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The hypothetical dark matter halo envelopping a galaxy looks like to a gaussian distribution of position probability density of a particle in a quantum coherent state.

How long should a wave function of this size take to collapse according to decoherence theory?

Then, can dark matter be ordinary matter in a quantum coherent state?

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  • $\begingroup$ Coherent or not it will still interact as much as matter in the noncoherent state. Which will lead to very fast decoherence... $\endgroup$ – OON Jun 2 '17 at 10:32
  • $\begingroup$ @OON: very fast at the galactic size can be very long. How long for a single wave function of a galactic size? $\endgroup$ – Sebastien Palcoux Jun 2 '17 at 10:36
  • $\begingroup$ Some apples are red and round , some stars are red and round, can stars be actually apples? $\endgroup$ – user126422 Jun 2 '17 at 11:43
  • $\begingroup$ @WillyBillyWilliams: I understand your point, but we can easily prove that stars are not apples. My point is: can we prove (so easily) that the dark matter halo is not a single wave function? $\endgroup$ – Sebastien Palcoux Jun 2 '17 at 11:53
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Contrary to what you seem to assume the spatial coherence is damped by interaction with environment the faster the longer distances you consider.

Following this paper if we couple the particle to the thermal bath of photons the relevant density matrix element goes like, \begin{equation} \rho(\vec{x},\vec{y},t)\simeq \rho(\vec{x},\vec{y},t_0) e^{-\Delta |\vec{y}-\vec{x}|^2(t-t_0)/2}, \end{equation} where $\Delta$ is a decoherence rate depending on the effective cross-section and the temperature of the bath.

For electrons interacting with just cosmological background radiation gives $\Delta\sim 10^{-10} cm^{-2} s^{-1}$. For $|\vec{y}-\vec{x}|\sim 10^5$ light years it gives characteristic time (in which the matrix element divides itself by $e$) of just $\sim 10^{-36} s$. And that's just from the cosmological background not taking into account all the stuff happening in galaxies which leaves no hope for your idea even in case of the neutral particles.

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  • $\begingroup$ Ok (+1), for an electron under CMB with $\Delta \sim 10^{-10} cm^{-2} s^{-1}$ and $|\vec{y}-\vec{x}| \sim 10^5 ly \sim 10^{23}cm$, then $\Delta |\vec{y}-\vec{x}|^2(t-t_0) \sim 1$ if $(t-t_0) \sim 10^{-(2 \cdot 23 - 10)} = 10^{-36}s$. Now I don't know if this result work for a galactic mass, due to the incompatibility QM/GR. $\endgroup$ – Sebastien Palcoux Jun 2 '17 at 17:34
  • $\begingroup$ Note that for $|\vec{y}-\vec{x}|\sim 2 μm$ (which is huge for an electron), then the characteristic time is $\sim 4 \cdot 10^{17}s$ (the age of the universe). The dark matter density in the hypothetical halo of the milky-way is $\sim 1$ electron mass /$mm^3 \sim 1$ proton mass /$cm^3$. Do you know what is $\Delta$ for a proton under CMB? For an hypothetical particle of galactic mass (ignoring GR)? $\endgroup$ – Sebastien Palcoux Jun 2 '17 at 18:43
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If we ignore GR, the answer is no, by using the decoherence time formula (19) of Zurek's paper p13, depending on the thermal de Broglie wavelength.

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