Category:Quantum Cosmology - Revision history

Cosmo All at 11:42, 23 April 2014

2014-04-23T11:42:52Z

2014-04-23T11:22:07Z

2014-04-23T11:20:42Z

‎Introduction to vacuum fluctuations

2014-04-23T11:17:49Z

‎Problem

2014-04-23T11:17:08Z

2014-04-23T11:16:40Z

2014-04-23T09:46:24Z

Created page with "(Craig J. Hogan, Quantum indeterminacy in local measurement of cosmic expansion, 1312.7797)"

New page

(Craig J. Hogan, Quantum indeterminacy in local measurement of cosmic expansion, 1312.7797)

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 The Klein-Gordon equation for each $k$-component is
 \[\ddot\varphi_{\vec{k}}+(k^2+m^2)\varphi_{\vec{k}}=0.\]
- Each (in general) complex function $\varphi_{\vec{k}}(t)$ satisfies the harmonic oscillator equation with the frequency $\omega_k=(k^2+m^2)^{1/2}$. The functions $\varphi_{\vec{k}}(t)$ are called the modes of the field $\varphi$.
+Each (in general) complex function $\varphi_{\vec{k}}(t)$ satisfies the harmonic oscillator equation with the frequency $\omega_k=(k^2+m^2)^{1/2}$. The functions $\varphi_{\vec{k}}(t)$ are called the modes of the field $\varphi$.
 <br/>
 To quantize the field, each mode $\varphi_{\vec{k}}(t)$ is quantized as a separate harmonic oscillator. We replace the classical "coordinates" $\varphi_{\vec{k}}$ and momenta $\pi_{\vec{k}}\equiv\dot\varphi^*_{\vec{k}}$ by operators $\hat\varphi_{\vec{k}}$ and $\hat\pi_{\vec{k}}$ with the equal-time commutation relations

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-     <p style="text-align: left;">The lower bound on system size is obtained where gravitational (problem \ref{QC_2}) and quantum (problem \ref{QC_3}) bounds intersect:
+     <p style="text-align: left;">The lower bound on system size is obtained where gravitational (problem [[#QC_2]]) and quantum (problem [[#QC_3]]) bounds intersect:
 	\[M_{Pl(min)}(=H_{Pl}^{-1}R_{Pl}^{-2})=R_{Pl}^3H_{Pl}^2\Rightarrow R_{Pl}\ge H_{Pl}^{-3/5}.\]
 The quantity $H_{Pl}^{-3/5}$ is a new scale of quantum indeterminacy associated with expanding cosmic space. Today, the minimum boundary scale is macroscopic but not astronomical: with $H=H_0$, it is about 60 meters. Again, this scale has been derived only from standard quantum mechanics and gravity, and depends only on $H$. The corresponding system mass is $M\approx10^7 GeV$ ($M_{Pl}=H_{Pl}^{1/5}$).</p>
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 === Problem  ===
 <p style= "color: #999;font-size: 11px">problem id: QC_5</p>