Abstract
In collapse models, the induced narrowing of the wavefunction typically leads to an increase in energy. For realistic non-relativistic models, the parameters of the model can be set such that the energy increase is small enough to be within experimental bounds. However, for relativistic versions of collapse models the energy increase is divergent. Here we show how to regulate this divergent behaviour by formulating a collapse model on a discrete Lorentzian spacetime. The result is a relativistic collapse model with finite energy production. This energy increase can be made sufficiently small with a reasonable choice for the discreteness scale of spacetime.
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Notes
- 1.
Weak measurements such as this can be brought about by the use of a projective measurement on an auxiliary quantum system which interacts with the system of interest.
- 2.
This can be motivated by considering a one-dimensional space of length L as a regular array of N points. The kronecker delta function can be written
$$\begin{aligned} \delta _{nm} = \frac{1}{N}\sum _{k = 1}^Ne^{2\pi i \frac{k}{N}(n-m)}, \end{aligned}$$(16)and so if we define \(x_n = Ln/N\) and \(p_k = 2\pi k/L\) so that \(\Delta x = a = L/N\), and \(\Delta p = 2\pi /L\), then
$$\begin{aligned} a^{-1}\delta _{nm} = \frac{1}{2\pi }\sum _{k=1}^N \Delta p e^{ip_k(x_n-x_m)}. \end{aligned}$$(17)This is the discretised version of the Dirac delta function \(\delta (x_n-x_m)\) in one dimension. Setting \(m=n\) results in \(\delta (x = 0)\sim a^{-1}\). We also note that, by the same argument \(\delta (p = 0) \sim L/(2\pi )\).
- 3.
I am unaware of any argument or experimental result ruling out such a large discreteness scale.
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Bedingham, D.J. (2021). Collapse Models, Relativity, and Discrete Spacetime. In: Allori, V., Bassi, A., Dürr, D., Zanghi, N. (eds) Do Wave Functions Jump? . Fundamental Theories of Physics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-030-46777-7_15
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