Phase amplitude conformal symmetry in Fourier transforms. (March 2015) - Kuwata, S
Abstract: For the Fourier transform ℑ : L 2 (R) → L 2 (R) of a complex-valued even or odd function ψ, it is found that the amplitude invariance | ℑ ψ | = | ψ | leads to a phase invariance or inversion as arg( ℑ ψ ) = ±arg ψ + θ ( θ = constant). The converse holds unless arg ψ = constant. The condition | ψ | = | ℑ ψ | is required in dealing with, for example, the minimum uncertainty relation between position and momentum. Without the evenness or oddness of ψ, | ℑ ψ | = | ψ | does not necessarily imply arg( ℑ ψ ) = ±arg ψ + θ, nor is the converse.