Abstract
A simple quantum model describing the onset of time is presented. This is combined with a simple quantum model of the onset of space. A major purpose is to explore the interpretational issues which arise. The state vector is a superposition of states representing different “instants.” The sample space and probability measure are discussed. Critical to the dynamics is state vector collapse: it is argued that a tenable interpretation is not possible without it. Collapse provides a mechanism whereby the universe size, like a clock, is narrowly correlated with the quantized time eigenvalues.
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Notes
A treatment of collapse choosing a universe within the context of the inflaton theory of cosmogenesis.
As pointed out there, if one wishes to have exponential grown of the volume in the universe, one may replace g(a †+a) in Eq. (30) by g(a †+a)2.
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I would like to thank Carl Rubino for help with Greek.
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Appendix: Calculation of \(\overline{\langle N\rangle^{2}/\langle1 \rangle}\)
Appendix: Calculation of \(\overline{\langle N\rangle^{2}/\langle1 \rangle}\)
We begin with the expression for |〈n|ψ,t〉 w |2 given in Eq. (38). The Poisson distribution appears there. For large n, the Poisson distribution [x n/n!]e −x can be replaced by its gaussian approximation (2πx)−1/2exp−[(n−x)2/2x], which is good for large x,n. Since it is the large t limit which is of interest, and large n’s are excited for large t, this is a good approximation. We make this replacement in Eq. (38), obtaining
We note that this approximation does not change the calculated values of \(\overline{\langle N\rangle}\) and \(\overline{\langle N^{2}\rangle}\), since
which is Eq. (39), and similarly
which is Eq. (41).
In Eq. (47), because the width of the gaussian \(\sqrt{ \alpha^{\prime*}(t)\alpha(t)}\) is small compared to the mean value α′(t)α(t), we make the approximation w−2λα′α≈w−2λn. With this approximation, and a change of variables to ξ(t)≡η(t)−η′(t), μ(t)≡η(t)+η′(t), Eq. (47) becomes
This approximation also does not change the results of the above calculations of \(\overline{\langle N\rangle}\) and \(\overline{\langle N^{2}\rangle}\). If the integral over Dw is performed in Eq. (50), the result is \(\prod_{t'=0}^{t}\delta[\xi(t')]\). Then, the Dξ integral can be performed, with the result
Setting μ=2η, this is identical to the expression for ∫Dw|〈n|ψ,t〉 w |2 in the middle integral of Eqs. (48), (49), which gives the correct expressions for \(\overline {\langle N\rangle}\) and \(\overline{\langle N^{2}\rangle}\).
Therefore, as our last approximation, we replace the gaussian in the second integral in Eq. (50) by a gaussian in n which gives these same correct expressions, obtaining
where \(\sigma^{2}\equiv\overline{\langle N^{2}\rangle }-\overline{\langle N\rangle}^{2}\).
Now, just as in Sect. 2.2.1, we can change variables to B(t′)’s from w(t′)’s, and integrate over all B(t′)’s except B(t), obtaining the equivalent of Eq. (51),
The rest of the calculation is straightforward. We immediately find the quantities
where 1 has been neglected compared to 4λtσ 2. Also, \(\overline{\langle N\rangle}/2\sigma^{2}\), whose contribution vanishes as t→∞, has been neglected compared to B, whose contribution is infinite in that limit.
Finally, using Eqs. (53), (54), we calculate
Although our result is \(\overline{\sigma_{N}^{2}}=\overline{\langle N^{2}\rangle}- \overline{\langle N\rangle^{2}/\langle1 \rangle} =0\), in view of the approximations made without assessing the errors involved, it is more conservative to conclude that \(\overline{\sigma _{N}^{2}}\sim t\) at worst, which still results in \(\overline{\sigma _{N}^{2}}/\overline{\langle N\rangle^{2}}\rightarrow0\) as t→∞.
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Pearle, P. Chronogenesis, Cosmogenesis and Collapse. Found Phys 43, 747–768 (2013). https://doi.org/10.1007/s10701-013-9714-8
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DOI: https://doi.org/10.1007/s10701-013-9714-8