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arXiv:0910.2453 · Journal of Mathematical Physics

Quadratic exponential vectors

Luigi Accardi, Ameur Dhahri

Published Dec 1, 2009• 11 citations
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Abstract

We give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in L2(Rd)∩L∞(Rd). We prove the linear independence and totality, in the quadratic Fock space, of these vectors. Using a technique different from the one used by Accardi et al. [Quantum Probability and Infinite Dimensional Analysis, Vol. 25, p. 262, (2009)], we also extend, to a more general class of test functions, the explicit form of the scalar product between two such vectors.

DOI
10.1063/1.3266166

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We give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in L2(Rd)∩L∞(Rd). We prove the linear independence and totality, in the quadratic Fock space, of these vectors. Using a technique different from the one used by Accardi et al. [Quantum Probability and Infinite Dimensional Analysis, Vol. 25, p. 262, (2009)], we also extend, to a more general class of test functions, the explicit form of the scalar product between two such vectors.
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.
A class of representations of the canonical commutation relations is studied, each of which is characterized by an expectation functional that is the exponential of a Euclidean-invariant quadratic form of the test functions. The underlying field operators are realized as the direct product of two Fock representations and the consequences of this realization are analyzed. Compatible Hamiltonians are constructed and an extensive study of the most general quadratic Hamiltonians is presented. In order to include thermodynamic examples, the analysis includes indefinite Hamiltonian spectra as well as the usual definite spectra. Finally, conditions are given for a theory to be local in the sense that all time derivatives of the field operator commute with one another at equal times but unequal spatial arguments.
For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.
Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain eμa2eλa†2e^{μa^2}e^{λa^{\dagger2}}eμa2eλa†2 of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.