Gauge transformation of phase space for quantized systems

The gauge transformation first proposed by Hermann Weyl in 1918 is a misnomer, which, according to C N Yang, should be phase transformation when correctly understood, and the subsequent gauge theory developed along this way. Here we find that for quantum systems that are actualized under the quantization condition x, p=iℏ, a true gauge transformation x → x/α, p →αp exists, which preserves, besides the volume of phase space, the energy spectrum, as exemplified by calculation on isotropic harmonic oscillators, hydrogen atom and a class of many- body problem. This discovery reminds us that the phase space as a whole may encode a scaling invariance, only the volume of phase space and its quantization matter, as unconsciously adopted by quantum mechanics and statistical mechanics. This is also in accordance with the de Broglie relation λ=h/p, and with the gist, as in the gauge theory in common- sense, that electromagnetic quantities are such that their characterization by numbers under a given coordinate system is independent of the scale, so are the quantum quantities.