Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice , which is a sublattice of the lattice of all quantum propositions and is determined by a quantum state e and a preferred determinate observable R. Topos-theoretic extension is made in the functor category of which base category is determined by R. Each true atom, which determines truth values, true or false, of all propositions in , generates also a multi-valued valuation function of which domain and range are and a Heyting algebra given by the subobject classifier in , respectively. All true propositions in are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category . Here, the base category includes all ’s as subcategories. Although has a structure apparently different from , a subobject semi-classifier of gives valuations completely equivalent to those in ’s.     

Lower Bound of Minimal Time Evolution in Quantum Mechanics

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S ², must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2×2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector P of the evolving quantum state with the vector of the 2×2 hermitian Hamiltonians and it is shown that the Hamiltonian H can be parameterized by two independent parameters and Θ.     

Weakly Non-Ergodic Statistical Physics

For weakly non ergodic systems, the probability density function of a time average observable is where is the value of the observable when the system is in state j=1,…L. p _j ^eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _j ^eq is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x ²〉∼t ^α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered . We briefly discuss possible physical applications in single particle experiments.     

A Note on the Extreme Points of Positive Quantum Operations

In this paper, an error in the proof of Theorem 4.9 in Gudder’s paper (Int. J. Theor. Phys. 47(1):268–279, 2008) is pointed out and it is proved that if such that E _i ∈ℂI∖{0} and E _j ∉ℂI for some i,j in {1,2,…,n}, then . This subject is supported by the NNSF of China (No. 10571113, 10871224).     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n ^−θ(d) where θ(d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n ^{−2(θ(2)+θ(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like and where θ _∞ is such that in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by where N _ab is the mean number of real roots in [a,b] and a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, 2007).     

NLO contributions to <Emphasis Type="Italic">B</Emphasis>→<Emphasis Type="Italic">KK</Emphasis><Superscript>*</Superscript> decays in the pQCD approach

We calculate the important next-to-leading-order (NLO) contributions to the B→KK ^* decays from the vertex corrections, the quark loops, and the magnetic penguins in the perturbative QCD (pQCD) factorization approach. The pQCD predictions for the CP-averaged branching ratios are , , and Br(B ⁰→K ⁺ K ^*−+K ⁻ K ^*+)≈1.3×10⁻⁷, which agree well with both the experimental upper limits and the predictions based on the QCD factorization approach. Furthermore, the CP violating asymmetries of the considered decay modes are also evaluated. The NLO pQCD predictions for and decays are and .     

An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model

In terms of the loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of 2 ^r eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the τ ₂-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the τ ₂-model commutes with the loop algebra, , and every regular Bethe state of the τ ₂-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the τ ₂-model generates an -degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.     

Constraints on unparticle interactions from particle and antiparticle oscillations

Unparticles have dramatic effects on particle and antiparticle oscillations in meson–antimeson and muonium–antimuonium systems. Unlike the usual tree-level contributions to meson oscillations from heavy-particle exchange, which results in a small Γ ₁₂, the unparticle may have sizeable contributions to both M ₁₂ and Γ ₁₂ due to the fractional dimension of the unparticle. If the unparticle effect dominates the contributions (which may happen in D ⁰– mixing) to the meson mixing parameters x and y, we find that . The mass difference Δm in meson mixing can provide interesting constraints on the unparticle interactions. The unparticle interaction can significantly enhance the CP asymmetry in meson mixing, which can be tested in more accurate experiments in the future. Interesting constraints on unparticle and particle interactions can also be obtained using muonion and antimuonion oscillation data.     

Spontaneous parity violation and minimal Higgs models

In this paper we present a model for the spontaneous breaking of parity with two Higgs doublets and two neutral Higgs singlets which are even and odd under -parity. The condition can be satisfied without introducing bidoublets, and it is induced by the breaking of -parity through the vacuum expectation value of the odd Higgs singlet. Examples of left–right symmetric and mirror fermions models in grand unified theories are presented. PACS 12.60.Cn; 14.80.Cp; 12.10.Dm     

Thermal unparticles: a new form of energy density in the universe

An unparticle with scaling dimension has peculiar thermal properties due to its unique phase space structure. We find that the equation of state parameter , the ratio of pressure to energy density, is given by providing a new form of energy in our universe. In an expanding universe, the unparticle energy density evolves dramatically differently from that for photons. For , even if at a high decoupling temperature T _D is very small, it is possible to have a large relic density at present photon temperature T _γ ⁰, large enough to play the role of dark matter. We calculate T _D and using photon–unparticle interactions for illustration.     

Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family of measures on a space of functions on the two-torus, parametrized by a polynomial P (the Wess-Zumino-Landau-Ginzburg model). The second is a family of measures on a space of maps from to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family of measures on the product of a space of connections on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of -valued three-forms on M. We show that these measures are positive, and that the measures are Borel probability measures. As an application we show that formulas arising from expectations in the measures reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures , where M is a homology three-sphere, will yield the Casson invariant of M. Dedicated to the memory of Raoul Bott Supported in part by NSF grant DMS 04/05670.     

Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator Chains

We examine the thermal conductivity and bulk viscosity of a one-dimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPU-α β system in a subset of its parameter space. We identify three distinct frequency regimes which we call the hydrodynamic regime, the perturbative regime and the collisionless regime. In the lowest frequency regime (the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as ω→0 the frequency dependent bulk viscosity, , and the frequency dependent thermal conductivity, , should diverge with the same power law dependence on ω. Thus, we can define the bulk Prandtl number, , where m is the particle mass and k _B is Boltzmann’s constant. This dimensionless ratio should approach a constant value as ω→0. We use mode-coupling theory to predict the ω→0 limit of Pr _ζ . Values of Pr _ζ obtained from simulations are in agreement with these predictions over a wide range of system parameters. In the middle frequency regime, which we call the perturbative regime, heat is transported by sound modes which are damped by four-phonon processes. This regime is characterized by an intermediate-frequency plateau in the value of . We find that the value of in this plateau region is proportional to T ⁻² where T is the temperature; this is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. The Boltzmann-Peierls approach fails, however, to give a nonvanishing bulk viscosity for all FPU-α β chains. We call the highest frequency regime the collisionless regime since at these frequencies the observing times are much shorter than the characteristic relaxation times of phonons.     

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds

The cotangent bundle T ^* X to a complex manifold X is classically endowed with the sheaf of k-algebras of deformation quantization, where k := is a subfield of . Here, we construct a new sheaf of k-algebras which contains as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of , we show that is well defined in .     

Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators which act on . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in can be expressed in terms of an exponential of a quadratic form of . On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992, Moscow, Russia Membre du CNRS     

First-Order Intertwining Operators with Position Dependent Mass and <Emphasis Type="Italic">η</Emphasis>-Weak-Pseudo-Hermiticity Generators

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM—Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding η-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models ( -symmetric and non- -symmetric) are given as products of the reference-target map.