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.

     

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.     

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.      

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.     

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.     

Recursion Relation for Wick Products of the CCR Algebra

In this paper we obtain an explicit recursion relation for the Wick products of the CCR algebra in terms of Wick products of lesser order and the Bose fields. From this formula we prove that the Fock (vacuum) state vanishes for the commutation of the Wick products of order n and the Bose fields,being , n > 1. Partially supported by Ministerio de Educación y Ciencia (Spain), MTM2007-65604.     

Why are there only three quantum noises?

In quantum stochastic calculus on the symmetric Fock space over L ²(ℝ₊), adapted processes of operators are integrated with respect to creation, annihilation and number processes. The main property which allows this integration is that the increments of integrators between s and t act only on Fock space over L ²([s, t]). In this article, we prove that there are no other process of closable operators on coherent vectors with this property. Thus the only possible integrators in quantum stochastic calculus are the creation, annihilation and number processes.     

The q-deformed harmonic oscillator, coherent states, and the uncertainty relation

For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the “coordinate-momentum” uncertainties in q-oscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the q-deformation results in a violation of the standard uncertainty relation; the product of the coordinate-and momentum-operator uncertainties is always less than 1/2. States with the minimum uncertainty, which tends to zero, correspond to the values of λ near the convergence radius of the q-exponential. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 315–322, May, 2006.     

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.     

Heisenberg representation of second-quantized fields in stationary external fields and nonlinear dielectric media

The Heisenberg formalism for the creation and annihilation operators of quantized fields in stationary external fields is developed. Fields with spin 0, 1/2, 1 are considered in external electromagnetic and scalar fields and in a field of stationary dielectric properties of a nonlinear medium. An elliptic operator that depends on the time as a parameter and whose eigenfunctions can be used to expand the field variables in the Heisenberg representation is constructed. The connection between the creation and annihilation Heisenberg operators and the operators found by diagonalizing the Hamiltonian by Bogolyubov transformations is established. Heisenberg equations of motion are obtained for external fields of arbitrary form. The phenomenological Hamiltonian that is widely used to describe parametric generation of light is derived in the framework of the quantum field theory, and the limits of applicability of the Hamiltonian are established.Technological Institute, St. Petersburg. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 431–451, December, 1993.     

Characterization of the QWN-conservation operator and applications

Based on the finding that the quantum white noise (QWN) conservation operator is a Wick derivation operator acting on white noise operators, we characterize the aforementioned operator by using an extended techniques of rotation invariance operators in a first place. In a second place, we use a new idea of commutation relations with respect to the QWN-derivatives. Eventually, we use the action on the number operator. As applications, we invest these results to study three types of Wick differential equations.     

An operator approach to some graph enumeration problems

An operator approach to some graph enumeration problems is developed together with the formal procedures related to the enumeration. Both annihilation and creation operators are defined for vertices, edges and Euler characteristics of a graph. An application to forest enumeration leads to compact expressions exhibiting the duality between the operators.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

Free oscillator realization of Laguerre polynomials

Theoretical and Mathematical Physics - We revisit the radial oscillator from the standpoint of a free oscillator realization. By using a free oscillator, namely, the creation/annihilation operators...     

Bogoliubov’s causal perturbative QED and white noise. Interacting fields

Theoretical and Mathematical Physics - We present Bogoliubov’s causal perturbative QFT with a single refinement: the creation–annihilation operators at a point, i.e., for a specific...     

Generalization of the Berezin Formula to the Noncommutative Case

With the help of functional integrals in Fock space, under some analytical assumptions, we construct representations for exponents of quadratic functions of creation and annihilation operators with noncommuting coefficients.     

Gaussian kernel operators on white noise functional spaces

The Gaussian kernel operators on white noise functional spaces, including second quantization, Fourier-Mehler transform, scaling, renormalization, etc. are studied by means of symbol calculus, and characterized by the intertwining relations with annihilation and creation operators. The infinitesimal generators of the Gaussian kernel operators are second order white noise operators of which the number operator and the Gross Laplacian are particular examples.     

Quantizing the KdV Equation

We consider the quantization procedure for the Gardner–Zakharov–Faddeev and Magri brackets using the fermionic representation for the KdV field. In both cases, the corresponding Hamiltonians are sums of two well-defined operators. Each operator is bilinear and diagonal with respect to either fermion or boson (current) creation/annihilation operators. As a result, the quantization procedure needs no space cutoff and can be performed on the entire axis. In this approach, solitonic states appear in the Hilbert space, and soliton parameters become quantized. We also demonstrate that the dispersionless KdV equation is uniquely and explicitly solvable in the quantum case.     

Ultratertiary Quantization of Thermodynamics

We show that if the Dirac–Bogoliubov rule for replacing the bosonic creation and annihilation operators with the c-numbers is used, then the ultratertiary quantization allows obtaining the Bardeen–Cooper–Schrieffer–Bogoliubov formulas.     

GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators dt and its dual, creation operators t*.