the two-level system dots or the coupling between the detector dot and the exterior electrodes. The system’s conductance presents Fano dips for both strong (fast detector) and weak coupling (slow detector) between the detector dot and the external electrodes. Due to stronger electronic correlations the noise characteristics in the case of a slow detector are much higher. This setup may be of interest for the practical realization of qubit states in quantum dots systems.     

Perfect-Fluid Sources for the Levi-Civita Metric II

A regular static interior solution of Einstein’s field equations representing a perfect fluid cylinder of finite radius is presented. The solution is matched to the Levi-Civita vacuum solution at a boundary where the pressure vanishes. The density and pressure are finite and positive inside the cylinder for a specific range of the mass parameter. The solution could thus represent a reasonable source for the Levi-Civita metric.