Beginning with the Pauli-Fierz theory, we construct a model for multi-graviton theory. Couplings between gravitons belonging to nearest-neighbor "theory spaces" lead to a discrete mass spectrum.Our model coincides with the Kaluza-Klein theory whose fifth dimension is latticized.We evaluate one-loop vacuum energy in models with a circular latticized extra dimension as well as with compact continuous dimensions. We find that the vacuum energy can take a positive value, if the dimension of the continuous space time is 6, 10, . . .. Moreover, since the amount of the vacuum energy can be an arbitrary small value according to the choice of parameters in the model, our models is useful to explain the small positive dark energy in the present universe.

We consider a model of quantum electrodynamics (QED) on a graph. The one-loop divergences in the model are investigated by use of the background field method.

The mass spectrum of a model constructed in a theory space is expressed by eigenvalues of the Laplacian on the graph structure of the theory space. The nature of the one-loop UV divergence in the vacuum energy is then controlled only by the degree matrix of the graph. Using these facts, we can construct models of induced gravity which do not suffer from divergences at the one-loop level.Comment: 14 pages, 5 figures, RevTeX

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We consider the third quantization in quantum cosmology of a minisuperspace extended by the Eisenhart‒Duval lift. We study the third quantization based on both Klein‒Gordon type and Dirac-type equations in the extended minisuperspace. Spontaneous creation of “universes” is investigated upon the quantization of a simple model. We find that the quantization of the Dirac-type wave function reveals that the number density of universes is expressed by the Fermi‒Dirac distribution. We also calculate the entanglement entropy of the multi-universe system.