Gravitational model of non-minimally coupled Brans Dicke (BD) scalar field Φ with dynamical unit time-like four vector field is used to study flat Robertson Walker (RW) cosmology in the presence of variable cosmological parameter V(Φ) = ΛΦ. Aim of the paper is to seek cosmological models which exhibit metric signature transition. The problem is studied in both classical and quantum cosmological approach with large values of BD parameter ω 1. Scale factor of RW metric is obtained as R(t) = 6 (3/Λ)~(1/2) cosh (t/4 (Λ/3)~(1/2)) which describes nonsingular inflationary universe in Lorentzian signature sector. Euclidean signature sector of our solution describes a re-collapsing universe and is obtained from analytic continuation of the Lorentzian sector by exchanging t → it as R(t) = 6 (3/Λ)~(1/2) cos (t/4 (Λ/3)~(1/2)). Dynamical vector field together with the BD scalar field are treated as fluid with time dependent barotropic index. They have regular (dark) matter dominance in the Euclidean (Lorentzian) sector. We solved Wheeler De Witt (WD) quantum wave equation of the cosmological system. Assuming a discrete non-zero ADM mass M_j = 4sqroot(2j + 1)(Λ/3)~(1/2) with j = 0,1,2,..., we obtained solutions of the WD equation as simple harmonic quantum Oscillator eigen functionals described by Hermite polynomials. Absolute values of these eigen functionals have nonzero values on the hypersurface R = 6 (3/Λ)~(1/2) in which metric field has signature degeneracy. Our eigen functionals describe nonzero probability of the space time with Lorentzian (Euclidean) signature for R > 6 (3/Λ)~(1/2) (R < 6 (3/Λ)~(1/2)). Maximal probability corresponds to the ground state j = 0.
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