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