Summary
A (2+1)-dimensional Riemannian space satisfying Einstein's equations is investigated as a model for the quantum theory of gravity. This model does not bear any dynamics as there is only one state in which the system can be found: flat space. However, the kinematical aspects of the model are not trivial. It is shown that Feynman's sum over histories leads to a one-dimensional Hilbert space of state vectors and an explicit representation of the physical state vector is given. Various general properties of the gravitational transition amplitude, in particular the canonical constraints, which were derived for the full fourdimensional theory in a previous paper, are verified with the explicit solution of this model.
Riassunto
Come modello per la teoria quantistica della gravitazione si studia uno spazio riemanniano a 2+1 dimensioni che soddisfa le equazioni di Einstein. Questo modello non comporta nessuna dinamica in quanto c'è un solo stato nel quale si può trovare il sistema: lo spazio piano. Però gli aspetti cinematici del modello non sono banali. Si dimostra che la somma di Feynman rispetto alle storie porta ad uno spazio hilbertiano unidimensionale dei vettori di stato e si dà una rappresentazione esplicita del vettore di stato fisico. Con la soluzione esplicita di questo modello si verificano varie proprietà generali dell'ampiezza di tranzizione gravitazionale, in particolare le costrizioni canoniche, che sono state dedotte in un articolo precedente per la teoria quadridimensionale completa.
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References
H. Leutwyler:Phys. Rev.,134, B 1155 (1965). This paper contains a more extensive list of the relevant literature and is quoted as I.
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We shall assume either that the surfacesx o=constant are closed and that therefore no such boundary conditions are needed or that sufficiently strong conditions for the behaviour of the space at spacelike infinity provide the necessary boundary conditions. See also ref
G. Darboux:Leçons sur la théorie générale des surfaces, III (Paris, 1894);L. Bianchi:Vorlesungen über Differentialgeometrie, chapt. VII (Leipzig, 1910);A. R. Forsyth:Lectures on Differential Geometry, chapt. X (Cambridge, 1920);W. Blaschke:Differentialgeometrie, II (Berlin, 1923).
Note that in the 3+1-dimensional case the lengthl o appears in the exponent of exp [iS/ħ] in the inverse second power.
The measureD σ g is discussed in Sect. 4.
P. A. M. Dirac:Proc. Roy. Soc. (London), A246, 333 (1950). See also ref.R. Arnowitt, S. Deser andC. W. Misner:Phys. Rev.,113, 745 (1959);
An entirely analogous situation arises if one studies the differential equation d2 y/dx 2=xy, whose solutions are Airy functions. Only one solution of this equation may be represented by an integral analogous to (3.13). The second independent solution does not admit of a Fourier transform as it diverges faster than any polynomial asx→∞.
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Leutwyler, H. A (2+1)-dimensional model for the quantum theory of gravity. Nuovo Cimento A (1965-1970) 42, 159–178 (1966). https://doi.org/10.1007/BF02856201
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DOI: https://doi.org/10.1007/BF02856201