Fourier transform identities in quantum mechanics and the quantum line

Abstract

We give a quantum-mechanical interpretation of some modular identities ( LF) 3=λ L 2 arising in conformal field theory and the theory of quantum groups in relation to the mapping class group of a torus. The interpretation follows by evaluating the identity in the case of a non-standard group of the real line. The operation L takes the form of the Fourier transform of wave functions in L 2( R ) with length scale √ℏ τ/ m. We find that this can be factorised as λ L= exp(− iτ p 2/2mℏ) exp(− i x 2/2ℏτ) P exp(− iτ p 2/2mℏ), where x, p mare the usual quantum mechanical position and momentum operators , P is the parity and λ=(1−i)/√2 is a normalization constant determined by verifying the identity on wavepackets. We understand this further in terms of the observation that the Fourier transformation is realized on the wavefunctions in quantum mechanics as the evolution by a quantum harmonic oscillator of period 2 πτ for 1 4 of a cycle.