Ground states and spectrum of quantum electrodynamics of nonrelativistic particles

A system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered. An ultraviolet cut-off is imposed on the quantized radiation field. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. The existence of the ground states of the Hamiltonian is established. It is shown that there exist asymptotic annihilation and creation operators. Hence the location of the absolutely continuous spectrum of the Hamiltonian is specified.

     

Connection of fields quantized in overlapping regions and the Unruh effect

For the example of a field quantized for positive values of the spatial coordinate (when particle creation certainly cannot occur) it is shown that incorrect use of the formula for the creation and annihilation operators leads to a transformation from the creation and annihilation operators to those in Minkowski space that corresponds to particle creation. It is shown that the connection of fields quantized in the Minkowski and Rindler spaces has an analogous nature, i.e., a creation effect cannot be observed in the Rindler space. The correspondence between subspaces of states of these fields is considered.Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 217–233, May, 1992.     

Parapositronium atom in the field of annihilation and optical photons as a nonlinear quantum system

A parapositronium atom in an optical laser field is described beyond the perturbation theory framework by a closed system of Heisenberg equations on operators of atoms and photons. Wwe consider the annihilation of the parapositronium atom, which starts from one or two quantum states; optical quantum transitions between these states are caused by one or two optical photons. Mean occupation numbers of these states are governed by a system of two nonlinear equations. We investigated particular stationary and nonstationary solutions of this system and find that annihilation photons substantially affect the annihilation process. We show that definite optical laser radiation may stabilize the parapositronium atom and make its lifetime hundreds of times longer than the lifetime of the free parapositronium atom in the 1s state. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 148–168, July, 2000.     

Conditions Sufficient for the Conservativity of a Minimal Quantum Dynamical Semigroup

Conditions sufficient for a minimal quantum dynamical semigroup (QDS) to be conservative are proved for the class of problems in quantum optics under the assumption that the self-adjoint Hamiltonian of the QDS is a finite degree polynomial in the creation and annihilation operators. The degree of the Hamiltonian may be greater than the degree of the completely positive part of the generator of the QDS. The conservativity (or the unital property) of a minimal QDS implies the uniqueness of the solution of the corresponding master Markov equation, i.e., in the unital case, the formal generator determines the QDS uniquely; moreover, in the Heisenberg representation, the QDS preserves the unit observable, and in the Schrödinger representation, it preserves the trace of the initial state. The analogs of the conservativity condition for classical Markov evolution equations (such as the heat and the Kolmogorov--Feller equations) are known as nonexplosion conditions or conditions excluding the escape of trajectories to infinity.     

Wick products of the CAR algebra

The purpose of this paper is to put in a precise mathematical (algebraic) form the Wick products of the CAR algebra. We state in detail the reduction of the ordinary product of Fermi fields in terms of a finite sum of monomials in the creation and annihilation operators in which all creation operators occur to the left of all annihilation operators (Wick-ordered) and the Fock (vacuum) state of the former.

     

Local Gaussian Dominance: An Anharmonic Excitation of Free Bosons

We prove the local Gaussian dominance condition for a Bose system whose Hamiltonian is diagonal with respect to the particle number operators. The proof is based on obtaining an upper bound estimate for the Bogoliubov inner product of the Bose creation and annihilation operators.     

Wick products of the CCR algebra

The purpose of this paper is to put in a precise mathematical (algebraic) form the Wick products of the CCR algebra. We state in detail the reduction of ordinary product of Bose fields in terms of a finite sum of monomials in the creation and annihilation operators in which all creation operators occur to the left of all annihilation operators (Wick‐ordered) and the Fock (vacuum) state of the former. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Inverse problem equations for a three-wave quantum system

Analytic properties of the Jost solutions of the auxiliary linear problem of a three-wave system, whose potentials are operator-valued functions, are studied. Creation and annihilation operators of elementary excitations and their bound states are constructed. Singular integral equations which let one reconstruct the local fields from the creation and annihilation operators mentioned are derived.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 53–69, 1986.     

Spectrum and states of the bcs hamiltonian in a finite domain. I. Spectrum

The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs, as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic form with respect to the operators of creation and annihilation.     

Quantum stochastic calculus and quantum Gaussian processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy [9]. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space Γ(? ⁿ ) over ? ⁿ . These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn [19]. They were recently investigated in the context of quantum information theory by Heinosaari et al. [7]. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.     

Anomalous scaling in statistical models of passively advected vector fields

We use the methods of the renormalization group and the operator product expansion to consider the problem of the stochastic advection of a passive vector field with the most general form of the nonlinear term allowed by the Galilean symmetry. The external velocity field satisfies the Navier-Stokes equation. We show that the correlation functions have anomalous scaling in the inertial range. The corresponding anomalous exponents are determined by the critical dimensions of tensor composite fields (operators) built from only the fields themselves. We calculate the anomalous dimensions in the leading order of the expansion in the exponent in the correlator of the external force in the Navier-Stokes equation (the oneloop approximation of the renormalization group). The anomalous exponents exhibit a hierarchy related to the anisotropy degree: the lower the rank of the tensor operator is, the lower its dimension. The leading asymptotic terms are determined by the scalar operators in both the isotropic and the anisotropic cases, which completely agrees with Kolmogorov’s hypothesis of local isotropy restoration.     

A modified Bogoliubov method applied to a simple boson model

We use a non-gauge-invariant modification of the exact Hamiltonian to obtain a new Hamiltonian-like operator for a simple exactly solvable boson model. The eigenvalues of the new operator are close to those of the original Hamiltonian. We make a one-body approximation of the new two-body operator in the spirit of the Bogoliubov approximation. Because only the number operator appears, the c-number approximation is not required individually for the creation or annihilation operators in the ground state. For the simple model, the results using the new approximation are closer to the exact results than the usual Bogoliubov results over a wide range of parameters. The improvement increases dramatically as the model interaction strength increases.     

Dynamically evolving Gaussian spatial fields

We discuss general non-stationary spatio-temporal surfaces that involve dynamics governed by velocity fields. The approach formalizes and expands previously used models in analysis of satellite data of significant wave heights. We start with homogeneous spatial fields. By applying an extension of the standard moving average construction we obtain models which are stationary in time. The resulting surface changes with time but is dynamically inactive since its velocities, when sampled across the field, have distributions centered at zero. We introduce a dynamical evolution to such a field by composing it with a dynamical flow governed by a given velocity field. This leads to non-stationary models. The models are extensions of the earlier discretized autoregressive models which account for a local velocity of traveling surface. We demonstrate that for such a surface its dynamics is a combination of dynamics introduced by the flow and the dynamics resulting from the covariance structure of the underlying stochastic field. We extend this approach to fields that are only locally stationary and have their parameters varying over a larger spatio-temporal horizon.     

Spectral transform for the sub-Laplacian on the Heisenberg group

We continue our analysis of nilpotent groups related to quantum mechanical systems whose Hamiltonians have polynomial interactions. For the spinless particle in a constant external magnetic field, the associated nilpotent group is the Heisenberg group. We solve the heat equation for the Heisenberg group by diagonalizing the sub-Laplacian. The unitary map to the Hilbert space in which the sub-Laplacian is a multiplication operator with positive spectrum is given. The spectral multiplicity is shown to be related to the irreducible representations of SU(2). A Lax pair, generated from the Heisenberg sub-Laplacian, is used to find operators unitarily equivalent to the sub-Laplacian, but not arising from the SL(2,R) automorphisms of the Heisenberg group. Department of Mathematics, supported in part by NSF. Department of Physics and Astronomy, supported in part by DOE.     

Central limit theorem for positively associated stationary random fields

Positively associated stationary random fields on d-dimensional integral lattice arise in various models of mathematical statistics, percolation theory, statistical physics, and reliability theory. In this paper, we shall be concerned with a field with covariance functions satisfying a more general condition than summability. A criterion for the validity of the central limit theorem (CLT) for partial sums of a field from this class is established. The sums are taken over an increasing nest of parallelepipeds or cubes. The well-known conjecture of Newman stated that for an associated stationary random field the above condition on the covariance function should force the CLT to hold. As was shown by N. Herrndorf and A. P. Shashkin, this conjecture fails already for d = 1. In the present paper, the uniform integrability of the squared partial sums is shown as being of key importance for the CLT to hold. Thus, an extension of Lewis’s theorem proved for a sequence of random variables is obtained. Also, it is indicated how to modify Newman’s conjecture for any d. A representation of variances of partial sums of a field by means of slowly varying functions of several arguments is used in an essential way.     

Nash and log-Sobolev inequalities for hypoelliptic operators

By estimating the intrinsic distance and using known heat kernel upper bounds, the global Nash inequality with exact dimension is established for a class of square fields with algebraic growth induced by vector fields satisfying the Hörmander condition. As an application, a sufficient condition is presented for the log-Sobolev inequality to hold. Typical examples for Gruschin type operators and generalized Kohn-Lapacians on Heisenberg groups are provided.     

Spectral representations and density operators for infinite-dimensional homogeneous random fields

A theory of spectral representations and spectral density operators of infinite-dimensional homogeneous random fields is established. Some results concerning the form of the spectral representation are given in the general infinite-dimensional case, while the results pertaining to the density operator are confined to Hilbert space valued fields. The concept of a purely non-deterministic (p.n.d.) field is defined, and necessary and sufficient conditions for the property of p.n.d. are obtained in terms of the spectral density operator. The theory is developed using some isomorphisms induced by families of self-adjoint operators in the linear second order space associated with the field. The method seems to lead to more direct results also in the random process case, and it sheds new light on concepts such as multiplicity of the field and rank of the spectral density operator.     

Representations of general commutation relations

General commutation relations involving creation, annihilation, and particle number operators are considered. Such commutation relations arise in the context of nonstandard Poisson brackets. All possible types of irreducible representations in which the particle number operator or the product of the creation and annihilation operators has a basis of orthonormal eigenvectors are constructed. The irreducible representations that involve the particle number operator reduce to one of four types and those that do not involve the particle number operator reduce to one of five types. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 369–383, December, 1997.     

A Study of Probability Measures Through Commutators

The first-order moments and two families of commutators are proven to determine uniquely the moments of a probability measure on ℝ ^d . These families are the commutators between the annihilation and creation operators, and the commutators between the annihilation and preservation operators. An explicit method for recovering the moments from these commutators and first-order moments is presented.