[Paper Review] Quantum mechanical perspectives and generalization of the fractional Fourier Transformation

Junhua Chen, Hong-Yi Fan|arXiv (Cornell University)|Jul 24, 2013

Mathematical Analysis and Transform Methods4 references3 citations

TL;DR

This paper generalizes the fractional Fourier transform (FrFT) using quantum mechanical formalism, specifically through unitary transformations generated by the number operator and the method of integration within normal-ordered products (IWOP). It derives a full family of generalized fractional transformations (GFrT) that are composable and additive, with explicit eigenfunctions, extending FrFT beyond its standard form to include new classes of linear, kernel-based transforms.

ABSTRACT

Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.

Motivation & Objective

• To generalize the fractional Fourier transform (FrFT) beyond its standard form by identifying the quantum mechanical principles underlying its structure.
• To establish criteria for constructing new generalized fractional transformations (GFrT) that preserve the key properties of additivity and compositability.
• To derive the eigenfunctions of arbitrary GFrT explicitly using quantum mechanical techniques.
• To reveal the role of unitary operators, particularly those based on the number operator and squeezing, in generating new classes of fractional transforms.
• To unify various known transforms (e.g., Hadamard, standard FrFT) under a single quantum mechanical framework via IWOP and representation transformation.

Proposed method

• Utilizes quantum mechanical representation transformation to map between different basis states, such as position |x⟩ and momentum |p⟩.
• Applies the method of integration within normal-ordered product (IWOP) to evaluate matrix elements and derive transformation kernels.
• Constructs generalized fractional transformations (GFrT) via unitary operators of the form exp[i(π/2−α)a†a] and generalized squeezing operators involving a² and (a†)².
• Derives the kernel of the transformation as ⟨p|K̃αM†|x⟩ using coherent state overlaps and Gaussian integrals with complex parameters.
• Introduces a parameter θ to generalize the squeezing operator, leading to a family of transforms parameterized by α and θ.
• Demonstrates additivity by showing Fα∘Fβ = Fα+β through explicit composition of integral kernels and verification via Gaussian integration identities.

Experimental results

Research questions

• RQ1 What quantum mechanical structure underlies the additivity and compositability of the standard fractional Fourier transform?
• RQ2 How can the fractional Fourier transform be generalized to include new classes of linear, kernel-based transformations while preserving its key properties?
• RQ3 What are the eigenfunctions of arbitrary generalized fractional transformations (GFrT), and how can they be derived systematically?
• RQ4 How do different choices of the squeezing parameter θ in the unitary operator affect the resulting transformation kernel and its physical interpretation?
• RQ5 Can the Hadamard transform and other known transforms be recovered as special cases within this generalized framework?

Key findings

• A full family of generalized fractional transformations (GFrT) is derived, with the standard FrFT as a special case when the squeezing parameter θ = 0.
• The generalized transformation kernel is explicitly given as ⟨p|K̃αM†|x⟩ = 1/√(2πi cosθ sinhα) × exp[i/2((x²+p²)/(tanhα cosθ) + (x²−p²)tanθ − 2xp/(sinhα cosθ))], which reduces to the standard FrFT kernel when θ→0.
• The additivity property Fα∘Fβ = Fα+β is rigorously proven for all GFrT, confirming that the transformations form an Abelian Lie group under composition.
• The eigenfunctions of arbitrary GFrT are derived in closed form, providing a complete spectral decomposition for these generalized transforms.
• When θ = π/2, the transformation reduces to the Hadamard transform of continuous variables, confirming its role as a special case in the generalized framework.
• The absence of eigenstates for certain GFrT (e.g., when the generator is a²e^{iθ} + e^{-iθ}(a†)²) is explained by the non-Hermitian nature of the generator, consistent with the lack of eigenfunctions.