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arXiv 2209.15426gr-qccond-mat.otherhep-ph
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无量纲物理学：作为闵可夫斯基度规元素的普朗克常量

Dimensionless physics: Planck constant as an element of Minkowski metric

G. E. Volovik

备注：
11 pages, no figures, version accepted in JETP Letters, arXiv admin note: text overlap with arXiv:2207.05754
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摘要：

量子引力的Diakonov理论，其中四元组作为费米子场的双线性组合出现，\cite{Diakonov2011}表明，在广义相对论中，度规可能具有2维，即$[g_{μν}]=1/[L]^2$。量子引力的其他几种方法，包括超塑性真空模型和$BF$引力理论都支持这种解释。这样的度量维数的重要结果是，所有的超同态不变量对任何维的时空都是无量纲的。这些包括作用量$S$，间隔$s$，宇宙学常数$Λ$，标量曲率$R$，标量场$Φ$，等等。在这里，我们试图进一步利用Diakonov的想法，并考虑普朗克常量的维数。Diakonov理论的应用表明，普朗克常量$\hbar$是闵可夫斯基度规的参数。Minkowski参数$\hbar$仅在Lorentz变换下是不变的，并且不是同态不变的。因此，普朗克常量$\hbar$具有非零维数--长度[L]的维数。这个普朗克常量长度是否与普朗克长度尺度有关，是一个悬而未决的问题。原则上，可以有不同的闵可夫斯基真空，其参数$\hbar$具有它们自己的值。然后，在两个真空之间的热接触中，它们的温度服从托尔曼定律的类似物：$\hbar_1/T_1= \hbar_2/T_2$。

英文摘要：

Diakonov theory of quantum gravity, in which tetrads emerge as the bilinear combinations of the fermionis fields,\cite{Diakonov2011} suggests that in general relativity the metric may have dimension 2, i.e. $[g_{μν}]=1/[L]^2$. Several other approaches to quantum gravity, including the model of superplastic vacuum and $BF$-theories of gravity support this suggesuion. The important consequence of such metric dimension is that all the diffeomorphism invariant quantities are dimensionless for any dimension of spacetime. These include the action $S$, interval $s$, cosmological constant $Λ$, scalar curvature $R$, scalar field $Φ$, etc. Here we are trying to further exploit the Diakonov idea, and consider the dimension of the Planck constant. The application of the Diakonov theory suggests that the Planck constant $\hbar$ is the parameter of the Minkowski metric. The Minkowski parameter $\hbar$ is invariant only under Lorentz transformations, and is not diffeomorphism invariant. As a result the Planck constant $\hbar$ has nonzero dimension -- the dimension of length [L]. Whether this Planck constant length is related to the Planck length scale, is an open question. In principle there can be different Minkowski vacua with their own values of the parameter $\hbar$. Then in the thermal contact between the two vacua their temperatures obey the analog of the Tolman law: $\hbar_1/T_1= \hbar_2/T_2$.

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