量子体系的相空间规范变换

外尔于1918年引入的规范变换实际上是相位变换而非真正的尺度变换,但规范不变性、规范理论等概念都沿袭了下来。我们发现,针对由量子化条件x,p]=ih而来的量子体系之本征值问题存在规范变换,或者说尺度变换,x→x/α,p→αp,该变换保体系的能量谱不变。量子谐振子、氢原子问题及一类多体问题的精确解析解证实了这一点。量子化条件x,p]=ih看来是个对量子力学很强的约束,不止于能量的量子化。这个规范变换提醒我们相空间的体积及其量子化才是物理的关键,这也是量子力学和统计物理在潜意识里一直沿用却未予关注的思路。有趣的是,从量子谐振子体系的相空间表述似乎不能导向这个结论。如同规范理论所断言的电磁学量在给定坐标系下的数值表征与标度无关,我们认为量子体系的物理量,如能量谱等,在给定坐标系下的数值表征亦应与标度无关。此尺度变换与德布罗意关系相恰。

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 conditionx,p]=ih,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 relationλ=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.