Quantum-Classical Correspondence in Wave Functions of the Universe

Abstract

By using the classically solvable minimal massless scalar minisuperspace model we study two exam­ ples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case. In quantum cosmologyl)-4) we describe the universe by a wave function which satisfies the Wheeler-DeWitt equation_ This is a second-order linear partial differential equation_ In the h -> 0 limit some of its solutions admit clear classical interpretation while others do not. In the present paper we study the semiclassical limit of wave functions in a classical­ ly solvable minisuperspace model and discuss the classical interpretation. In § 2 we present the model and its classical solutions. The model is the minimal massless scalar theoryS) coupled to the Einstein gravity with all the spatial degrees. of freedom suppressed, i.e., in the minisuperspace approximation. By using a convenient set of variables we write down the most general complex classical solutions which include real Lorentzian and Euclidean solutions as special cases. (For another classically solva­ ble model, see Ref.· 6).) In § 3 we show the first set of examples for wave functions of the universe. In the semiclassical sense these wave functions correspond to a set of Lorent­ zian and Euclidean solutions which pass a certain fixed point in the minisuperspace (the configuration space), and therefore can be interpreted as a superposition of universes. In § 4 we rewrite the above examples in the path integral representation, which includes slight refinement of the formula in previous works. We also study the conse­ quence of Hartle and Hawking's initial condition 3 ) which states that only the histories starting from the vanishing cosmic scale factor should be summed. In the present model . this condition gives us a wave function which contains no universe. In § 5 we present the second example of wave functions, in which appears a region that is neither Lorentzian nor Euclidean. We use a path integral representation written in terms of a new set of canonical variables. This is transformed to the wave function in th~ original variables by using the generating function of the canonical transformation. The result shows that the wave function in the saddle point approximation is given by (neither Lorentzian nor Euclidean) complex classical solutions in some region. In § 6 we discuss the picture that the universe is created from nothing by the tunneling