标度不变引力与物理学理论的不守恒解释
Scale Invariant Gravitation and Unambiguous Interpretation of Physical Theories
摘要：
我们传统的物理单位制是基于局部的或微观的{ \it维}量，为了方便或美学的原因，这些量被{ \it定义}为时空无关的。更一般的单位选择可能需要时空中基本物理量（“常数”）的变化。引力理论通常不满足保形对称性，也就是说，它对长度单位的局部变化不是不变的。因此，与引力的爱因斯坦-希尔伯特作用量（$S_{EH}$）相关的{ \it无量纲}作用量$ϕ_{EH}=S_{EH}/\hbar$，对于长度单位的局部变化不是不变的;对于无量纲量来说，这显然是一个不令人满意的特征。在这里，我们通过增加额外的条款，占时空变化的物理“常数”在任意单位系统的阶段。在这样的单位系统中，所有的维度量都隐含地依赖于时空;这是通过度规的保角变换来实现的，通过适当的度规依赖的维度量的重新标度来增加。所得到的修改的无量纲作用是尺度不变的，即独立于单位系统，如所期望的。引力，无量纲物理量，和量子力学之间的深刻联系，阐明和隐含的歧义在解释维度量强调。
英文摘要：
Our conventional system of physical units is based on local or microscopic {\it dimensional} quantities which are {\it defined}, for convenience or otherwise aesthetic reasons, to be spacetime-independent. A more general choice of units may entail variation of fundamental physical quantities (`constants') in spacetime. The theory of gravitation generally does not satisfy conformal symmetry, i.e. it is not invariant to local changes of the unit of length. Consequently, the {\it dimensionless} action associated with the Einstein-Hilbert action ($S_{EH}$) of gravitation, $ϕ_{EH}=S_{EH}/\hbar$, is not invariant to local changes of the length unit; clearly an unsatisfactory feature for a dimensionless quantity. Here we amend the phase by adding extra terms that account for spacetime variation of the physical `constants' in arbitrary unit systems. In such a unit system, all dimensional quantities are implicitly spacetime-dependent; this is achieved by a conformal transformation of the metric augmented by appropriate metric-dependent rescalings of the dimensional quantities. The resulting modified dimensionless action is scale-invariant, i.e. independent of the unit system, as desired. The deep connection between gravitation, dimensionless physical quantities, and quantum mechanics, is elucidated and the implicit ambiguity in interpretations of dimensional quantities is underlined.
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