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Quadratic Fock space calculus (III): Inner product of the quadratic exponential vectors

Habib Rebei, Amor Rebey

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.2298/fil2515003r

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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.
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
The relevant structure of a Bose Fock space using the notion of free commutative algebra with unit element and with a scalar product is incorporated. The squeezed states are then the exponentials of quadratic forms and can be written in the normal form of a general squeeze group. The connection of this group to polarization of the initial Hilbert space is established and its infinitesimal generator is computed. In this way, a common denominator for a series of papers is provided in the topic of light squeezing and last but not least a mathematical formalism is provided that may prove convenient in the general treatment of quantum optics.
I start by briefly overviewing quantum mechanics as it is standardly formulated and by discussing the question of whether this standard formulation needs to be supplemented by an interpretation. The overview is based mainly on Von Neumann (1932, 1955) and its aim is not to give the reader a crash course in quantum mechanics, but to present those parts of the standard formalism which I use in this book. For a more complete treatment, the reader may consult the standard textbooks on quantum mechanics or, for instance, Sudbery (1986) or Redhead (1987). The standard formulation The standard formulation of quantum mechanics can be introduced in four steps. The first step is that in quantum mechanics one describesthe physics of a system by means of a Hilbert space ℋ. This Hilbert space is a complex linear vector space on which an inner product is defined. Let's adopt the convention (please consult the Glossary at the end of the book for notational conventions) that α refers to a system and that is the Hilbert space that is associated with this system. Let ∣ℋ α 〉 denote a vector in ℋ α and let 〈ψ α ∣φ α 〉 be the inner product between the vectors ∣ψ α 〉 and ∣φ α 〉.
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.
This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].