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Mathematische Nachrichten

Wick products of the CCR algebra

B. J. González, E. R. Negrín

Published May 25, 2011• 0 citations
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Abstract

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

DOI
10.1002/mana.200810253

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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
Abstract This work deals with the optimal control of one dimensional quantum harmonic oscillator under an external field characterized by a linear dipole function. The penalty term is taken as kinetic energy. The objective operator whose expectation value is desired to get a prescribed target value is taken as the square of the position operator. The dipole function hypothesis in the external field is valid only for weak fields otherwise hyperpolarizability terms which contain powers of the field amplitude higher than 1 should be considered. The weak field assumption enables us to develop a first order perturbation approach to get approximate solutions to the wave and costate equations. These solutions contain the field amplitude and another unknown, so‐called deviation constant, through some certain integrals. By inserting these expressions into the connection equation which functionally relates the field amplitude to the wave and costate function it is possible to produce an integral equation. Same manipulations on the objective equation results in an algebraic equation to determine the deviation parameter. The algebraic deviation equation produces incompatibility which can be relaxed by including second order perturbative term of wave function. The integral and algebraic equations mentioned above are asymptotically solved. Their global solutions are left to future works since the main purpose of this work is to obtain the so‐called field and deviation equations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known computer algebra systems.