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6
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84927821868
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L. P. Grishchuk and Ya. B. Zel'dovich, in Quantum Structure of Space and Time, edited by M. Duff and C. Isham, (Cambridge University Press, Cambridge, 1982).
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34
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84927821867
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Although there seems to be no dispute about this result, the exact form of the boundary condition that selects this wave function is not clear. The nonperturbative, radial part of psi should certainly satisfy the outgoing-wave boundary condition, and it was argued in Refs. [29,30] that the rest of psi should be fixed by the regularity condition, | ψ | < inf. However, it was later realized [57] that this is not sufficient: the wave function obtained from psi by acting with PHI-particle creation operators still satisfies all the boundary conditions. Sasaki et al.. [58,59] emphasized that the boundary condition should reflect the fact that the bubble nucleates from vacuum and not from some other excited state. However, the specific form of the boundary condition they suggest is not suitable for a thin-wall bubble. It is possible that psi can be completely fixed only by requiring that it respects the Lorentz invariance of the false vacuum.
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37
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84927821866
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The wave function (3.18) includes expanding and contracting components with equal amplitudes, and it is natural to interpret it as describing a contracting and reexpanding de Sitter universe (3.9). An alternative view [60,19], is to interpret ψ-(a) and its time reverse ψ+(a) as describing the same nucleating universe, but with a different choice made for the direction of the time coordinate. This interpretation may be problematic due to interference between the two components of psi.
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40
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84927821865
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I assume that V ( φ ) grows slower than exp( 6 | φ | ) at φ -> +- inf.
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42
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84927821864
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Whether or not this actually happens, depends on how fast U diverges towards the boundary. The WKB approximation assumes that | del2S | << | ( del S )2|. With del S app U1/2, this implies | del U | << | U |3/2.
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43
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84927821863
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We note in passing, that the wave function (6.9) in the α -> - inf limit is also of the semi-classical form exp(iS) and that the WKB approximation becomes increasingly accurate as α -> - inf.
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47
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84927821862
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Two-dimensional closed manifolds split into two cobordism classes, corresponding to even and odd Euler characteristics, respectively. The Euler characteristic for a sphere is E=2, for a torus is E=0, and each additional handle reduces it by 2. An example of a manifold with an odd Euler characteristic is a sphere with opposite points identified (E=1). Since two-manifolds belonging to different cobordism classes cannot be connected by an interpolating three-manifold, the transition amplitude between them is zero. One can therefore consistently assume that only manifolds with even Euler characteristic are included in superspace, as I did in the text.
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49
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84927821861
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S. del Campo and A. Vilenkin (unpublished).
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50
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In Ref. [17] I defined the regular boundary as consisting of singular configurations which can be obtained by slicing regular Euclidean four-geometries, but the relation to Morse functions was not spelled out. This is somewhat imprecise and has lead to misinterpretations [61-63]. Consider, for example, a manifold scrM of topology S2times S2 with the metric d s2= R12d Ω12+ R22d Ω22, where R1, R2=const and d Ωa2= d θa2+sinθa2d φa2. A possible slicing of scrM can be obtained by settling θ1=const. This gives three-mainfolds of topology S1times S2, where S1 is a circle of radius 0 <= r <= R1. One could have thought that the configuration with r=0 belongs to the regular boundary. However, it is easily understood that such configurations cannot be obtained as critical slices using a Morse function. In order to give slices of θ1=const, the Morse function should be a function only of θ1, but then det( partialμpartialνf ) = 0, and the critical points are always degenerate. To see what is wrong with degenerate critical points, consider a doughnut (torus) lying on a horizontal surface and imagine slicing it with horizontal planes. The slice at the bottom is a circle. But if the torus is slightly tilted, the circular slice disappears, and the bottom slice is an isolated point. The circular slice is degenerate in the sense that it is present only for a very special slicing. It can be shown that critical points of a Morse function are always isolated points.
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52
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84927821859
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It is interesting to note that discontinuities resulting from string reconnection are preserved at later times by the classical string evolution. They are known as ``kinks'' and propagate around the string at the speed of light.
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56
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84927815750
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M. Henneaux and C. Teitelboim [, ] have shown that a consistent quantum theory of a particle in an external field can be constructed using the particle's proper time as a time coordinate. It would be interesting to extend this approach to interacting particles. The interaction vertices would then correspond to topology-changing events in quantum cosmology.
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(1982)
Ann. Phys. (N.Y.)
, vol.143
, pp. 127
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57
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84927821858
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This target space is what Banks called E-space in Ref. [64].
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65
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T. Banks, in Physicalia Magazine 12, Special issue in honor of the 60th birthday of R. Brout (Ghent, 1990).
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