声学度规与普朗克常数
Acoustic metric and Planck constants
AI总结：
本文基于AK涌现四分标理论，利用超流液氦中声子的有效声学度规引入声学普朗克常数，证明其长度量纲约为原子间距，从而支持相对论量子真空中普朗克常数与普朗克长度同量级的猜想。
摘要：
基于Akama-Diakonov（AK）的涌现四分标（emergent tetrads）理论，文献[Volovik2023b]提出可以引入两个普朗克常数，它们是对应的闵可夫斯基度规分量的参数。在AK理论中，间隔$ds$是无量纲的，因此度规元素以及普朗克常数具有非零量纲。普朗克常数$\hbar$具有时间量纲，而第二个普朗克常数$\hslash$具有长度量纲。自然会将$\hslash$与普朗克长度$l_{\rm P}$进行比较，后者通过$l_{\rm P}^2= \hslash G$与牛顿常数相联系。然而，由于量子真空的微观（跨普朗克）物理尚不清楚，这一联系仍是一个未解决的问题。本文利用超流玻色液体（如$^4$He）中声波量子（声子）所涌现的有效引力来研究这一问题，该体系的微观物理是已知的：即原子物理。有效声学度规的元素由该玻色液体的参数决定，并且与AK理论中一样，间隔$ds$是无量纲的。可以将有效的“声学普朗克常数”作为声学度规的元素引入，即$g^{μν}_{\rm ac}= {\rm diag}(-\hbar_{\rm ac}^2,\hslash_{\rm ac}^2,\hslash_{\rm ac}^2,\hslash_{\rm ac}^2)$。由此得到声学普朗克常数$\hslash_{\rm ac}$具有长度量纲，并且在液氦中其数量级为该液体中的原子间距离，$\hslash_{\rm ac} \sim a$。这支持了如下图景：相对论量子真空中的普朗克常数在数量级上等于普朗克长度，$\hslash\sim l_{\rm P} \sim G$。我们还利用声学度规考察了膨胀宇宙中普朗克常数可能对哈勃参数的依赖关系。
英文摘要：
Based on Akama-Diakonov (AK) theory of emergent tetrads, it was suggested\cite{Volovik2023b} that one can introduce two Planck constants, which are the parameters of the corresponding components of Minkowski metric. In the AK theory, the interval $ds$ is dimensionless, as a result the metric elements and thus the Planck constants have nonzero dimensions. The Planck constant $\hbar$ has dimension of time, and the second Planck constant $\hslash$ has dimension of length. It is natural to compare $\hslash$ with the Planck length $l_{\rm P}$, which is related to the Newton constant as $l_{\rm P}^2= \hslash G$. However, this connection remains an open question, because the microscopic (trans-Planckian) physics of the quantum vacuum is not known. Here we study this question using the effective gravity emerging for sound wave quanta (phonons) in superfluid Bose liquid, such as $^4$He, where the microscopic physics is known: it is atomic physics. The elements of the effective acoustic metric are determined by the parameters of this Bose liquid, and as in the AK theory, the interval $ds$ is dimensionless. One may introduce the effective "acoustic Planck constants" as elements of acoustic metric, $g^{μν}_{\rm ac}= {\rm diag}(-\hbar_{\rm ac}^2,\hslash_{\rm ac}^2,\hslash_{\rm ac}^2,\hslash_{\rm ac}^2)$. Then one obtains that the acoustic Planck constant $\hslash_{\rm ac}$ has dimension of length, and in liquid helium it is on the order of the interatomic distance in this liquid, $\hslash_{\rm ac} \sim a$. This supports the scenario in which the Planck constant in the relativistic quantum vacuum is on the order of the Planck length, $\hslash\sim l_{\rm P} \sim G$. We also use the acoustic metric for consideration of the possible dependence of the Planck constant on the Hubble parameter in expanding Universe.
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