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• W1569690821 abstract "It is shown how the Wheeler-Dewitt and Hamilton-Jacobi equations for a two-dimensional minisuperspace may be solved in full generality if the superpotential of the wavefunction is a separable function of the minisuperspace null coordinates. In this c8se, the system may be viewed as a constrained oscillatorghost-oscillator model. These solutions describe the quantum cosmology of a renormalizahle two-dimensional dilaton gravity theory and the quantum dynamics of the event horizon in Hindler space-time. PACS NUMBERS: 04.60.-m,98.80.Hw,04.6O.Kz,04.7O.Dy ‘Electronic address: jimOfnas09.fnal.gov e Operated by Univerrltier Research Association Inc. under contract with the United States Department of Energy In the canonical quantization of General Relativity, the classical Hamiltonian constraint 3-1 = 0 is viewed as a quantum mechanical operat,or that annihilates the physical states 111! of the vniverse: 7-N = 0. This functional differential equation governs the dynamics of the wavefunction in an infinite-dimensional configuration space known as superspace. Currently it is not known how to solve this equation in full generality, but progress can be made by imposing a high degree of symmetry on the system and considering a dimensional reduction of super-space to a finite-dimensional sector known as minisuperspace. If the minisuperspace is two-dimensional, the Hamiltonian constraint (Wheeler-DeWitt equation [l]) has the generic form of a hyperbolic, second-order partial differential equation: 1 a* a2 w-w ad ap* 4m*(a,p) I rk = 0, (1) where the ‘superpotential’ m*(a, p) is some function of the minisuperspace coordinates (a, p). Equations of this form also arise in a number of other fundamental problems in physics and cosmology and it is therefore important to develop techniques that allow exact solutions to be derived in a straightforward manner. The purpose of this letter is to illustrate how this equation may be solved for a wide class of m2(a,p). We assume that the super-potential is positive definite, although the analysis is easily extended to negative potentials. Examples include minisuperspaces corresponding to a (1 + 1)-dimensional dilaton gravity theory and the event horizon of the Rindler space-time. If the function m*(q p) is independent of the minisuperspace coordinates, Eq. (1) may be transformed into the canonical form a* [ 1 --1 *=o, auav where u = m(a+P) and v = m(a p) are null coordinates over minisuperspace. This equation has been studied previously by Page 121 and one family of solutions is given by iIr* = e -ibu+k/b, where+krra.rbi+rary-complex constant:-This-familyforms the basis for the general solution which can be expressed as the two-dimensional integral * en = J &L(b, b’)Q b, where the density L(b,b’) is an arbitrary function. The wavefunction is bounded and square-integrable if L(b, b’) is finite and only supported in a compact region when Imb < 0 [2]. In this case, Cauchy’s theorem implies that the two-dimensional integral may be replaced by the line integral J +oO a pen = dwq%, -00 where M(b) is an arbitrary function. In general, however, it is rather difficult to evaluate the integral (3) analytically. On the other hand, new exact solutions may be generated after further coordinate" @default.
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• W1569690821 date "1994-09-19" @default.
• W1569690821 modified "2023-09-27" @default.
• W1569690821 title "A Class of integrable minisuperspaces" @default.
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• W1569690821 hasPublicationYear "1994" @default.
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