Boson expansion of lie algebras: The fermion pair algebra

A unitary boson expansion theory for Lie algebras with ladder representations is discussed and applied to the fermion pair algebra SO(2n).     

Boson exchange quasipotentials for a pseudoscalar fermion—antifermion bound state

An explicit form of equations for a relativistic wave function of a fermion-antifermion pair with J^P = 0^– is obtained in the framework of covariant single-time approach. The cases of vector, scalar, pseudovector, and pseudoscalar boson exchange are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 29–31, July, 1991.     

Semi-classical quantum mechanics and stochastic calculus of variations

Within the framework of the general theory of stochastic calculus of variations, we examine mainly the notion of second variation in the stochastic mechanics of E. Nelson, a representative of quantum mechanics in which the concept of path for particles keep a sense. We show that the two approaches used in classical calculus of variation to know if a path is not only an extremum but also the minimum of the action, namely, the local one (weak minimum) and the global one (strong minimum), can be generalized to include the quantum-mechanical paths. Thus, we can prove that locally, a solution of the classical equation of motion is really the minimum, even in a large class of quantum paths containing the semi-classical trajectories. By introducing a stochastic version of the excess function of Weierstrass, we show the analogous global property. There, of course, one can speak of the principle of least action in a strict sense. Several explicit examples are discussed.     

Unitary evolutions and horizontal lifts in quantum stochastic calculus

Unitarity is proved for a class of solutions of quantum stochastic differential equations with unbounded coefficients. The resulting processes are then used to construct algebraic quantum diffusions. Applications include an existence proof for a class of diffusions on the non-commutative two-torus and a geometric interpretation for diffusions driven by the classical Poisson process.     

Itô's stochastic calculus and the bosonic string

A new perturbative technique for solving a scalar φ^2P theory consists of expanding a φ^2(1+δ) interaction in powers of δ. The Green functions are computed as a power series in δ by applying a linear differential operator to the Green functions of a specially constructed intermediate Lagrangian. We confront this linear procedure with the quadratic requirement of perturbative unitarity. We verify up to order δ³ that unitarity is indeed satisfied, by virtue of the precise structure of the intermediate Lagrangian. Unitarity gives constraints on that structure, but does not fix it uniquely.     

Covariant stochastic calculus in the sense of Itô

The Langevin equation in the sense of Itô on a manifold is put into a manifestly covariant form, which reduces to the original form in harmonic coordinates and directly corresponds to the covariant Fokker-Planck equation and a covariant form of the functional integral in prepoint discretization.     

Forward and backward adapted quantum stochastic calculus and double product integrals

We show that iterated stochastic integrals can be described equivalently either by the conventional forward adapted, or by backward adapted quantum stochastic calculus. By using this equivalence, we establish two properties of triangular (causal) and rectangular double quantum stochastic product integrals, namely a necessary and sufficient condition for their unitarity, and the coboundary relation between the former and the latter.     

Stochastic dynamics: A review of stochastic calculus of variations

We present the main results of a variational calculus for Markovian stochastic processes which allows us to characterize the dynamics of probabilistic systems by extremal properties for some functionals of processes. They generalize, by construction, the main variational formulations of classical dynamics. This framework is used for the dynamical analysis of Nelson's stochastic mechanics, an approach to quantum mechanics in which the concept of trajectory for particles still makes sense. The semiclassical limit is formulated in terms of the second variation of the starting functional. We also use the proposed stochastic calculus of variations in the context of statistical mechanics of systems far from equilibrium, namely, to solve the Onsager-Machlup problem.On leave from Département de Physique Théorique, Université de Genève, CH-I2II, Genève 4, Switzerland.     

Constructing the davies process of resonance fluorescence with quantum stochastic calculus

The starting point is a given semigroup of completely positive maps on the 2×2 matrices. This semigroup describes the irreversible evolution of a decaying two-level atom. By using the integral-sum kernel approach to quantum stochastic calculus, the two-level atom is coupled to an environment, which in this case will be interpreted as the electromagnetic field. The irreversible time evolution of the two-level atom then stems from the reversible time evolution of the atom and the field together. Mathematically speaking, a Markov dilation of the semigroup has been constructed. The next step is to drive the atom by a laser and to count the photons emitted into the field by the decaying two-level atom. For every possible sequence of photon counts, a map is constructed that gives the time evolution of the two-level atom implied by that sequence. The family of maps obtained in this way forms a so-called Davies process. In his book, Davies describes the structure of these processes, which brings us into the field of quantum trajectories. Within the model presented in this paper, the jump operators are calculated and the resulting counting process is briefly described.     

Quantum stochastic differential equations for boson and fermion systems—method of non-equilibrium thermo field dynamics

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of non-equilibrium thermo field dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.     

The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case

We study the fluctuations of free energy, energy and entropy in the high temperature regime for the Sherrington-Kirkpatrick model of spin glasses. We introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter, thus formulating the thermodynamic formalism in terms of random processes. The well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments, a generalization of a result proved by Aizenman, Lebowitz and Ruelle [1].     

QoS-aware resource allocation and fault tolerant operation in hybrid SDN using stochastic network calculus

Mobile edge computing (MEC) is a key feature of next-generation mobile networks aimed at providing a variety of services for different applications by performing related processing tasks closer to the users. With the advent of the next-generation mobile networks, researchers have turned their attention to various aspects of edge computing in an effort to leverage the new capabilities offered by 5G. So, the integration of software defined networking (SDN) and MEC techniques was seriously considered to facilitate the orchestration and management of Mobile Edge Hosts (MEH). Edge clouds can be installed as an interface between the local servers and the core to provide the required services based on the known concept of the SDN networks. Nonetheless, the problem of reliability and fault tolerance will be of great importance in such networks. The paper introduced a dynamic architecture that focuses on the end-to-end mobility support required to maintain service continuity and quality of service. This paper also presents an SDN control plane with stochastic network calculus (SNC) framework to control MEC data flows. In accordance with the entrance processes of different QoS-class data flows, closed-form problems were formulated to determine the correlation between resource utilization and the violation probability of each data flow. Compared to other solutions investigated in the literature, the proposed approach exhibits a significant increase in the throughput distributed over the active links of mobile edge hosts. It also proved that the outage index and the system’s aggregate data rate can be effectively improved by up to 32%.     

Condensates and Correlated Boson Systems

We study two-body correlations in a many-boson system with a hyperspherical approach, where we can use arbitrary scattering length and include two-body bound states. As a special application we look on Bose-Einstein condensation and calculate the stability criterium in a comparison with the experimental criterium and the theoretical criterium from the Gross-Pitaevskii equation.     

Boson expansions and broken symmetry

A new boson mapping for the group SU(2) × SU(2) is used to generate a perturbation expansion in a small parameter, valid in the deformed regime of the two-dimensional Moszkowski model, without violating angular-momentum conservation at any stage. The corrections to the rotational and vibrational energies and transition matrix elements are obtained to an accuracy one order beyond the RPA. These results are used as a criterion for testing the validity of two methods employing conventional boson expansions about a symmetry-broken minimum and correcting for the symmetry violation a posteriori. It is shown that both methods—that of Marshalek and Weneser, and that of Bes et al., adapted here to boson expansions—reproduce the results of the criterion.     

Boson expansions and Hartree-Bogoliubov theory

If the boson operators in boson expansions of the Hamiltonian are suitably replaced by c-numbers regarded as variational parameters, solutions of the Hartree-Bogoliubov or Hartree-Fock-Bogoliubov equations are obtained. The c-number replacement in the boson equations of motion yields solutions of the corresponding time-dependent self-consistent field equations.