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Communications in Theoretical Physics

Some Addenda About the General Formula of Normal Product Calculation for Boson Exponential Quadratic Operators

Jian-Wei Pan, Yong-de Zhang, Xiang‐Bin Wang, Lei Ma

Published Dec 15, 1996• 3 citations
Publisher

Abstract

In this paper, we studied the general formula of normal product calculation for boson exponential quadratic operators (EQO) which is first obtained by WANG et al. (J. Phys. A: Math. Gen. 27 (1994) 6563]. It is shown that the boundness of boson EQO will result in a restriction on the applicable bounds of this formula. According to the restriction condition, the conclusion about unity operator given by MA et al. [Commun. Theor. Phys. (Beijing,China) 21 (1994) 119] should be made some corresponding corrections. At last, we extended this formula to the case of fermion for the convenience of later applications.

DOI
10.1088/0253-6102/26/4/479

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In this paper, we studied the general formula of normal product calculation for boson exponential quadratic operators (EQO) which is first obtained by WANG et al. (J. Phys. A: Math. Gen. 27 (1994) 6563]. It is shown that the boundness of boson EQO will result in a restriction on the applicable bounds of this formula. According to the restriction condition, the conclusion about unity operator given by MA et al. [Commun. Theor. Phys. (Beijing,China) 21 (1994) 119] should be made some corresponding corrections. At last, we extended this formula to the case of fermion for the convenience of later applications.
By means of the transformation relation between the ordinary form of boson exponential quadratic operators (BEQO) and its anti-normal product form, we present an effective method to conveniently calculate arbitrary matrix elements of BEQO. By this method, many important matrix elements can be calculated analytically. As a direct application, we obtain the exact solutions of the density matrix and partition function for general boson quadratic Hamiltonian without any information about the energy level.
Making use of the transformation relation between the ordinary form and the antinormal product form of boson exponential quadratic operators (BEQO's), we present an effective method which can be conveniently used to calculate arbitrary matrix elements of BEQO's. By this method, some important matrix elements have been calculated analytically. As a preliminary application, we obtain the exact solution of the density matrix and partition function for the general boson quadratic Hamiltonian without any information for the energy level. As a natural extension, we also obtain the partition function for a general fermion quadratic system.
This paper falls within the first phase of the Fock quadratic quantization program, where we define the appropriate framework for the program, namely the quadratic Fock space. During this phase, our goal is to synthesize foundational results and standardize notations in alignment with those employed in the series of papers focusing on the one-mode case (J Math Anal App 439(1): 135-153, 2016). The primary focus of this paper is on the inner product of quadratic exponential vectors. We begin by revisiting the expression for the inner product of two one-mode quadratic exponential vectors as established in a previous work (J Math Anal App 439(1): 135-153, 2016). We then extend this result to a more general scenario, specifically the continuous form of quadratic exponential vectors. Although the result is known, our exposition deviates from the one outlined in (J Math Phys 51:2, 2010). Specifically, we offer several modifications. Our contribution entails introducing a distinct approach customized for the one-mode scenario, along with providing a reference that consolidates the fundamental findings related to this topic.