1. P. Ivanov, S. V. Chernov P.N. Lebedev Physical Institute
A quasi-classical stochastic wavefunction of the Universe from third quantization procedure
P. Ivanov, S. V. Chernov
P.N. Lebedev Physical Institute
PRD, volume 92, id

2. Founding fathers

3. ADM formalism and canonical quantization procedure

5. Tunneling and no-boundary proposals for wavefunction of the Universe

6. Third quantization of WdW equation with Lambda term and n massless scalar fields

7. Time dependant creation and annihilation operators and Bogolubov coefficients

8. Asymptotic solution of WdW equation

9. An amount of created Universes

10. Wigner function

12. Quasi-classical wavefunction of the Universe

14. Density matrix
It can be shown that the stochastic wavefunction is equivalent to the presence of a density matrix with c-numbered matrix elements in position and momentum representations. For that we make the Fourier tranform of the wavefunction to obtain

15. The diagonal elements in position representation are trivial, giving again the Hartle-Hawking value, while in the position representation it gives a non-trivial distributions over field velocities. For example, we have
where we remind that
These distributions can be used to specify most probable initial conditions for
classical evolution of the Universe.

16. CONCLUSIONS
The third quantization procedure may provide ‘initial condition’ for wavefunction of the Universe. Although this has been
shown in framework of a particular model, we suspect that this effect could operate in all models, where a copious production of universes takes place.
This wavefunction is necessarily stochastic, with its average value being equal to zero. At
least in our model its r.m.s value is proportional to the absolute value of Hartle- Hawking wavefunction.