Abstract
Loop quantum cosmology(LQC) is the symmetric model of loop quantum gravity. In this paper, we generalize the structure of loop quantum cosmology to the theories with arbitrary spacetime dimensions. The isotropic and homogenous cosmological model in n+1 dimensions is quantized by the loop quantization method. Interestingly, we find that the underlying quantum theories are divided into two qualitatively different sectors according to spacetime dimensions. The effective Hamiltonian and modified dynamical equations of n+1 dimensional LQC are obtained. Moreover, our results indicate that the classical big bang singularity is resolved in arbitrary spacetime dimensions by a quantum bounce. We also briefly discuss the similarities and differences between the n+1 dimensional model and the 3+1 dimensional one. Our model serves as a first example of higher dimensional loop quantum cosmology and offers possibility to investigate quantum gravity effects in higher dimensional cosmology.
Highlights
Loop quantum gravity (LQG) is a quantum gravity theory trying to quantize general relativity (GR) with nonperturbative techniques [8,9,10,11]
Loop quantum cosmology (LQC), which is the cosmological application of LQG, has received interest
We start from the classical connection dynamics of n + 1 dimensional general relativity together with symmetry reduction procedures, and using the nonperturbative loop quantization method, we find that the dynamical evolution of the n + 1 dimensional Universe is fully determined by a difference equation
Summary
Loop quantum gravity (LQG) is a quantum gravity theory trying to quantize GR with nonperturbative techniques [8,9,10,11]. It is naturally to ask if it is possible to generalize the structure of LQC to the higher spacetime dimensions This is not an easy task, essentially because LQG is a quantization scheme based on the connection dynamics formalism. The SU (2) connection dynamics is only well defined in 3 and 4 dimensions and cannot be directly generalized to the higher dimensional gravity theories This difficulty has been overcome by Thiemann et al in a series of papers [20,21,22,23]. The main idea of [20] is that in n + 1 dimensional GR, in order to obtain a well-defined connection dynamics, one should adopt S O(n + 1) connections AaI J rather than the speculated S O(n) connections With this higher dimensional connection dynamics in hand, Thiemann et al successfully generalize the LQG to arbitrary spacetime dimensions.
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