An Approach to Loop Quantum Cosmology Through Integrable Discrete Heisenberg Spin Chains
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An Approach to Loop Quantum Cosmology Through Integrable Discrete Heisenberg Spin Chains Christine C. Dantas
Received: 14 September 2012 / Accepted: 16 November 2012 / Published online: 6 December 2012 © Springer Science+Business Media New York 2012
Abstract The quantum evolution equation of Loop Quantum Cosmology (LQC)— the quantum Hamiltonian constraint—is a difference equation. We relate the LQC constraint equation in vacuum Bianchi I separable (locally rotationally symmetric) models with an integrable differential-difference nonlinear Schrödinger type equation, which in turn is known to be associated with integrable, discrete Heisenberg spin chain models in condensed matter physics. We illustrate the similarity between both systems with a simple constraint in the linear regime. Keywords Loop quantum gravity · Loop quantum cosmology · Integrable systems
1 Introduction Loop Quantum Gravity (LQG) [1–4] aims to describe the physics of spacetime at Planck-length scales. In the last few years, research in this area has progressed towards predictions that area and volume observables are fundamentally quantized. In addition, LQG has been explored in symmetry reduced models, which take in consideration simple cosmological spacetimes, a research area known as Loop Quantum Cosmology (LQC) [5, 6]. The quantum evolution equation of LQG—the quantum Hamiltonian constraint— is a difference equation for the wavefunction (i.e., a discrete recursion relation [28]), instead of a differential equation like the Wheeler-de Witt equation (see, e.g., Ref. [7]). A series of consistency studies indicates that solutions to the quantum evolution equation, under certain conditions, do not lead to singularities: a minimum observable volume leads to a “bounce” near classical singularities and quantum states C.C. Dantas () Departamento de Ciência e Tecnologia Aeroespacial (CTA), Divisão de Materiais (AMR), Instituto de Aeronáutica e Espaço (IAE), São Jose dos Campos, Brazil e-mail: [email protected]
Found Phys (2013) 43:236–242
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evolve deterministically through those singularities (e.g., [7, 8, 12]). Recently, the nature of the bounce and its robustness have been addressed under several approaches: numerically; by use of approximation methods (“effective equation” techniques); or by the analysis of simplified exact analytical solutions (for reviews, see, e.g., [9, 10] and [11] and references therein). However, as symmetry-reduced versions of LQG, LQC models assume certain results directly from full LQG. For instance, the smallest nonzero area eigenvalue of LQG is the assumed step size (the so called “area gap”) of the LQC difference equation. It has been shown that the Wheeler-de Witt equation can be made to agree with LQC (in the case of a k = 0, Λ = 0 FRW cosmological model, coupled to a massless scalar field), as the area gap diminishes. However, the approximation is not uniform in the chosen interval of “internal time”, and in fact if the area gap is set to zero, LQC does not admit any limit, being an intrinsically discrete
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